fix problems with html output

[light-and-matter.git] / lm / em / f.rbtex

<%
  require "../eruby_util.rb"
%>

<%
  chapter(
    '26',
    %q{The Atom and E=mc$^2$},
    'ch:atom',
    %q{Marie and Pierre Curie were the first to purify radium in
        significant quantities. Radium's intense radioactivity made possible
        the experiments that led to the modern planetary model of the atom, in
        which electrons orbit a nucleus made of protons and neutrons.},
    {'opener'=>'curies','sidecaption'=>true}
  )
%>

<% begin_sec("Atoms",0,'atoms') %>
        \epigraph{I was brought up to look at the atom as a nice, hard fellow,
        red or grey in color according to taste.}{Rutherford}

  <% begin_sec("The chemical elements",nil,'chemical-elements') %>\index{elements, chemical}

        How would one find out what types of atoms there were?
        Today, it doesn't seem like it should have been very
        difficult to work out an experimental program to classify
        the types of atoms. For each type of atom, there should be a
        corresponding element, i.e., a pure substance made out of
        nothing but that type of atom. Atoms are supposed to be
        unsplittable, so a substance like milk could not possibly be
        elemental, since churning it vigorously causes it to split
        up into two separate substances: butter and whey. Similarly,
        rust could not be an element, because it can be made by
        combining two substances: iron and oxygen. Despite its
        apparent reasonableness, no such program was carried out
        until the eighteenth century.

        By 1900, however, chemists had done a reasonably good job of
        finding out what the elements were. They also had determined
        the ratios of the different atoms' masses fairly accurately.
        A typical technique would be to measure how many grams of
        sodium (Na) would combine with one gram of chlorine (Cl) to
        make salt (NaCl). (This assumes you've already decided based
        on other evidence that salt consisted of equal numbers of Na
        and Cl atoms.) The masses of individual atoms, as opposed to
        the mass ratios, were known only to within a few orders of
        magnitude based on indirect evidence, and plenty of
        physicists and chemists denied that individual atoms were
        anything more than convenient symbols.
<% marg(50) %>
        \begin{minipage}[t]{52mm}
        \begin{equation*}
        \frac{m_\zu{He}}{m_\zu{H}}=3.97
        \end{equation*}
        \begin{equation*}
        \frac{m_\zu{Ne}}{m_\zu{H}}=20.01
        \end{equation*}
        \begin{equation*}
        \frac{m_\zu{Sc}}{m_\zu{H}}=44.60
        \end{equation*}
        \docaption{Examples of masses of atoms compared to that of hydrogen. Note
        how some, but not all, are close to integers.} %
        \label{fig:hmassratios} %
        \end{minipage}
<% end_marg %>



The following
table gives the atomic masses of all the elements, on a standard scale in which 
the mass of hydrogen is very close to 1.0.
The absolute calibration of the whole
scale was only very roughly known for a long time, but was eventually tied down,
with the mass of a hydrogen atom being determined to be about
$1.7\times10^{-27}$ kg.

{\footnotesize
\begin{tabular}{|ll|ll|ll|ll|}
\hline
Ag      & 107.9 &Eu     & 152.0 &Mo     & 95.9  &Sc     & 45.0  \\
Al      & 27.0  &F      & 19.0  &N      & 14.0  &Se     & 79.0  \\
Ar      & 39.9  &Fe     & 55.8  &Na     & 23.0  &Si     & 28.1  \\
As      & 74.9  &Ga     & 69.7  &Nb     & 92.9  &Sn     & 118.7 \\
Au      & 197.0 &Gd     & 157.2 &Nd     & 144.2 &Sr     & 87.6  \\
B       & 10.8  &Ge     & 72.6  &Ne     & 20.2  &Ta     & 180.9 \\
Ba      & 137.3 &H      & 1.0   &Ni     & 58.7  &Tb     & 158.9 \\
Be      & 9.0   &He     & 4.0   &O      & 16.0  &Te     & 127.6 \\
Bi      & 209.0 &Hf     & 178.5 &Os     & 190.2 &Ti     & 47.9  \\
Br      & 79.9  &Hg     & 200.6 &P      & 31.0  &Tl     & 204.4 \\
C       & 12.0  &Ho     & 164.9 &Pb     & 207.2 &Tm     & 168.9 \\
Ca      & 40.1  &In     & 114.8 &Pd     & 106.4 &U      & 238   \\
Ce      & 140.1 &Ir     & 192.2 &Pt     & 195.1 &V      & 50.9  \\
Cl      & 35.5  &K      & 39.1  &Pr     & 140.9 &W      & 183.8 \\
Co      & 58.9  &Kr     & 83.8  &Rb     & 85.5  &Xe     & 131.3 \\
Cr      & 52.0  &La     & 138.9 &Re     & 186.2 &Y      & 88.9  \\
Cs      & 132.9 &Li     & 6.9   &Rh     & 102.9 &Yb     & 173.0 \\
Cu      & 63.5  &Lu     & 175.0 &Ru     & 101.1 &Zn     & 65.4  \\
Dy      & 162.5 &Mg     & 24.3  &S      & 32.1  &Zr     & 91.2  \\
Er      & 167.3 &Mn     & 54.9  &Sb     & 121.8         & & \\
\hline
\end{tabular}
}

  <% end_sec('chemical-elements') %>
  <% begin_sec("Making sense of the elements",nil,'making-sense-of-elements') %>
        As the information accumulated, the challenge was to find a
        way of systematizing it; the modern scientist's aesthetic
        sense rebels against complication. This hodgepodge of
        elements was an embarrassment. One contemporary observer,
        William \index{Crookes, William}Crookes, described the
        elements as extending ``before us as stretched the wide
        Atlantic before the gaze of Columbus, mocking, taunting and
        murmuring strange riddles, which no man has yet been able to
        solve.'' It wasn't long before people started recognizing
        that many atoms' masses were nearly integer multiples of the
        mass of hydrogen, the lightest element. A few excitable
        types began speculating that hydrogen was the basic building
        block, and that the heavier elements were made of clusters
        of hydrogen. It wasn't long, however, before their parade
        was rained on by more accurate measurements, which showed
        that not all of the elements had atomic masses that were
        near integer multiples of hydrogen, and even the ones that
        were close to being integer multiples were off by one percent or so.
<%
  fig(
    'periodictable',
    %q{%
      A modern periodic table. Elements in the same column have
              similar chemical properties. The modern atomic numbers, discussed on
              p.~\pageref{subsec:atomic-number}, were not known in Mendeleev's time, since the table could be
              flipped in various ways.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>
\index{periodic table}

        Chemistry professor Dmitri \index{Mendeleev, Dmitri}Mendeleev,
        preparing his lectures in 1869, wanted to find some way to
        organize his knowledge for his students to make it more
        understandable. He wrote the names of all the elements on
        cards and began arranging them in different ways on his
        desk, trying to find an arrangement that would make sense of
        the muddle. The row-and-column scheme he came up with is
        essentially our modern periodic table. The columns of the
        modern version represent groups of elements with similar
        chemical properties, and each row is more massive than the
        one above it. Going across each row, this almost always
        resulted in placing the atoms in sequence by weight as well.
        What made the system significant was its predictive value.
        There were three places where Mendeleev had to leave gaps in
        his checkerboard to keep chemically similar elements in the
        same column. He predicted that elements would exist to fill
        these gaps, and extrapolated or interpolated from other
        elements in the same column to predict their numerical
        properties, such as masses, boiling points, and densities.
        Mendeleev's professional stock skyrocketed when his three
        elements (later named gallium, scandium and germanium) were
        discovered and found to have very nearly the properties he had predicted.

        One thing that Mendeleev's table made clear was that mass
        was not the basic property that distinguished atoms of
        different elements. To make his table work, he had to
        deviate from ordering the elements strictly by mass. For
        instance, iodine atoms are lighter than tellurium, but
        Mendeleev had to put iodine after tellurium so that it would
        lie in a column with chemically similar elements.

  <% end_sec('making-sense-of-elements') %>
  <% begin_sec("Direct proof that atoms existed",nil,'direct-proof-of-atoms') %>

        The success of the kinetic theory of heat was taken as
        strong evidence that, in addition to the motion of any
        object as a whole, there is an invisible type of motion all
        around us: the random motion of atoms within each object.
        But many conservatives were not convinced that atoms really
        existed. Nobody had ever seen one, after all. It wasn't
        until generations after the kinetic theory of heat was
        developed that it was demonstrated conclusively that atoms
        really existed and that they participated in continuous
        motion that never died out.

        The smoking gun to prove atoms were more than mathematical
        abstractions came when some old, obscure observations were
        reexamined by an unknown Swiss patent clerk named Albert
        Einstein. A botanist named \index{Brownian motion}Brown,
        using a microscope that was state of the art in 1827,
        observed tiny grains of pollen in a drop of water on a
        microscope slide, and found that they jumped around randomly
        for no apparent reason. Wondering at first if the pollen
        he'd assumed to be dead was actually alive, he tried looking
        at particles of soot, and found that the soot particles also
        moved around. The same results would occur with any small
        grain or particle suspended in a liquid. The phenomenon came
        to be referred to as Brownian motion, and its existence was
        filed away as a quaint and thoroughly unimportant fact,
        really just a nuisance for the microscopist.

        It wasn't until 1906 that \index{Einstein, Albert!and
        Brownian motion}Einstein found the correct interpretation
        for Brown's observation: the water molecules were in
        continuous random motion, and were colliding with the
        particle all the time, kicking it in random directions.
        After all the millennia of speculation about atoms, at last
        there was solid proof. Einstein's calculations dispelled all
        doubt, since he was able to make accurate predictions of
        things like the average distance traveled by the particle in
        a certain amount of time.  (Einstein received the Nobel
        Prize not for his theory of relativity but for his papers on
        Brownian motion and the photoelectric effect.)


