fix problems with html output

[light-and-matter.git] / me / cl / b.rbtex

<%
  require "../eruby_util.rb"
%>
<%
  chapter(
    '12',
    %q{Simplifying the Energy Zoo},
    'ch:energy-zoo',
    %q{Do these forms of energy have anything in common?},
    {'opener'=>'energy-collage','width'=>'fullpage','anonymous'=>true}
  )
%>


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__incl(text/clb)
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<% begin_sec("E=mc$^2$",4,'mass-energy-equivalence',{'optional'=>true}) %>\index{mass!equivalent to energy}\index{energy!equivalent to mass}

In section \ref{sec:ultrarelativistic} we found the relativistic expression for kinetic energy
in the limiting case of an ultrarelativistic particle, i.e., one with a speed very
close to $c$: its energy is proportional
to the ``stretch factor'' of the Lorentz transformation, $s=\sqrt{(1+v)/(1-v)}$ (in units with $c=1$),
for $v\rightarrow+c$ and $1/s$ for $v\rightarrow-c$.
What about intermediate cases, like $v=c/2$?

<%
  fig(
    'bell-jar',
    %q{%
      The match is lit inside the bell jar. It burns, and energy escapes from the jar
      in the form of light. After it stops burning, all the same atoms are still in the
      jar: none have entered or escaped. The figure shows the outcome expected before
      relativity, which was that the mass measured on the balance would remain exactly the same.
      This is not what happens in reality.
    },
    {
      'width'=>'wide',   
      'sidecaption'=>true
    }
  )
%>

When we are forced to tinker with a time-honored theory, our first instinct should
always be to tinker as conservatively as possible. Although we've been forced to
admit that kinetic energy doesn't vary as $v^2/2$ at relativistic speeds,
the next most conservative thing we could do would be to assume that the \emph{only} change
necessary is to replace the factor of $v^2/2$ in the nonrelativistic expression for
kinetic energy with some other function, which would have to act like $s$ or $1/s$ for $v\rightarrow\pm c$. I suspect that
this is what Einstein thought when he completed his original paper on relativity in 1905,
because it wasn't until later that year that he published a second paper showing that this
still wasn't enough of a change to produce a working theory. We now know that there is
something more that needs to be changed about prerelativistic physics, and this is the
assumption that mass is only a property of material particles such as atoms (figure \figref{bell-jar}).
Call this the ``atoms-only hypothesis.''

Now that we know the correct relativistic way of finding the energy of a ray of
light, it turns out that we can use that to find what we were originally seeking,
which was the energy of a material object. The following discussion closely
follows Einstein's. 

Suppose that a material object O of mass $m_\zu{o}$,
initially at rest in a certain frame A, emits two rays of light, each with energy $E/2$. By
conservation of energy, the object must have lost an amount of energy equal to $E$.
By symmetry, O remains at rest.

We now switch to a different frame of reference B moving at some arbitrary speed
corresponding to a stretch factor $S$. The change of frames means that we're
chasing one ray, so that its energy is scaled down to $(E/2)S^{-1}$, while running
away from the other, whose energy gets boosted to $(E/2)S$. In frame B, as in A,
O retains the same speed after emission of the light. But observers in frames
A and B disagree on how much energy O has lost, the discrepancy being
\begin{equation*}
  E\left[\frac{1}{2}(S+S^{-1})-1\right] \qquad .
\end{equation*}
Let's consider the case where B's velocity relative to A is small. Expanding the above 
expression in a Taylor series in $v$, the discrepancy in
O's energy loss is approximately
\begin{equation*}
  \frac{1}{2}Ev^2/c^2 \qquad .
\end{equation*}
The interpretation is that when O reduced its energy by $E$ in order to make
the light rays, it reduced its \emph{mass} from $m_\zu{o}$ to $m_\zu{o}-m$,
where $m=E/c^2$. Rearranging factors, we have Einstein's famous
\begin{equation*}
  E=mc^2 \qquad .
\end{equation*}
This derivation entailed an approximation, and redoing it without the approximation
entails some complexity.\footnote{See Ohanian, ``Einstein's $E=mc^2$ mistakes,'' \url{arxiv.org/abs/0805.1400}, 
and Jammer, Concepts of Mass in Contemporary Physics and Philosophy.}                              
It turns out, however, to be valid in general.

We find that mass is not simply a built-in property of the particles that
make up an object, with the object's mass being the sum of the masses of its particles.
Rather, mass and energy are equivalent, so that if the experiment of figure \figref{bell-jar} is
carried out with a sufficiently precise balance, the reading will drop because of
the mass equivalent of the energy emitted as light.

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. Conversely, an ordinary amount of energy
corresponds to an extremely small mass, and this is why nobody detected the non-null
result of experiments like the one in figure \figref{bell-jar} hundreds of years ago.

The big event here is mass-energy equivalence, but we can also
harvest a result for the energy of a material particle moving at a certain speed. 
Plugging in $S=\sqrt{(1+v)/(1-v)}$ to the equation above
for the energy discrepancy of object O between frames A and B, we find $m(\gamma-1)c^2$. This is the
difference between O's energy in frame B and its energy when it is at rest, but since mass
and energy are equivalent, we assign it energy $mc^2$ when it is at rest. The result is that
the energy is
\begin{equation*}
  E = m\gamma c^2 \qquad .
\end{equation*}

\pagebreak

\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 %>

\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}{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}
<% 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}{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}


<% end_sec() %> % E=mc$^2$

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__incl(text/energy_zoo_summary)
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<% begin_hw_sec %>

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

<% begin_hw('throw-down-and-up') %>__incl(hw/throw-down-and-up)<% end_hw() %> 

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

<% begin_hw('slidingmagnets') %>
(a) You release a magnet on a tabletop near a big piece
of iron, and the magnet slides across the table to the iron.
Does the magnetic potential energy increase, or decrease?
Explain. \hwendpart
(b) Suppose instead that you have two repelling
magnets. You give them an initial push towards each other,
so they decelerate while approaching each other. Does the
magnetic potential energy increase or decrease? Explain.

<% end_hw() %>

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

<% begin_hw('grasshopper') %>
<% hw_solution %> A grasshopper with a mass of 110 mg falls from rest
from a height of 310 cm. On the way down, it dissipates 1.1
mJ of heat due to air resistance. At what speed, in m/s,
does it hit the ground?
<% end_hw() %>

\pagebreak

<% begin_hw('two-gases-leaking') %>__incl(hw/two-gases-leaking)<% end_hw() %>

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

<% begin_hw('rolling-hoop',2) %>Problem \ref{hw:rolling-hoop} has been deleted.<% end_hw() %>

<% begin_hw('pe-ke-yin-yang') %>__incl(hw/pe-ke-yin-yang)<% end_hw() %>

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

<% begin_hw('niagara-hydroelectric') %>__incl(hw/niagara-hydroelectric)<% end_hw() %>

\pagebreak

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

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

<% begin_hw('hoop',2) %>__incl(hw/hoop)<% end_hw() %> 

<% marg(50) %>
<% fig('skater-rolls-off-pipe','Problem \ref{hw:skater-rolls-off-pipe}.') %>
<% end_marg %>
<% begin_hw('skater-rolls-off-pipe',2) %>__incl(hw/skater-rolls-off-pipe)<% end_hw() %>

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

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

\vfill

<% begin_hw('jack-on-hill',2) %>__incl(hw/jack-on-hill)<% end_hw() %>

\pagebreak

<% marg(0) %>
<%
  fig(
    'hw-atwood',
    %q{Problem \ref{hw:atwood-energy}.},
    {'suffix'=>'2'}
  )
%>
<% end_marg %>

<% begin_hw('atwood-energy') %>__incl(hw/atwood-energy)<% end_hw() %>

<% end_hw_sec %>

 %%========================================== toc decoration ===============================================

<% if false then figure_in_toc("baseball-pitch") end %>
<% end_chapter() %>