Accelerator Movie

This notebook contains a movie of a linear accelarator as seen by an electron being accelerated by a constant E field.  The accelerator itself is a tunnel 10 units long 2 units wide and 2 units high. The cross-section of the accelerator is formed by a semicircular arch on a rectangle.  The figure below shows the accelerator tunnel as viewed by one standing at the end.

Accelerator Tunnel

The equations governing the acceleration of an electron by a constant electric field can be solved exactly.  The position x, the observer's time t, and the § value are given in terms of the electron's proper time tau:

x = c*tau0*(Cosh[tau/tau0] - 1)
t = tau0*Sinh[tau/tau0]
§ = Tanh[tau/tau0]

where tau0 = m*c/(e*E)

In our example we choose to have § = .99 at the end of the tunnel (x =10).

tauovrtau0max = ArcTanh[.99]

ctau0 = 10/(Cosh[tauovrtau0max]-1)

We bring in our Lorenz package:

<<Lorenzp.m

make our tunnel:

tunnel = Tunnel[0.,10.,-1.,1.,0.,1.,1.];

set the film size:

filmsize = {{-1.,1.},{-1.,1.}};

set the number of frames:

nframes = 21;

define velocity and position of each frame:

bb[i_Integer] := bb[i] = Tanh[tauovrtau0max*(i-1)/(nframes-1)];
xx[i_Integer] := xx[i] = ctau0*(Cosh[tauovrtau0max*(i-1)/(nframes-1)] - 1);
Table[{xx[i],bb[i]},{i,nframes}]

and generate our movie:

Do[LorenzCamera[bb[i],{xx[i],0.,1.},filmsize,tunnel],{i,nframes }]

We note that the openning at the far end of the tunnel shrinks initially as we move down the tunnel.  This effect is produced by the forced convergence of light in the direction of our motion.  Ultimately, however, our motion towards the portal forces it to open up.  

We plot below the apparent position of the apex of the tunnel portal as a function of proper time to see this effect.

bb[ii_] := Tanh[tauovrtau0max*(ii-1)/(nframes-1)];
xx[ii_] := ctau0*(Cosh[tauovrtau0max*(ii-1)/(nframes-1)] - 1);
zeta[ii_] := Sqrt[1-bb[ii]^2]/(10.-xx[ii] + bb[ii]*Sqrt[1.+(10.-xx[ii])^2]);
ParametricPlot[{xx[ii],zeta[ii]},{ii,1.,nframes},
	AxesLabel->{"Position in Tunnel","Portal Image"}]