\startdqs

\begin{dq}
        How could knowledge of the size of an individual aluminum
        atom be used to infer an estimate of its mass, or vice versa?
\end{dq}
 %
\begin{dq}
        How could one test Einstein's interpretation of Brownian motion by observing
        it at different temperatures?
\end{dq}

  <% end_sec('direct-proof-of-atoms') %>
<% end_sec('atoms') %>
<% begin_sec("Quantization of charge",0,'charge-quantization') %>\index{quantization}\index{charge!quantization of}
        Proving that atoms actually existed was a big accomplishment,
        but demonstrating their existence was different from
        understanding their properties. Note that the Brown-Einstein
        observations had nothing at all to do with electricity, and
        yet we know that matter is inherently electrical, and we
        have been successful in interpreting certain electrical
        phenomena in terms of mobile positively and negatively
        charged particles. Are these particles atoms? Parts of
        atoms? Particles that are entirely separate from atoms? It
        is perhaps premature to attempt to answer these questions
        without any conclusive evidence in favor of the charged-particle
        model of electricity.

        Strong support for the charged-particle model came from a
        1911 experiment by physicist Robert \index{Millikan, Robert}Millikan
        at the University of Chicago. Consider a jet
        of droplets of perfume or some other liquid made by blowing
        it through a tiny pinhole. The droplets emerging from the
        pinhole must be smaller than the pinhole, and in fact most
        of them are even more microscopic than that, since the
        turbulent flow of air tends to break them up. Millikan
        reasoned that the droplets would acquire a little bit of
        electric charge as they rubbed against the channel through
        which they emerged, and if the charged-particle model of
        electricity was right, the charge might be split up among so
        many minuscule liquid drops that a single drop might have a
        total charge amounting to an excess of only a few charged
        particles --- perhaps an excess of one positive particle on
        a certain drop, or an excess of two negative ones on another.
<% marg(60) %>
<%
  fig(
    'millikan',
    %q{A young Robert Millikan.}
  )
%>
        \spacebetweenfigs
        %
<%
  fig(
    'oildrop',
    %q{A simplified diagram of Millikan's apparatus.}
  )
%>
        
<% end_marg %>

        Millikan's ingenious apparatus, \ref{fig:oildrop},
        consisted of two metal plates, which could be electrically
        charged as needed. He sprayed a cloud of oil droplets into
        the space between the plates, and selected one drop through
        a microscope for study. First, with no charge on the plates,
        he would determine the drop's mass by letting it fall
        through the air and measuring its terminal velocity, i.e.,
        the velocity at which the force of air friction canceled out
        the force of gravity. The force of air drag on a slowly
        moving sphere had already been found by experiment to be
        $bvr^2$, where $b$ was a constant.
        Setting the total force equal to zero when the drop is at
        terminal velocity gives
        \begin{equation*}
                   bvr^2 - mg = 0\qquad   ,
        \end{equation*}
        and setting the known density of oil equal to the drop's
        mass divided by its volume gives a second equation,
        \begin{equation*}
                   \rho = \frac{m}{\frac{4}{3}\pi r^3}\qquad   .
        \end{equation*}
        Everything in these equations can be measured directly
        except for $m$ and $r$, so these are two equations in two
        unknowns, which can be solved in order to determine
        how big the drop is.

        Next Millikan charged the metal plates, adjusting the amount
        of charge so as to exactly counteract gravity and levitate
        the drop. If, for instance, the drop being examined happened
        to have a total charge that was negative, then positive
        charge put on the top plate would attract it, pulling it up,
        and negative charge on the bottom plate would repel it,
        pushing it up. (Theoretically only one plate would be
        necessary, but in practice a two-plate arrangement like this
        gave electrical forces that were more uniform in strength
        throughout the space where the oil drops were.) 
        When the drop was being levitated, the gravitational and
        electric forces canceled, so $qE=mg$. Since the mass of the drop,
        the gravitational
        field $g$, and the electric field $E$ were all known, the charge
        $q$ could be determined.

        Table \figref{millikanratios} shows a few of the results from
        Millikan's 1911 paper. (Millikan took data on both
        negatively and positively charged drops, but in his paper he
        gave only a sample of his data on negatively charged drops,
        so these numbers are all negative.) Even a quick look at the
        data leads to the suspicion that the charges are not simply
        a series of random numbers. For instance, the second charge
        is almost exactly equal to half the first one. Millikan
        explained the observed charges as all being integer
        multiples of a single number, $1.64\times10^{-19}$ C. In the
        second column, dividing by this constant gives numbers that
        are essentially integers, allowing for the random errors
        present in the experiment. Millikan states in his paper that
        these results were a
<% marg(80) %>
        \begin{minipage}[t]{52mm}
        \begin{tabular}{ll}
                                        & $q$\\
                                        & $/(1.64$\\
                $q$ (C)                & $\times10^{-19}\ \zu{C})$ \\
                \hline
                $-1.970\times10^{-18}$        & $-12.02$ \\
                $-0.987\times10^{-18}$        & $-6.02$ \\
                $-2.773\times10^{-18}$        & $-16.93$ \\
        \end{tabular}
        \docaption{A few samples of Millikan's data.} %
        \label{fig:millikanratios} %
        \end{minipage}
        
<% end_marg %>

        \begin{quote}
        \ldots direct and tangible demonstration \ldots of the correctness of
        the view advanced many years ago and supported by evidence
        from many sources that all electrical charges, however
        produced, are exact multiples of one definite, elementary
        electrical charge, or in other words, that an electrical
        charge instead of being spread uniformly over the charged
        surface has a definite granular structure, consisting, in
        fact, of \ldots specks, or atoms of electricity, all precisely
        alike, peppered over the surface of the charged body.
        \end{quote}

        In other words, he had provided direct evidence for the
        charged-particle model of electricity and against models in
        which electricity was described as some sort of fluid. The
        basic charge is notated $e$, and the modern value is
        $e=1.60\times10^{-19}$ C.
        The word ``\emph{quantized}'' is used in physics to describe
        a quantity that can only have certain numerical values, and
        cannot have any of the values between those. In this
        language, we would say that Millikan discovered that charge
        is quantized. The charge $e$ is referred to as the quantum of charge.

        <% self_check('quantizedmoney',<<-'SELF_CHECK'
               Is money quantized? What is the quantum of money?
          SELF_CHECK
  ) %>

        \begin{optionaltopic}{A historical note on Millikan's fraud}
        Very few undergraduate physics textbooks mention
         the well-documented fact that although
        Millikan's conclusions were correct, he was guilty of scientific
        fraud. His technique was difficult and painstaking to perform, and his
        original notebooks, which have been preserved, show that the data were
        far less perfect than he claimed in his published scientific papers.
        In his publications, he stated categorically that every single oil
        drop observed had had a charge that was a multiple of $e$, with no
        Exceptions or omissions. But his notebooks are replete with notations
        such as ``beautiful data, keep,'' and ``bad run, throw out.'' Millikan,
        then, appears to have earned his Nobel Prize by advocating a correct
        position with dishonest descriptions of his data.
        \end{optionaltopic}

<% end_sec('charge-quantization') %> % Quantization of charge
<% begin_sec("The electron",nil,'electron') %>
  <% begin_sec("Cathode rays",nil,'cathode-rays') %>\index{cathode rays}

        Nineteenth-century physicists spent a lot of
        time trying to come up with wild, random ways to play with
        electricity. The best experiments of this kind were the ones
        that made big sparks or pretty colors of light.

        One such parlor trick was the cathode ray. To produce it,
        you first had to hire a good glassblower and find a good
        vacuum pump. The glassblower would create a hollow tube and
        embed two pieces of metal in it, called the electrodes,
        which were connected to the outside via metal wires passing
        through the glass. Before letting him seal up the whole
        tube, you would hook it up to a vacuum pump, and spend
        several hours huffing and puffing away at the pump's hand
        crank to get a good vacuum inside. Then, while you were
        still pumping on the tube, the glassblower  would melt the
        glass and seal the whole thing shut. Finally, you would put
        a large amount of positive charge on one wire and a large
        amount of negative charge on the other. Metals have the
        property of letting charge move through them easily, so the
        charge deposited on one of the wires would quickly spread
        out because of the repulsion of each part of it for every
        other part. This spreading-out process would result in
        nearly all the charge ending up in the electrodes, where
        there is more room to spread out than there is in the wire.
        For obscure historical reasons a negative electrode is
        called a cathode and a positive one is an anode.

        Figure \ref{fig:crt} shows the light-emitting stream that was
        observed. If, as shown in this figure, a hole was made in
        the anode, the beam would extend on through the hole until
        it hit the glass. Drilling a hole in the cathode, however
        would not result in any beam coming out on the left side,
        and this indicated that the stuff, whatever it was, was
        coming from the cathode. The rays were therefore christened
        ``cathode rays.'' (The terminology is still used today in
        the term ``cathode ray tube'' or ``CRT'' for the picture
        tube of a TV or computer monitor.)

  <% end_sec('cathode-rays') %>
  <% begin_sec("Were cathode rays a form of light, or of matter?",nil,'nature-of-cathode-rays') %>

        <% marg(10) %>
<%
  fig(
    'crt',
    %q{Cathode rays observed in a vacuum tube.}
  )
%>
<% end_marg %>

        Were cathode rays a form of light, or matter? At first no
        one really cared what they were, but as their scientific
        importance became more apparent, the light-versus-matter
        issue turned into a controversy along nationalistic lines,
        with the Germans advocating light and the English holding
        out for matter. The supporters of the material interpretation
        imagined the rays as consisting of a stream of atoms ripped
        from the substance of the cathode.

