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Science Forum Index » Physics - Electromagnetic Forum » E of a moving charge
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| blackhead |
 Posted: Fri Aug 31, 2007 11:25 am |
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Guest
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Here's my derivation of the retarded Electric field of a charge moving
in one dimension with constant velocity:
Let q be a charge moving along the positive x-axis from the origin,
O, at t = 0 and E = k/x^2 directed towards O propagate at velocity c,
where k = q/ (4 pi epsilon0) and ' represent a quantity evaluated at
the retarded time t' = t - x'/c.
If q moves dx' in a time dt', then dE = -2k dx'/x'^3 at t = x'/c + t'
during dt = dt' + dx'/c.
dE/dt = -2k dx' / ( x'^3 ( dt'+ dx'/c ))
= -2k /( x'^3(1/u' + 1/c )) (1) where u' = dx'/dt'
Note that x' is still dependent upon t.
If q moves with u' = u1 = constant, then x' = u1 t', = u1 (t - x'/c)
So x' = t/ (1/u1 + 1/c) and substituting into (1) gives:
dE/dt = -2k ( 1/u1 + 1/c)^2 / t^3
At t = infinity, x = infinity, E = 0 so integrating:
Integrate( E=E, E=0, 1, dE) = integrate( t = t, t=infinity, -2k ( 1/u1
+ 1/c) / t^3 , dt)
E(t) = k (1/u1 + 1/c)^2/ t^2, = k (u1 + c)/ ( c^2 ( u1 t)^2)
= k (1 + B)^2/ x^2 where B = u1/c
Which is different from what is given in most textbooks:
E(t) = k (1 - B^2)/ x^2
Where have I gone wrong in my derivation? |
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