Charge in motion (Homework 3-323)

2. A point charge q moves at a constant velocity V in the z-direction. Suppose at time t=0 the charge is at the origin. At a later time t’ at a field point x=x0, y=z=0:

a. Find the scalar and vector potential.

b. What coordinate components does the electric field have?

c. What coordinate components does the magnetic field have?

d. Find the component Ex.

Chartesian system for charge

At $t=0$ the charge is at origin 0. At $t’$ the retarded time is


The z coordinate at $t’$ is



The scalar potential is

$V(x_0,t’ )=q/(4πϵ_0 )*1/sqrt{(x_0^2+(vt_r )^2)}=q/(4πϵ_0 )*1/sqrt{(x_0^2+v^2 (t’-x_0/c)^2)}$

The current density is

$J ⃗=qv*z ̂$

The (magnetic) vector potential is

$A ⃗(x_0,t’ )=μ_0/4π*qv/sqrt{(x_0^2+v^2 (t’-x_0/c)^2 )}*z ̂$    (same considerations as above for distance)

The electric field is (eq. 10.3)

$E ⃗=-∇V-(∂A ̂)/∂t=-dV/(dx_0 )*x ̂+(…)*z ̂$   so it has $x ̂$  and $z ̂$  components

The magnetic field is $(x_0→x,y_0→y,z_0→z)$

$B ⃗=∇×A ⃗|=i*(dA_z)/dy-j*(dA_z)/dx=-(dA_z)/(dx_0 )*y ̂ $ so it has only $y ̂$  component


$E_x=-dV/(dx_0 )=-1/2*q/(4πϵ_0 )*frac{(2x_0-(2v^2)/c (t’-x_0/c))}{(x_0^2+v^2 (t’-x_0/c)^2 )^{3/2}}$