        One of our defining characteristics of matter is that
        material objects cannot pass through each other. Experiments
        showed that cathode rays could penetrate at least some small
        thickness of matter, such as a metal foil a tenth of a
        millimeter thick, implying that they were a form of light.

        Other experiments, however, pointed to the contrary
        conclusion. Light is a wave phenomenon, and one distinguishing
        property of waves is demonstrated by speaking into one end of a
        paper towel roll. The sound waves do not emerge from the
        other end of the tube as a focused beam. Instead, they begin
        spreading out in all directions as soon as they emerge. This
        shows that waves do not necessarily travel in straight
        lines. If a piece of metal foil in the shape of a star or a
        cross was placed in the way of the cathode ray, then a
        ``shadow'' of the same shape would appear on the glass,
        showing that the rays traveled in straight lines. This
        straight-line motion suggested that they were a stream of
        small particles of matter. 

        These observations were inconclusive, so what was really
        needed was a determination of whether the rays had mass and
        weight. The trouble was that cathode rays could not simply
        be collected in a cup and put on a scale. When the cathode
        ray tube is in operation, one does not observe any loss of
        material from the cathode, or any crust being deposited on the anode.

         Nobody could think of a good way to weigh cathode rays, so
        the next most obvious way of settling the light/matter
        debate was to check whether the cathode rays possessed
        electrical charge. Light was known to be uncharged. If the
        cathode rays carried charge, they were definitely matter and
        not light, and they were presumably being made to jump the
        gap by the simultaneous repulsion of the negative charge in
        the cathode and attraction of the positive charge in the
        anode. The rays would overshoot the anode because of their
        momentum. (Although electrically charged particles do not
        normally leap across a gap of vacuum, very large amounts of
        charge were being used, so the forces were unusually intense.)

  <% end_sec('nature-of-cathode-rays') %>
  <% begin_sec("Thomson's experiments",nil,'thompson-experiment') %>\index{Thomson, J.J.!cathode ray experiments}
        <% marg(42) %>
<%
  fig(
    'thomson',
    %q{J.J. Thomson.}
  )
%>
<% end_marg %>
        Physicist J.J. Thomson at Cambridge carried out a series of
        definitive experiments on cathode rays around the year 1897.
        By turning them slightly off course with electrical forces, \ref{fig:deflect},
        he showed that they were indeed
        electrically charged, which was strong evidence that they
        were material. Not only that, but he proved that they had
        mass, and measured the ratio of their mass to their charge,
        $m/q$. Since their mass was not zero, he concluded that they
        were a form of matter, and presumably made up of a stream of
        microscopic, negatively charged particles. When Millikan
        published his results fourteen years later, it was
        reasonable to assume that the charge of one such particle
        equaled minus one fundamental charge, $q=-e$, and from the
        combination of Thomson's and Millikan's results one could
        therefore determine the mass of a single cathode ray particle.
        
        %
<%
  fig(
    'deflect',
    %q{%
      Thomson's experiment proving cathode rays had electric charge
              (redrawn from his original paper). The cathode, c, and anode, A, are
              as in any cathode ray tube. The rays pass through a slit in the anode,
              and a second slit, B, is interposed in order to make the beam thinner
              and eliminate rays that were not going straight. Charging plates D and
              E shows that cathode rays have charge: they are attracted toward the
              positive plate D and repelled by the negative plate E.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

        The basic technique for determining $m/q$ was simply to
        measure the angle through which the charged plates bent the
        beam. The electric force acting on a cathode ray particle
        while it was between the plates was
        \begin{equation*}
                        F_{elec}        =  qE      \qquad   .  
        \end{equation*}
        By Newton's second law, $a=F/m$, we can find
        $m/q$:
        \begin{equation*}
                        \frac{m}{q}        =      \frac{E}{a}
        \end{equation*}

        There was just one catch. Thomson needed to know the cathode
        ray particles' velocity in order to figure out their
        acceleration. At that point, however, nobody had even an
        educated guess as to the speed of the cathode rays produced
        in a given vacuum tube. The beam appeared to leap across the
        vacuum tube practically instantaneously, so it was no simple
        matter of timing it with a stopwatch!

        Thomson's clever solution was to observe the effect of both
        electric and magnetic forces on the beam. The magnetic force
        exerted by a particular magnet would depend on both the
        cathode ray's charge and its speed:
        \begin{equation*}
                        F_{mag}        =      qvB
        \end{equation*}

        Thomson played with the electric and magnetic forces until
        either one would produce an equal effect on the beam,
        allowing him to solve for the speed,

        \begin{equation*}
                        v    =     \frac{E}{B}  \qquad   .  
        \end{equation*}

        Knowing the speed (which was on the order of 10\% of the
        speed of light for his setup), he was able to find the
        acceleration and thus the mass-to-charge ratio $m/q$.
        Thomson's techniques were relatively crude (or perhaps more
        charitably we could say that they stretched the state of the
        art of the time), so with various methods he came up with
        $m/q$ values that ranged over about a factor of two, even
        for cathode rays extracted from a cathode made of a single
        material. The best modern value is
        $m/q=5.69\times10^{-12}$ kg/C, which is consistent with the low
        end of Thomson's range.

  <% end_sec('thompson-experiment') %>
  <% begin_sec("The cathode ray as a subatomic particle: the electron",nil,'cathode-rays-are-subatomic') %>\index{electron}
        What was significant about Thomson's experiment was not the
        actual numerical value of $m/q$, however, so much as the
        fact that, combined with Millikan's value of the fundamental
        charge, it gave a mass for the cathode ray particles that
        was thousands of times smaller than the mass of even the
        lightest atoms. Even without Millikan's results, which were
        14 years in the future, Thomson recognized that the cathode
        rays' $m/q$ was thousands of times smaller than the
        $m/q$ ratios that had been measured for electrically charged
        atoms in chemical solutions. He correctly interpreted this
        as evidence that the cathode rays were smaller building
        blocks --- he called them \emph{electrons} --- out of which
        atoms themselves were formed. This was an extremely radical
        claim, coming at a time when atoms had not yet been proven
        to exist! Even those who used the word ``atom'' often
        considered them no more than mathematical abstractions, not
        literal objects. The idea of searching for structure inside
        of ``unsplittable'' atoms was seen by some as lunacy, but
        within ten years Thomson's ideas had been amply verified by
        many more detailed experiments.


\startdqs

\begin{dq}
        Thomson started to become convinced during his experiments
        that the ``cathode rays'' observed coming from the cathodes
        of vacuum tubes were building blocks of atoms --- what we
        now call electrons. He then carried out observations with
        cathodes made of a variety of metals, and found that $m/q$
        was roughly the same in every case, considering his limited
        accuracy. Given his suspicion, why did it make sense to try
        different metals? How would the consistent values of $m/q$
        test his hypothesis?
\end{dq}
 %
\begin{dq}
        My students have frequently asked whether the $m/q$ that
        Thomson measured was the value for a single electron, or for
        the whole beam. Can you answer this question?
\end{dq}
 %
\begin{dq}
        Thomson found that the $m/q$ of an electron was thousands
        of times smaller than that of charged atoms in chemical
        solutions. Would this imply that the electrons had more
        charge? Less mass? Would there be no way to tell? Explain.
        Remember that Millikan's results were still many years in
        the future, so $q$ was unknown.
\end{dq}
 %
\begin{dq}
        Can you guess any practical reason why Thomson couldn't
        just let one electron fly across the gap before disconnecting
        the battery and turning off the beam, and then measure the
        amount of charge deposited on the anode, thus allowing him
        to measure the charge of a single electron directly?
\end{dq}
 %
\begin{dq}
        Why is it not possible to determine $m$ and $q$
        themselves, rather than just their ratio, by observing
        electrons' motion in electric and magnetic fields?
\end{dq}

 %%----------------------------------------------
  <% end_sec('cathode-rays-are-subatomic') %>
  <% begin_sec("The raisin cookie model",nil,'raisin-cookie') %>
        Based on his experiments, Thomson proposed a picture of the
        atom which became known as the \index{atom!raisin-cookie
        model of}\index{raisin cookie model}raisin cookie model.
        In the neutral atom, \ref{fig:raisincookie}, there are four
        electrons with a total charge of $-4e$, sitting in a sphere
        (the ``cookie'') with a charge of $+4e$ spread throughout
        it. It was known that chemical reactions could not change
        one element into another, so in Thomson's scenario, each
        element's cookie sphere had a permanently fixed radius,
        mass, and positive charge, different from those of other
        elements. The electrons, however, were not a permanent
        feature of the atom, and could be tacked on or pulled out to
        make charged ions. Although we now know, for instance, that
        a neutral atom with four electrons is the element beryllium,
        scientists at the time did not know how many electrons the
        various neutral atoms possessed.
<% marg(0) %>
<%
  fig(
    'raisincookie',
    %q{%
      The raisin cookie model of the atom with four units of charge,
              which we now know to be beryllium.
    }
  )
%>
<% end_marg %>

        This model is clearly different from the one you've learned
        in grade school or through popular culture, where the
        positive charge is concentrated in a tiny nucleus at the
        atom's center. An equally important change in ideas about
        the atom has been the realization that atoms and their
        constituent subatomic particles behave entirely differently
        from objects on the human scale. For instance, we'll see
        later that an electron can be in more than one place at one
        time. The raisin cookie model was part of a long tradition
        of attempts to make mechanical models of phenomena, and
        Thomson and his contemporaries never questioned the
        appropriateness of building a mental model of an atom as a
        machine with little parts inside. Today, mechanical models
        of atoms are still used (for instance the tinker-toy-style
        molecular modeling kits like the ones used by Watson and
        Crick to figure out the double helix structure of DNA), but
        scientists realize that the physical objects are only aids
        to help our brains' symbolic and visual processes think about atoms.

        Although there was no clear-cut experimental evidence for
        many of the details of the raisin cookie model, physicists
        went ahead and started working out its implications. For
        instance, suppose you had a four-electron atom. All four
        electrons would be repelling each other, but they would also
        all be attracted toward the center of the ``cookie'' sphere.
        The result should be some kind of stable, symmetric
        arrangement in which all the forces canceled out. People
        sufficiently clever with math soon showed that the electrons
        in a four-electron atom should settle down at the vertices
        of a pyramid with one less side than the Egyptian kind, i.e.,
        a regular tetrahedron. This deduction turns out to be wrong
        because it was based on incorrect features of the model, but
        the model also had many successes, a few of which we will now discuss.

\begin{eg}{Flow of electrical charge in wires}
                One of my former students was the son of an electrician, and
        had become an electrician himself. He related to me how his
        father had remained refused to believe all his life that
        electrons really flowed through wires. If they had, he
        reasoned, the metal would have gradually become more and
        more damaged, eventually crumbling to dust.

                His opinion is not at all unreasonable based on the fact
        that electrons are material particles, and that matter
        cannot normally pass through matter without making a hole
        through it. Nineteenth-century physicists would have shared
        his objection to a charged-particle model of the flow of
        electrical charge. In the raisin-cookie model, however,
        the electrons are very low in mass, and therefore presumably
        very small in size as well. It is not surprising that they
        can slip between the atoms without damaging them.
\end{eg}
\begin{eg}{Flow of electrical charge across cell membranes}
                Your nervous system is based on signals carried by charge
        moving from nerve cell to nerve cell. Your body is
        essentially all liquid, and atoms in a liquid are mobile.
        This means that, unlike the case of charge flowing in a
        solid wire, entire charged atoms can flow in your nervous system
\end{eg}

\begin{eg}{Emission of electrons in a cathode ray tube}
                Why do electrons detach themselves from the cathode of a
        vacuum tube? Certainly they are encouraged to do so by the
        repulsion of the negative charge placed on the cathode and
        the attraction from the net positive charge of the anode,
        but these are not strong enough to rip electrons out of
        atoms by main force --- if they were, then the entire
        apparatus would have been instantly vaporized as every atom
        was simultaneously ripped apart!

                The raisin cookie model leads to a simple explanation. We
        know that heat is the energy of random motion of atoms. The
        atoms in any object are therefore violently jostling each
        other all the time, and a few of these collisions are
        violent enough to knock electrons out of atoms. If this
        occurs near the surface of a solid object, the electron may
        can come loose. Ordinarily, however, this loss of electrons
        is a self-limiting process; the loss of electrons leaves the
        object with a net positive charge, which attracts the lost
        sheep home to the fold. (For objects immersed in air rather
        than vacuum, there will also be a balanced exchange of
        electrons between the air and the object.)

                This interpretation explains the warm and friendly yellow
        glow of the vacuum tubes in an antique radio. To encourage
        the emission of electrons from the vacuum tubes' cathodes,
        the cathodes are intentionally warmed up with little heater coils.
\end{eg}


\startdqs

\begin{dq}
        Today many people would define an ion as an atom (or
        molecule) with missing electrons or extra electrons added
        on. How would people have defined the word ``ion'' before
        the discovery of the electron?
\end{dq}
 %
\begin{dq}
        Since electrically neutral atoms were known to exist,
        there had to be positively charged subatomic stuff to cancel
        out the negatively charged electrons in an atom. Based on
        the state of knowledge immediately after the Millikan and
        Thomson experiments, was it possible that the positively
        charged stuff had an unquantized amount of charge? Could it
        be quantized in units of +e? In units of +2e? In units of +5/7e?
\end{dq}

  <% end_sec('raisin-cookie') %> % The raisin cookie model
<% end_sec('electron') %> % The electron

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
__incl(text/nucleus)
__incl(text/biological_effects_of_radiation)
__incl(text/nucleosynthesis)
<% end_sec('nucleus') %>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


<% begin_sec("Relativistic mass and energy",nil,'reldynamics') %>
The radioactive decay processes described in this chapter do not conserve mass.
For example, you're probably reading this in a building that has smoke detectors
containing the isotope $^{241}\zu{Am}$, an alpha emitter. The decay products are
a helium atom plus an atom of $^{237}\zu{Np}$. In units of $10^{-27}\ \kgunit$,
the masses involved are:

\noindent\begin{tabular}{lr}
  $^{241}\zu{Am}$ & 400.28421 \\
  $^{237}\zu{Np}$ & 393.62768 \\
  $^4\zu{He}$     & 6.64647
\end{tabular}

The final state has a total mass of 400.27415 of these units, meaning a loss of
0.01006. This is an example of Einstein's famous $E=mc^2$ at work: some mass has
been converted into energy. In fact this type of mass-energy conversion is not
just a property of nuclear decay. It is an example of a much wider set of
phenomena in relativity. Let's see how quantities like mass, force, momentum,
and energy behave relativistically.

  <% begin_sec("Momentum",nil,'rel-momentum') %>\index{momentum!relativistic}
Consider the following scheme for traveling faster than the speed of light.
The basic idea can be demonstrated by dropping a ping-pong ball and a baseball
stacked on top of each other like a snowman. They separate slightly in mid-air,
and the baseball therefore has time to hit the floor and rebound before it
collides with the ping-pong ball, which is still on the way down. The result is
a surprise if you haven't seen it before: the ping-pong ball flies off at high
speed and hits the ceiling! A similar fact is known to people who investigate the
scenes of accidents involving pedestrians. If a car moving at 90 kilometers per
hour hits a pedestrian, the pedestrian flies off at nearly double that speed, 180
kilometers per hour. Now suppose the car was moving at 90 percent of the speed of
light. Would the pedestrian fly off at 180\% of $c$?

\vspace{15mm}

To see why not, we have to back up a little and think about where this speed-doubling
result comes from. 
For any collision, there is a special frame of reference, the center-of-mass frame,
in which the two colliding objects approach each other,
collide, and rebound with their velocities reversed. In the center-of-mass frame,
the total momentum of the objects is zero both before and after the collision.

<%
  fig(
    'unequalcollision',
    %q{%
      An unequal collision, viewed in the center-of-mass frame, 1, and
      in the frame where the small ball is initially at rest, 2. The motion is shown as it
      would appear on the film of an old-fashioned movie camera, with an equal amount of time separating
      each frame from the next. Film 1 was made by a camera that tracked the center of mass, film 2 by
      one that was initially tracking the small ball, and kept on moving at the same speed after the collision.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

Figure \subfigref{unequalcollision}{1} shows such a frame of reference for objects of very unequal mass.
Before the collision, the large ball is moving relatively slowly toward the top of the page, but
because of its greater mass, its momentum cancels the momentum of the smaller ball, which is
moving rapidly in the opposite direction. The total momentum is zero. After the collision, the
two balls just reverse their directions of motion. We know that this is the
right result for the outcome of the collision because
it conserves both momentum and kinetic energy, and everything not forbidden is compulsory, i.e.,
in any experiment, there is only one possible outcome, which is the one that obeys all the
conservation laws.

<% self_check('unequalcollisioncons',<<-'SELF_CHECK'
How do we know that momentum and kinetic energy are conserved
in figure \subfigref{unequalcollision}{1}?
  SELF_CHECK
  ) %>

Let's make up some numbers as an example. Say the small ball is 1 kg, the big
one 6 kg. In frame 1, let's make the velocities as follows:

<% if is_print then
print %q(
\\newcommand{\\smallvelocitytable}[6]{%
\\noindent\\hspace{5mm}\\begin{tabular}{|p{4mm}|p{40mm}|p{40mm}|}
\\hline
 & before the collision
       & after the collision  \\\\
\\hline
\\anonymousinlinefig{#5} & #1 & #2  \\\\
\\anonymousinlinefig{#6} & #3 & #4  \\\\
\\hline
\\end{tabular}
}
)
end
%>

<%
def small_vel_table(a,b,c,d)
if is_print then
  return "\\smallvelocitytable{#{a}}{#{b}}{#{c}}{#{d}}{../../../share/relativity/figs/smallball}{../../../share/relativity/figs/bigball}"
end
if is_web then
return %Q(
\\noindent\\begin{tabular}{|p{4mm}|p{40mm}|p{40mm}|}
\\hline
 & before the collision
       & after the collision  \\\\
\\hline
small ball & #{a} & #{b}  \\\\
big ball & #{c} & #{d}  \\\\
\\hline
\\end{tabular}
)
end
end
%>

<% print small_vel_table(-0.6,0.6,0.1,-0.1) %>

Figure \subfigref{unequalcollision}{2} shows the same collision in a frame of reference where
the small ball was initially at rest.
 To find all the velocities in this frame, we
just add 0.6 to all the ones in the previous table.

<% print small_vel_table(0,1.2,0.7,0.5) %>

\noindent In this frame, as expected, the small ball flies off with a velocity, 1.2, that
is almost twice the initial velocity of the big ball, 0.7. In this example the ratio of the
two balls' masses was 6, but if the ratio of the masses is made larger and larger, the ratio
of the velocities gets closer and closer to 2.

