Radiation from an accelerated charge.

Version 0.2 12-3-06
To do:
(i) Sort out question of transverse wave front.
(ii) Generalize to oscillating source.
(iii) Better way to generate multiple field lines?
(iv) Version in polar coordinates

c – speed of light; m – slope of field line;

c=slider([1,10])

function(m,A)=tan(A)

function(Y,x,m,b)=m*x+b

Position of accelerated charge – b; bM is endpoint and D the time interval. Maximum charge velocity is bM/D, so c should be bigger than this.

b_M=1,D=1

function(b,s)=branch(if(0,s<-0.01),if(b_M*[1-(1/D^2*[s-D]^2)],-0.01<s<D),if(b_M,s=D),if(b_M,s>D))

vector(0,function(b,n))

vector(0,0)

Position of propagating wave fronts

function(X_O,s,A)=c*cos(A)*s

function(X_o,s,A)=-(c*cos(A)*s)

function(Y_O,s,A)=c*sin(A)*s

Field before charge begins to accelerate:

function(Y_0,x,A)=[if(function(Y,x,function(m,A),0),abs(x)>max([function(X_O,n,A),function(X_o,n,A)]))]

function(Y_0,x,T),in(T,set([-5]*pi/12,[-4]*pi/12,[-3]*pi/12,[-2]*pi/12,[-1]*pi/12,1*pi/12,2*pi/12,3*pi/12,4*pi/12,5*pi/12,-pi/120))

x=0

Field from instantaneous position of charge:

function(Y_i,x,A)=[if(function(Y,x,function(m,A),function(b,n)),min([function(X_b,n,A),function(X_B,n,A)])<x<max([function(X_b,n,A),function(X_B,n,A)]))]

function(Y_i,x,T),in(T,set([-5]*pi/12,[-4]*pi/12,[-3]*pi/12,[-2]*pi/12,[-1]*pi/12,1*pi/12,2*pi/12,3*pi/12,4*pi/12,5*pi/12,-pi/120))

What initially seems like the right thing is not transverse:

function(Y,x,function(m,pi/4),function(b,n-x/(c*cos(pi/4)))),0<x<function(X_O,n)

Transverse wavefront by hand; B is the y-intercept of and Xb/B the intersection point with the transverse wave front. (B changes sign for x><0, so Xb changes as well.) Some of the extraneous definitions (Xo,BB) are because GC chokes on putting - signs in conditionals arg lists -- bug.

function(B,s,A)=function(Y_O,s,A)+1/function(m,A)*function(X_O,s,A)

function(B_B,s,A)=-[function(Y_O,s,A)+1/function(m,A)*function(X_O,s,A)]

function(X_b,s,A)=[function(B,s,A)-function(b,s)]*sin([2*A])/2

function(X_B,s,A)=[-[function(B,s,A)+function(b,s)]]*sin([2*A])/2

function(Y_b,x,A)=[if(function(Y,x,-1/function(m,A),function(B,n,A)),min([function(X_b,n,A),function(X_O,n,A)])<x<max([function(X_b,n,A),function(X_O,n,A)]))]

function(Y_B,x,A)=[if(function(Y,x,-1/function(m,A),function(B_B,n,A)),min([function(X_B,n,A),function(X_o,n,A)])<x<max([function(X_B,n,A),function(X_o,n,A)]))]

function(Y_b,x,T),in(T,set([-5]*pi/12,[-4]*pi/12,[-3]*pi/12,[-2]*pi/12,[-1]*pi/12,1*pi/12,2*pi/12,3*pi/12,4*pi/12,5*pi/12,-pi/120))

function(Y_B,x,T),in(T,set([-5]*pi/12,[-4]*pi/12,[-3]*pi/12,[-2]*pi/12,[-1]*pi/12,1*pi/12,2*pi/12,3*pi/12,4*pi/12,5*pi/12,-pi/120))



Author: David A. Craig <http://web.lemoyne.edu/~craigda/>

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