Liénard-Wiechert potentials, (Homework 4-323)

2. An electron moves in uniform circular motion in the X-Y plane at angular frequency ω. The circular motion is centered on the origin. At t=0 the electron is a x=x0, y=z=0. Find the Liénard-Wiechert potentials for points on the z-axis. (This is a classic problem.)

3. Consider the small electric dipole of Griffiths 11.1.2.

a. In which direction is the electric field always zero? Which direction is the electric field maximum?

b. For the mid-plane (z=0), how are the electric field and magnetic fields polarized?

Circular Lienard

We have

$q=-e$;   $r=z$;   $x(t)=r’*cos⁡ ωt$  and $y(t)=r’*sin⁡ ωt$    with $x_0=r’$  $(at t=0)$
$v=ωr’=ωx_0$    and $r ⃗v ⃗=0$    since $r ⃗$  is perpendicular to $v ⃗$
$v ⃗(t_r )=ω ⃗  ×(r’ ) ⃗(t_r )=-ωy(t_r )*i ̂=-ω|(r’ ) ⃗ |*sin⁡[ω(t-z/c) ]*i ̂=-ωx_0  sin⁡ [ω(t-z/c) ]*i ̂$

since $t_r=t-r/c=t-z/c$

Therefore

$V(r ⃗,t)=1/(4πϵ_0 )*qc/sqrt{((c^2 t^2-r ⃗v ⃗ )-(c^2-v^2 )(c^2 t^2-r^2 ))}=$

$=1/(4πϵ_0 )*qc/sqrt{((c^2 t^2 )-(c^2-ω^2 x_0^2 )(c^2 t^2-z^2 ))}$

$A ⃗(r ⃗,t)=v ⃗/c^2$  $V(r ⃗,t)=-ωx_0  sin⁡ [ω(t-z/c) ]*i ̂ *V(r,t)$

3. Consider the small electric dipole of Griffiths 11.1.2.

a. In which direction is the electric field always zero? Which direction is the electric field maximum?

b. For the mid-plane (z=0), how are the electric field and magnetic fields polarized? 

Electric Dipole

The electric field of a radiating dipole is (equation 11.18)

$E=-(μ_0 p_0 ω^2)/4π*(sinθ/r)*cos⁡[ω(t-r/c) ] θ ̂ $

Which shows that E is maximum for $sin⁡ θ=1$  or $θ=π/2$ which means on directions that are perpendicular to the dipole (y axis). E is zero for the observer situated on the same line on which the dipole is (z axis).

For the mid plane (z=0, observer along y axis) the electric field is in the direction of θ ̂ in the plane (zy) which means E is oscillating perpendicular to y axis and B field is in the ϕ ̂ direction (eq.11.19) which means B is oscillating along x in the plane (zx).