If all those velocities were in meters per second, then that's exactly what would happen. But
what if all these velocities were in units of the speed of light? Now it's no longer a good
approximation just to add velocities. We need to combine them according to the relativistic
rules. For instance,  in problem \ref{hw:six-tenths-c-twice}
on p.~\pageref{hw:six-tenths-c-twice} you showed that
combining a velocity of 0.6 times the
speed of light with another velocity of 0.6 results in 0.88, not 1.2. The results are very different:

<% print small_vel_table(0,0.88,0.67,0.51) %>

% initial velocities in cm = 0.600, .124

<%
  fig(
    'unequalrel',
    %q{%
      A 6-kg ball moving at 88\% of the speed of light hits a 1-kg ball. The balls
      appear foreshortened due to the relativistic distortion of space.
    },
    {
      'width'=>'wide',
      'sidecaption'=>true
    }
  )
%>

We can interpret this as follows. Figure \subfigref{unequalcollision}{1} is one in which the
big ball is moving fairly slowly. This is very nearly the way the scene would be seen by an
ant standing on the big ball. According to an observer in frame \figref{unequalrel}, however,
both balls are moving at nearly the speed of light after the collision. Because of this, the
balls appear foreshortened, but the distance between the two balls is also shortened. To this
observer, it seems that the small ball isn't pulling away from the big ball very fast.

Now here's what's interesting about all this. The outcome shown in figure \subfigref{unequalcollision}{2}
was supposed to be the only one possible, the only one that satisfied both conservation of energy
and conservation of momentum. So how can the \emph{different} result shown in figure
\figref{unequalrel} be possible? The answer is that relativistically, momentum must not equal
$mv$. The old, familiar definition is only an approximation that's valid at low speeds. If
we observe the behavior of the small ball in figure \figref{unequalrel}, it looks as though it
somehow had some extra inertia. It's as though a football player tried to knock another player
down without realizing that the other guy had a three-hundred-pound bag full of lead shot
hidden under his uniform --- he just doesn't seem to react to the collision as much as he should.
This extra inertia is described\footnote{See p.~\pageref{rel-momentum-proof} for a proof.} by redefining momentum as
\begin{equation*}
        p = m \mygamma v \qquad .
\end{equation*}
At very low velocities, $\mygamma$ is close to 1, and the result is very nearly $mv$, as demanded
by the correspondence principle. But at very high velocities, $\mygamma$ gets very big --- the
small ball in figure \figref{unequalrel} has a $\mygamma$ of 2.1, and therefore has more than double
the inertia that we would expect nonrelativistically.\index{correspondence principle!for relativistic momentum}

This also explains the answer to another paradox often posed by beginners at relativity.
Suppose you keep on applying a steady force to an object that's already moving at $0.9999c$.
Why doesn't it just keep on speeding up past $c$? The answer is that force is the rate of
change of momentum. At $0.9999c$, an object already has a $\mygamma$ of 71, and therefore
has already sucked up 71 times the momentum you'd expect at that speed. As its velocity gets closer and
closer to $c$, its $\mygamma$ approaches infinity. To move at $c$, it would need an infinite
momentum, which could only be caused by an infinite force.

m4_include(../share/relativity/eg/bertozzi.tex)

 %
  <% end_sec('rel-momentum') %>
  <% begin_sec("Equivalence of mass and energy",nil,'mass-energy') %>\index{energy!equivalence to mass}\index{mass!equivalence to energy}\label{mass-energy-equivalence}
Now we're ready to see why mass and energy must be equivalent as claimed
in the famous $E=mc^2$. So far we've only considered collisions
in which none of the kinetic energy is converted into any other form
of energy, such as heat or sound.
Let's consider what happens if a blob of putty moving at
velocity $v$ hits another blob that is initially at rest,
sticking to it.  The nonrelativistic result is
that to obey conservation of momentum the two blobs must fly
off together at $v/2$. Half of the initial kinetic energy
has been converted to heat.\footnote{A double-mass object moving
at half the speed does not have the same kinetic energy. Kinetic
energy depends on the square of the velocity, so cutting the velocity
in half reduces the energy by a factor of 1/4, which, multiplied
by the doubled mass, makes 1/2 the original energy.}

Relativistically, however, an interesting thing happens. A
hot object has more momentum than a cold object! This is
because the relativistically correct expression for momentum
is $m\mygamma v$, and the more rapidly moving atoms in the hot
object have higher values of $\mygamma$.
In our collision, the final combined blob must therefore be
moving a little more slowly than the expected $v/2$, since
otherwise the final momentum would have been a little
greater than the initial momentum. To an observer who
believes in conservation of momentum and knows only about
the overall motion of the objects and not about their heat
content, the low velocity after the collision would seem
to be the result of a magical change in the mass, as if the mass
of two combined, hot blobs of putty was more than the sum of
their individual masses.

We know that the masses of all the atoms in the blobs
must be the same as they always were. The change is due to
the change in $\mygamma$ with heating, not to a change in mass.
The heat energy, however, seems to be acting as if it was
equivalent to some extra mass. If the quantity of heat is $E$, then
it turns out that the extra mass $m$ is such that $E=mc^2$ (proof, p.~\pageref{rel-energy-proof}).


But this whole argument was based on the fact that heat is a
form of kinetic energy at the atomic level. Would $E=mc^2$
apply to other forms of energy as well? Suppose a rocket
ship contains some electrical energy stored in a
battery. If we believed that $E=mc^2$ applied to forms of
kinetic energy but not to electrical energy, then
we would have to believe that the pilot of the rocket could
slow the ship down by using the battery to run a heater!
This would not only be strange, but it would violate the
principle of relativity, because the result of the
experiment would be different depending on whether the ship
was at rest or not. The only logical conclusion is that all
forms of energy are equivalent to mass. Running the heater
then has no effect on the motion of the ship, because the
total energy in the ship was unchanged; one form of energy (electrical)
was simply converted to another (heat).

The equation $E=mc^2$
tells us how much energy is equivalent to how much mass: the conversion factor is the square
of the speed of light, $c$. Since $c$ a big number, you get a really really big number
when you multiply it by itself to get $c^2$. This means that even a small amount of mass
is equivalent to a very large amount of energy. 
<% marg(-90) %>
<%
  fig(
    'newspaper-eclipse',
    %q{%
      A New York Times headline from November 10, 1919, describing
      the observations discussed in example \ref{eg:eclipse}.
    }
  )
%>
<% end_marg %>

<%
  fig(
    'eclipse',
    %q{Example \ref{eg:eclipse}.},
    {
      'width'=>'wide'
    }
  )
%>

\begin{eg}{Gravity bending light}\label{eg:eclipse}
Gravity is a universal attraction between things that have mass, and since the energy
in a beam of light is equivalent to some very small amount of mass, we expect that
light will be affected by gravity, although the effect should be very small.
The first important experimental confirmation of relativity
came in 1919 when stars next to the sun during a solar eclipse were
observed to have shifted a little from their ordinary
position. (If there was no eclipse, the glare of the sun
would prevent the stars from being observed.) Starlight had
been deflected by the sun's gravity. Figure \figref{eclipse} is a
photographic negative, so the circle that appears bright is actually the
dark face of the moon, and the dark area is really the bright corona of
the sun. The stars, marked by lines above and below them, appeared at
positions slightly different than their normal ones.
\end{eg}

\begin{eg}{Black holes}\index{black hole}
A star with sufficiently strong gravity can prevent light
from leaving. Quite a few black holes have been detected via
their gravitational forces on neighboring stars or clouds of gas and dust.
\end{eg}


You've learned about conservation of mass and conservation of energy, but
now we see that they're not even separate conservation laws.
As a consequence of the theory of relativity,  mass and energy are equivalent, and
are not separately conserved --- one can be converted into the other. Imagine that
a magician waves his wand, and changes a bowl of dirt into a bowl of lettuce. You'd be
impressed, because you were expecting that both dirt and lettuce would be conserved
quantities. Neither one can be made to vanish, or to appear out of thin air. However,
there are processes that can change one into the other. A farmer changes dirt into
lettuce, and a compost heap changes lettuce into dirt. At the most fundamental
level, lettuce and dirt aren't really different things at all; they're just collections
of the same kinds of atoms --- carbon, hydrogen, and so on.
Because mass and energy are like two different sides of the same coin, we may speak of
mass-energy, a single conserved quantity, found by adding up all the mass and energy,
with the appropriate conversion factor: $E+mc^2$.\index{mass-energy!conservation of}

\begin{eg}{A rusting nail}\label{eg:rustingnail}
\egquestion
An iron nail is left in a cup of water
until it turns entirely to rust. The energy released is
about 0.5 MJ. In theory, would a sufficiently
precise scale register a change in mass? If so, how much?

\eganswer
 The energy will appear as heat, which will be lost
to the environment. The total mass-energy of the cup,
water, and iron will indeed be lessened by 0.5 MJ. (If it
had been perfectly insulated, there would have been no
change, since the heat energy would have been trapped in the
cup.) The speed of light is
$c=3\times10^8$ meters per second, so converting to mass units, we have
\begin{align*}
                m         &=    \frac{E}{c^2}  \\
                        &= \frac{0.5\times10^6\ \junit}{\left(3\times10^8\ \munit/\sunit\right)^2} \\
                         &=    6\times10^{-12}\  \text{kilograms}   \qquad   .
\end{align*}
The change in mass is too small to measure with any
practical technique. This is because the square of the speed
of light is such a large number.
\end{eg}

\begin{eg}{Electron-positron annihilation}\index{positron}
Natural radioactivity in the earth produces positrons, which are like electrons but have the
opposite charge. A form of antimatter, positrons annihilate with electrons to produce gamma
rays, a form of high-frequency light. Such a process would have been considered impossible
before Einstein, because conservation of mass and energy were believed to be separate
principles, and this process eliminates 100\% of the original mass. The amount of energy
produced by annihilating 1 kg of matter with 1 kg of antimatter is
\begin{align*}
 E &= mc^2\\
   &= (2\ \kgunit)\left(3.0\times10^8\ \munit/\sunit\right)^2\\
   &= 2\times10^{17}\ \junit \qquad ,
\end{align*}
which is on the same order of magnitude as a day's energy consumption for the
entire world's population!

Positron annihilation forms the basis for the medical imaging technique called
a PET (positron emission tomography) scan, in which a positron-emitting chemical
is injected into the patient and map\-ped by the emission of gamma rays from the parts
of the body where it accumulates.
\end{eg}
<% marg(140) %>
<%
  fig(
    'pet',
    %q{Top: A PET scanner. Middle: Each positron annihilates with an electron, producing two gamma-rays that fly off back-to-back.
       When two gamma rays are observed simultaneously in the ring of detectors, they are assumed to come from the same
       annihilation event, and the point at which they were emitted must lie on the line connecting the two detectors.
       Bottom: A scan of a person's torso. The body has concentrated the radioactive tracer around the stomach, indicating
       an abnormal medical condition.}
  )
%>
<% end_marg %>

One commonly hears some misinterpretations of $E=mc^2$, one being that the equation tells us
how much kinetic energy an object would have if it was moving at the speed of light. This
wouldn't make much sense, both because the equation for kinetic energy has $1/2$ in it, $KE=(1/2)mv^2$, and
because a material object can't be made to move at the speed of light. However, this naturally leads to the
question of just how much mass-energy a moving object has. We know that when the object is at rest, it
has no kinetic energy, so its mass-energy is simply equal to the energy-equivalent of its mass, $mc^2$,
\begin{equation*}
  \massenergy = mc^2 \ \text{when}\ v=0 \qquad ,
\end{equation*}
where the symbol $\massenergy$ (cursive ``E'') stands for mass-energy. The point of using the new symbol is simply
to remind ourselves that we're talking about relativity, so an object at rest has $\massenergy=mc^2$, not
$E=0$ as we'd assume in classical physics.

Suppose we start accelerating the object with a constant force. A constant force means a constant
rate of transfer of momentum, but $p=m\mygamma v$ approaches infinity as $v$ approaches $c$, so the object
will only get closer and closer to the speed of light, but never reach it. Now what about the work being
done by the force? The force keeps doing work and doing work, which means that we keep on using up
energy. Mass-energy is conserved, so the energy being expended must equal the increase in the object's
mass-energy. We can continue this process for as long as we like, and the amount of mass-energy
will increase without limit. We therefore conclude that an object's mass-energy approaches infinity
as its speed approaches the speed of light,
\begin{equation*}
  \massenergy \rightarrow \infty\ \text{when}\ v \rightarrow c \qquad .
\end{equation*}

\index{mass-energy!of a moving particle}
Now that we have some idea what to expect, what is the actual equation for the mass-energy? 
As proved in my book \emph{Simple Nature}, it is
\begin{equation*}
  \massenergy =m\mygamma c^2 \qquad .
\end{equation*}

<% self_check('mass-energy',<<-'SELF_CHECK'
Verify that this equation has the two properties we wanted.
  SELF_CHECK
  ) %>

\begin{eg}{KE compared to $mc^2$ at low speeds}\label{eg:massenergy-low-speed}
\egquestion An object is moving at ordinary nonrelativistic speeds. Compare its
kinetic energy to the energy $mc^2$ it has purely because of its mass.

\eganswer The speed of light is a very big number, so $mc^2$ is a huge number of
joules. The object has a gigantic amount of energy because of its mass, and only
a relatively small amount of additional kinetic energy because of its motion.

Another way of seeing this is that at low speeds, $\mygamma$ is only a tiny bit
greater than 1, so $\massenergy$ is only a tiny bit greater than $mc^2$.
\end{eg}

\begin{eg}{The correspondence principle for mass-energy}\index{mass-energy!correspondence principle}\index{correspondence principle!for mass-energy}
\egquestion Show that the equation $\massenergy=m\mygamma c^2$ obeys the correspondence principle.

\eganswer As we accelerate an object from rest, its mass-energy becomes greater than
its resting value. We interpret this excess mass-energy as the object's
kinetic energy,
\begin{align*}
  KE   &= \massenergy(v)-\massenergy(v=0) \\
       &= m\mygamma c^2 - m c^2 \\
       &= m(\mygamma-1)c^2 \qquad .
\end{align*}
In example \ref{eg:gamma-for-low-v} on page \pageref{eg:gamma-for-low-v}, we found
$\mygamma\approx 1+v^2/2c^2$, so
\begin{align*}
  KE   &\approx m(1+\frac{v^2}{2c^2}-1)c^2 \\
       &= \frac{1}{2}mv^2 \qquad ,
\end{align*}
which is the nonrelativistic expression. As demanded by the correspondence principle,
relativity agrees with nonrelativistic physics at speeds that are small compared to
the speed of light.
\end{eg}

  <% end_sec('mass-energy') %>
<% end_sec('reldynamics') %> % Relativistic mass and energy

<% begin_sec("Proofs",nil,'rel-p-and-e-proofs',{'optional'=>true}) %>
In section \ref{sec:reldynamics} I gave physical arguments to the effect that relativistic momentum should be greater than
$mv$ and that an energy $E$ should be equivalent relativistically to some amount of mass $m$.
In this section I'll prove that the relativistic equations are as claimed: $p=m\mygamma v$ and $E=mc^2$.
The structure of the proofs is essentially the same as in two famous 1905 papers by Einstein,
``On the electrodynamics of moving bodies'' and ``Does the inertia of a body depend upon its energy content?''
If you're interested in reading these arguments as Einstein originally wrote them, you can find English
translations at \verb@www.fourmilab.ch@. We start off by proving two preliminary results relating
to Doppler shifts.

  <% begin_sec("Transformation of the fields in a light wave",nil,'eb-lorentz') %>
On p.~\pageref{subsec:doppler-light} I showed that when a light wave is observed in two different frames
in different states of motion parallel to the wave's direction of motion, the frequency is observed to
be Doppler-shifted by a factor $D(v)=\sqrt{(1-v)/(1+v)}$, where $c=1$ and $v$ is the relative velocity of the two frames. 
But a change in frequency is not the only change we expect. We also expect the \emph{intensity} of the wave to
change, since a combination of electric and magnetic fields
observed in one frame of reference becomes some other set of fields in a different frame (p.~\pageref{relativity-requires-magnetism}).
There are equations that express this transformation from $\vc{E}$ and $\vc{B}$ to $\vc{E}'$ and $\vc{B}'$, but they're
a little complicated, so instead we'll just determine what happens in the special case of an electromagnetic wave.

Since the transformation of $\vc{E}$ and $\vc{B}$ to $\vc{E}'$ and $\vc{B}'$ is a universal thing, we're free to
imagine that the wave was created in any way we wish. 
Suppose that it was created by a uniform sheet of charge in the $x$-$y$ plane, oscillating in the $y$ direction with
amplitude $A$ and frequency $f$.
This will clearly produce electromagnetic waves propagating in the $+z$ and $-z$ directions, and
by an argument similar to that of problem \ref{hw:field-of-sheet-by-scaling} on p.~\pageref{hw:field-of-sheet-by-scaling},
we know that these waves' intensity will not fall off at all with distance from the sheet. Since magnetic fields are produced
by currents, and the currents produced by the motion of the sheet are proportional to $Af$, the amplitude of the magnetic field in the
wave is proportional to $Af$. The oscillating magnetic field induces an electric field, and since electromagnetic waves always have
$E=Bc$, the oscillating part of the electric field is also proportional of $Af$.

An observer moving away from the sheet sees a sheet that is both oscillating more slowly ($f$ is Doppler-shifted to $fD$)
and receding. But the recession has no effect, because the fields don't fall off with distance. Also, $A$ stays the
same, because the Lorentz transformation has no effect on lengths perpendicular to the relative motion of the two frames.
Since the fields are proportional to $Af$, the fields seen by the receding observer are attenuated by a factor of $D$.
  <% end_sec('eb-lorentz') %> % Transformation of the fields in a light wave
  <% begin_sec("Transformation of the volume of a light wave",nil,'em-vol-transformation') %>
Since the fields in an electromagnetic wave are changed by a factor of $D$ when we change frames, we might expect that the wave's energy
would change by a factor of $D^2$. But the square of the field only gives the energy per unit volume, and the volume
changes as well. The following argument shows that the volume increases by a factor of $1/D$.

If an electromagnetic wave-train has duration $\Delta t$, we already know that its duration changes by a factor of $1/D$ when we change to a different
frame of reference. But the speed of light is the same for all observers, so if the length of the wave-train is
$\Delta z$, all observers must agree on the value of $\Delta z/\Delta t$, and $1/D$ must also be the factor 
by which $\Delta z$ scales up.\footnote{At first glance, one might think that this length-scaling factor would simply be $\mygamma$, and
that the volume would be reduced rather than increased.
But $\mygamma$ is only the scale-down factor for the length of a thing compared to that thing's length in a frame where it is
at rest. Light waves don't have a frame in which they're at rest. One can also see this from 
the geometry of figure \figref{doppler-geometry} on p.~\pageref{fig:doppler-geometry}.
The diagram is completely symmetric with respect to its treatment of time and space, so if we flip it
across its diagonal, interchanging the roles of $z$ and $t$, we obtain the same result for the wave-train's spatial extent $\Delta z$.}
Since the Lorentz transformation doesn't change $\Delta x$ or $\Delta y$, the volume of the wave-train is also increased by
a factor of $1/D$.


