Field of a Moving Charge

A charged particle with charge q₁ moves with constant velocity v along the z-axis (see Figure 1.1). Its potentials are

Where s = [(1-β²) (x²-y²) + (x-vt)²]^1/2 and β = v/c

Figure 1.1

a) Show that ϕ and A satisfy the Lorentz condition

b) Calculate the fields E and B at point P (x, y, z) and time t. Show first that B = (v/c) × E, and calculate E explicitly. Show that E is parallel to n.

c) Assume at P a second particle with charge q₂ moving with the same velocity v as the first. Calculate the force on q₂.

SOLUTION

a) Differentiating, we obtain

(1)

and, since v lies only in the z direction (see Figure 1.2)

(2)

b) To calculate B, we recall from the definition of A that

(3)

where we have used the fact that

Now,

(4)

Figure 1.2

where  is the vector from the charge to point P and therefore is parallel to n. The same results can be obtained by calculating E in the moving frame (with the charge q₁) and then taking the Lorentz transformation.

c) The force acting on charge q₂ can be calculated as the force acting on q₂ in the field of q₁ from (4):

(5)

where we used (3). Substituting E from (4), we find

We can express through the angle θ between R and ẑ (see Figure 1.2):

where we used x² + y² = R² sin θ, and so s = |R| (1- β² sin² θ)^1/2. Now, F may be written as a sum of the projections perpendicular and parallel to the z direction: