# 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.

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

$t_r=t’-x_0/c$

The z coordinate at $t’$ is

$z=v*t_r=v(t’-x_0/c)$

a)

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)
b,c)

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

d)

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