  <% end_sec('em-vol-transformation') %> % Transformation of the volume of a light wave
  <% begin_sec("Transformation of the energy of a light wave",nil,'doppler-energy') %>\label{light-wave-energy-doppler}
Combining the two preceding results, we find that when we change frames of reference, the energy \emph{density} (per unit volume) of a light
wave changes by a factor of $D^2$, but the volume changes by $1/D$, so the result is that the wave's energy changes by a factor of $D$.
In Einstein's words, ``It is remarkable that the energy and the frequency of a [wave-train] vary with the state of motion of the observer in accordance with the same law,''
i.e., that both scale by the same factor $D$. Einstein had a reason to be especially interested in this fact. In the same
``miracle year'' of 1905, he also published a paper in which he hypothesized that light had both particle and wave properties,
with the energy $E$ of a light-particle related to the frequency $f$ of the corresponding light-wave by $E=hf$, where $h$ was a constant. (More
about this in ch.~\ref{ch:light-as-a-particle}.)
If $E$ and $f$ had not both scaled by the same factor, then the relation $E=hf$ could not have held in all frames of reference.
  <% end_sec('doppler-energy') %> % Transformation of the energy of a light wave
  <% begin_sec("$E=mc^2$",nil,'rel-energy-proof') %>\label{rel-energy-proof}
Suppose that a material object O, initially at rest, emits two light rays, each with energy $E$, in the $+z$ and $-z$ directions. O could be a lantern with windows
on opposite sides, or it could be an electron and an antielectron annihilating each other to produce a pair of gamma rays.
In this frame, O loses energy $2E$ and the light rays gain $2E$, so energy is conserved.

We now switch to a new frame of reference moving at a certain velocity $v$ in the $z$ direction relative to the original frame.
We assume that O's energy is different in this frame, but that the change in its energy amounts to multiplication by some unitless factor $x$,
which depends only on $v$, since there is nothing else it could depend on that could allow us to form a unitless quantity. In this frame the light rays have energies $ED(v)$ and
$ED(-v)$. If conservation of energy is to hold in the new frame as it did in the old, we must have $2xE=ED(v)+ED(-v)$.
After some algebra, we find $x=1/\sqrt{1-v^2}$. In other words, an object with energy $E$ in its rest frame
has energy $\mygamma E$ in a frame moving at velocity $v$ relative to the first one. Since $\mygamma$ is never zero,
it follows that even an object at rest has some nonzero energy. We define this energy-at-rest as its mass,
i.e., $E=m$ in units where $c=1$.
<% end_sec('rel-energy-proof') %>
<% begin_sec("$p=m\\mygamma v$",nil,'rel-momentum-proof') %>\label{rel-momentum-proof}
Defining an object's energy-at-rest as its mass only works if this same mass is also a valid measure of inertia.
More specifically, we should be able to use this mass
to construct a self-consistent logical system in which (1) momentum is conserved, (2) conservation of momentum
holds in all frames of reference, and (3) $p\approx mv$ for $v<<c$, satisfying the correspondence principle.

Let a material object P, at rest and having mass $2E$, be completely annihilated, creating two beams of light, each with
energy $E$, flying off in opposite directions. A real-world example would be if P consisted of an electron
and an antielectron. As shown on p.~\pageref{fig:maxwellian-momentum-of-light}, light has momentum.
Because beams of light can be split up or recombined without violating conservation of momentum, a light
wave's momentum must be proportional to its energy, $|p|=yE$, where the constant of proportionality $y$ is
found in problem \ref{hw:ultrarelativistic} on p.~\pageref{hw:ultrarelativistic} but not needed here. Let the momentum of
a material object be $mvx$, where our goal is to prove $x=\mygamma$. In this frame of reference, $v=0$, and
conservation of momentum follows by symmetry.

We now change to a new frame of reference, moving at some speed $v$ along the line of emission of the two light rays.
In this frame, conservation of momentum requires $2Evx=yE/D-yED$. We therefore have $vx/y=(1/D-D)/2$, which can be
shown with a little algebra to equal $v\mygamma$. Since only $x$ can depend on $v$, not $y$, and the correspondence
principle requires $x\approx 1$ for $v<<c$, we find that $x=\mygamma$, as claimed.

Problem \ref{hw:rel-momentum-cons-frame-invariant} on p.~\pageref{hw:rel-momentum-cons-frame-invariant} checks that
this result also works correctly for a system consisting of material particles.
<% end_sec('rel-momentum-proof') %>
<% end_sec('rel-p-and-e-proofs') %> % Proofs
<% begin_sec("Two tests of Newton's third law",nil,'kreuzer',{'optional'=>true}) %>
$E=mc^2$ states that a certain amount of energy $E$ is equivalent to a certain amount of mass $m$.
But mass pops up in physics in several different guises:
the mass measured
by an object's inertia, the ``active'' gravitational mass $m_a$ that determines
the gravitational forces it makes on other
objects, and the ``passive'' gravitational mass $m_p$ that measures how strongly it feels
gravity. Einstein's reason for
predicting the same behavior for $m_a$ and $m_p$ was that anything else would have violated Newton's
third law for gravitational
forces.

Suppose instead
that an object's energy content contributes only to $m_p$, not to $m_a$. 
Atomic nuclei get something like 1\% of their mass from the energy of the electric fields inside their nuclei,
but this percentage varies with the number of protons, so if we have objects $m$ and $M$ with different chemical
compositions, it follows that in this theory $m_p/m_a$ will not be the same as $M_p/M_a$, and in this non-Einsteinian version of relativity,
Newton's third law is violated.

This was tested in\index{Newton's laws of motion!third law!test of}
a Princeton PhD-thesis experiment by Kreuzer\footnote{Kreuzer, Phys. Rev. 169 (1968) 1007}
in 1966. Kreuzer carried out an experiment, figure \figref{kreuzer}, using masses made of two different substances. The first substance was teflon.
The second substance was
a mixture of the liquids trichloroethylene and dibromoethane, with the proportions chosen so as to give a passive-mass
density as close as possible to that of teflon, as determined by the neutral buoyancy of the teflon masses suspended inside the liquid.
If the active-mass densities of these substances are not strictly proportional to their passive-mass densities, then moving the chunk of
teflon back and forth in figure \subfigref{kreuzer}{2} would change the gravitational force acting on the nearby small sphere.
No such change was observed, and
the results verified $m_p/m_a=M_p/M_a$ to within one part in $10^6$, in agreement with Einstein and Newton. If electrical energy had not contributed at all to
active mass, then a violation of the third law would have been detected at the level of about one part in $10^2$.
% my x in genrel book is deviation from unit contribution by E field to m_a; x is limited to about 10^-4, but E field only makes up only about 10^-2 of rest mass

<% marg() %>
<%
  fig(
    'kreuzer',
    %q{1. A balance that measures the gravitational attraction between masses $M$ and $m$. (See section \ref{sec:weighing-the-earth} for
          a more detailed description.) When the two masses $M$ are inserted, the fiber twists.
       2. A simplified diagram of Kreuzer's modification. The moving teflon mass is submerged in a liquid with nearly the same density.
       3. Kreuzer's actual apparatus.}
  )
%>
<% end_marg %>


The Kreuzer result was improved in 1986 by Bartlett and van Buren\footnote{Phys. Rev. Lett. 57 (1986) 21}
using data gathered by bouncing laser beams off of a mirror left behind on the moon by the Apollo astronauts,
as described p.~\pageref{sec:battat}.
Since the moon
has an asymmetrical distribution of iron and aluminum, a theory with $m_p/m_a \ne M_p/M_a$ would cause it to have an anomalous acceleration along
a certain line. The lack of any such observed acceleration limits violations of
Newton's third law to about one part in $10^{10}$.
% not just 1/x=10^8, for same reasons as in the other comment above

<% end_sec('kreuzer') %> % Two tests of Newton's third law
\begin{summary}

\begin{vocab}

\vocabitem{alpha particle}{a form of radioactivity consisting of helium nuclei}

\vocabitem{beta particle}{a form of radioactivity consisting of electrons}

\vocabitem{gamma ray}{a form of radioactivity consisting of a very
high-frequency form of light}

\vocabitem{proton}{a positively charged particle, one of the types
that nuclei are made of}

\vocabitem{neutron}{an uncharged particle, the other types that nuclei are made of}

\vocabitem{isotope}{one of the possible varieties of atoms of a given
element, having a certain number of neutrons}

\vocabitem{atomic number}{the number of protons in an atom's nucleus;
determines what element it is}

\vocabitem{atomic mass}{the mass of an atom}

\vocabitem{mass number}{the number of protons plus the number of
neutrons in a nucleus; approximately proportional to its atomic mass}

\vocabitem{strong nuclear force}{the force that holds nuclei together
against electrical repulsion}

\vocabitem{weak nuclear force}{the force responsible for beta decay}

\vocabitem{beta decay}{the radioactive decay of a nucleus via the
reaction
                $\text{n} \rightarrow \text{p} + \zu{e}^- +  \bar{\nu}$
or
                $\text{p} \rightarrow \text{n} + \zu{e}^+ + \nu$; so called because an electron
or antielectron is also known as a beta particle}

\vocabitem{alpha decay}{the radioactive decay of a nucleus via emission
of an alpha particle}

\vocabitem{fission}{the radioactive decay of a nucleus by splitting into two parts}

\vocabitem{fusion}{a nuclear reaction in which two nuclei stick together
to form one bigger nucleus}

\vocabitem{$\mu$Sv}{a unit for measuring a person's exposure to radioactivity}

\end{vocab}

\begin{notation}
\notationitem{$\text{e}^-$}{an electron}
\notationitem{$\text{e}^+$}{an antielectron; just like an electron, but with positive charge}

\notationitem{n}{a neutron}

\notationitem{p}{a proton}

\notationitem{$\nu$}{a neutrino}
\notationitem{$\bar{\nu}$}{an antineutrino}
\notationitem{$\massenergy$}{mass-energy}
\end{notation}

\pagebreak

\begin{othernotation}

\notationitem{$Z$}{atomic number (number of protons in a nucleus)}

\notationitem{$N$}{number of neutrons in a nucleus}

\notationitem{$A$}{mass number $(N+Z)$}

\end{othernotation}

\begin{summarytext}


Quantization of charge: Millikan's oil drop experiment
showed that the total charge of an object could only be an
integer multiple of a basic unit of charge $(e)$. This
supported the idea the the ``flow'' of electrical charge was
the motion of tiny particles rather than the motion of some
sort of mysterious electrical fluid.

Einstein's analysis of Brownian motion was the first
definitive proof of the existence of atoms. Thomson's
experiments with vacuum tubes demonstrated the existence of
a new type of microscopic particle with a very small ratio
of mass to charge. Thomson correctly interpreted these as
building blocks of matter even smaller than atoms: the first
discovery of subatomic particles. These particles are called electrons.

The above experimental evidence led to the first useful
model of the interior structure of atoms, called the raisin
 cookie model. In the raisin cookie model, an atom
consists of a relatively large, massive, positively charged
sphere with a certain number of negatively charged
electrons embedded in it.

Rutherford and Marsden observed that some alpha particles
from a beam striking a thin gold foil came back at angles up
to 180 degrees. This could not be explained in the
then-favored raisin-cookie model of the atom, and led to
the adoption of the planetary model of the atom, in which
the electrons orbit a tiny, positively-charged nucleus.
Further experiments showed that the nucleus itself was a
cluster of positively-charged protons and uncharged neutrons.

Radioactive nuclei are those that can release energy. The
most common types of radioactivity are alpha decay (the
emission of a helium nucleus), beta decay (the transformation
of a neutron into a proton or vice-versa), and gamma decay
(the emission of a type of very-high-frequency light). Stars
are powered by nuclear fusion reactions, in which two light
nuclei collide and form a bigger nucleus, with the release of energy.

Human exposure to ionizing radiation is measured in units of
microsieverts ($\mu$Sv). The typical person is exposed to about 2000 $\mu$Sv
worth of natural background radiation per year.

\end{summarytext}

\begin{exploring}
\begin{reading}{The First Three Minutes}{Steven Weinberg}
This book describes the first three minutes of the universe's
existence.
\end{reading}
\end{exploring}

\end{summary}

<% begin_hw_sec %>

<% begin_hw('foodtoatoms') %>
Use the nutritional information on some packaged food to
make an order-of-magnitude estimate of the amount of
chemical energy stored in one atom of food, in units of
joules. Assume that a typical atom has a mass of $10^{-26}$
kg. This constitutes a rough estimate of the amounts of
energy there are on the atomic scale. [See chapter \ref{ch:scaling}
for help on how to do order-of-magnitude
estimates. Note that a nutritional ``calorie'' is really a 
kilocalorie.]\answercheck
<% end_hw() %>

<% begin_hw('beta-decay-charge-conservation') %>__incl(hw/beta-decay-charge-conservation)<% end_hw() %>

<% begin_hw('pu-decay-products') %>__incl(hw/pu-decay-products)<% end_hw() %>


<% begin_hw('uelec') %>
        (a) Recall that the gravitational energy of two gravitationally
        interacting spheres is given by $PE=-Gm_1m_2/r$, where $r$ is the
        center-to-center distance. What would be the analogous
        equation for two electrically interacting spheres? Justify
        your choice of a plus or minus sign on physical grounds,
        considering attraction and repulsion.\\
        (b) Use this
        expression to estimate the energy required to pull apart a
        raisin-cookie atom of the one-electron type, assuming a
        radius of $10^{-10}$ m.\answercheck\\
        (c) Compare this with the result of
         problem \ref{hw:foodtoatoms}.
<% end_hw() %>

\enlargethispage{\baselineskip}

<% begin_hw('neonaccel') %>
        A neon light consists of a long glass tube full of neon,
        with metal caps on the ends.  Positive charge is placed on
        one end of the tube, negative on the other.  The
        electric forces generated can be strong enough to strip
        electrons off of a certain number of neon atoms.  Assume for
        simplicity that only one electron is ever stripped off of
        any neon atom.  When an electron is stripped off of an atom,
        both the electron and the neon atom (now an ion) have
        electric charge, and they are accelerated by the forces
        exerted by the charged ends of the tube.  (They do not feel
        any significant forces from the other ions and electrons
        within the tube, because only a tiny minority of neon atoms
        ever gets ionized.)  Light is finally produced when ions are
        reunited with electrons.  Give a numerical comparison of the magnitudes and
        directions of the accelerations of the electrons and ions.
        [You may need some data from page \pageref{datatable}.]
        \answercheck
<% end_hw() %>

<% begin_hw('hatomscancellation') %>__incl(hw/hatomscancellation)<% end_hw() %>

<% marg(80) %>
<%
  fig(
    'hw-cathode-ray-gun',
    %q{Problem \ref{hw:crt-speed}.}
  )
%>
<% end_marg %>

<% begin_hw('crt-speed') %>__incl(../../share/em-dc/hw/crt-speed)<% end_hw() %>

<% marg(-100) %>
<%
  fig(
    'hw-tandem-accel',
    %q{Problem \ref{hw:tandem}.}
  )
%>
<% end_marg %>
<% begin_hw('tandem') %>
The figure shows a simplified diagram of a device called
a tandem accelerator, used for accelerating beams of ions up
to speeds on the order of 1\% of the speed of light. The
nuclei of these ions collide with the nuclei of atoms in a
target, producing nuclear reactions for experiments studying
the structure of nuclei. The outer shell of the accelerator
is a conductor at zero voltage (i.e., the same voltage as the
Earth). The electrode at the center, known as the ``terminal,''
is at a high positive voltage, perhaps millions of volts.
Negative ions with a charge of $-1$ unit (i.e., atoms with one
extra electron) are produced offstage on the right,
typically by chemical reactions with cesium, which is a
chemical element that has a strong tendency to give away
electrons. Relatively weak electric and magnetic forces are
used to transport these $-1$ ions into the accelerator, where
they are attracted to the terminal. Although the center of
the terminal has a hole in it to let the ions pass through,
there is a very thin carbon foil there that they must
physically penetrate. Passing through the foil strips off
some number of electrons, changing the atom into a positive
ion, with a charge of $+n$ times the fundamental charge. Now
that the atom is positive, it is repelled by the terminal,
and accelerates some more on its way out of the accelerator.
(a) Find the velocity, $v$, of the emerging beam of positive
ions, in terms of $n$, their mass $m$, the terminal voltage
$V$, and fundamental constants. Neglect the small change in
mass caused by the loss of electrons in the 
stripper foil.\answercheck\hwendpart
(b) To fuse protons with protons, a minimum beam velocity of
about 11\% of the speed of light is required. What terminal
voltage would be needed in this case?\answercheck\hwendpart
<% end_hw() %>

<% begin_hw('enterprise-ke') %>__incl(../../share/relativity/hw/enterprise-ke)<% end_hw() %>

<% begin_hw('freeneutron') %>__incl(../../share/relativity/hw/freeneutron)<% end_hw() %>

<% begin_hw('vintermsofp',2) %>__incl(../../share/relativity/hw/vintermsofp)<% end_hw() %>

<% begin_hw('ultrarelativistic') %>__incl(../../share/relativity/hw/ultrarelativistic)<% end_hw() %>

\enlargethispage{\baselineskip}

<% begin_hw('relativity-max-tension') %>__incl(../../share/relativity/hw/relativity-max-tension)<% end_hw() %>

<% begin_hw('double-voltage') %>__incl(../../share/relativity/hw/double-voltage)<% end_hw() %>

<% begin_hw('rel-momentum-cons-frame-invariant',2) %>
Problem \ref{hw:energy-frames} on p.~\pageref{hw:energy-frames} (with the solution
given in the back of the book) demonstrates that in Newtonian mechanics, conservation of momentum is the
necessary and sufficient condition if conservation of energy is to hold in all frames of reference. The purpose
of this problem is to generalize that idea to relativity (in one dimension).

Let a system contain two interacting particles, each with unit mass. Then if energy is conserved in a particular
frame, we must have $\mygamma_1+\mygamma_2=\mygamma'_1+\mygamma'_2$, where the primes indicate the quantities after interaction.
Suppose that we now change to a new frame, in motion relative to the first one at a velocity $\epsilon$ that is
much less than 1 (in units where $c=1$).
The velocities all change according to the result of
problem \ref{hw:rel-vel-addition} on p.~\pageref{hw:rel-vel-addition}.
Show that energy is conserved in the new frame if and only if momentum is conserved.

Hints: (1) Since $\epsilon$ is small, you can take $1/(1+\epsilon)\approx 1-\epsilon$.
(2) Rather than directly using the result of problem \ref{hw:rel-vel-addition}, it is easier to
eliminate the velocities in favor of the corresponding Doppler-shift factors $D$, which simply
multiply when the velocities are combined. (3) The identity $v\mygamma=(1/D-D)/2$ is handy here.
<% end_hw() %>

<% end_hw_sec %>

<% begin_ex("Sports in Slowlightland","A") %>
\noindent In Slowlightland, the speed of light is 20 mi/hr = 32 km/hr = 9 m/s. Think of an example of how
 relativistic effects would work in sports. Things can get very complex very quickly,
 so try to think of a simple example that focuses on just one of the following effects:
\begin{itemize}
\item relativistic momentum
\item relativistic kinetic energy
\item relativistic addition of velocities (See problem \ref{hw:rel-vel-addition}, with the answer given on p.~\pageref{soln:rel-vel-addition}.)
\item time dilation and length contraction
\item Doppler shifts of light (See section \ref{sec:doppler-and-clock}.)
\item equivalence of mass and energy
\item time it takes for light to get to an athlete's eye
\item deflection of light rays by gravity
\end{itemize}
<% end_ex %>

<% begin_ex("Nuclear Decay","B") %>
__incl(text/ex_chart_of_nuclei)
<% end_ex %>

<% begin_ex("Misconceptions about Relativity","C") %>
__incl(../../share/relativity/text/relativity_misconceptions)
<% end_ex %>

<% end_chapter() %>