Box and Impulsive Magnetic Field

Two opposite walls of a rigid box are uniformly charged with surface charge densities σ and -σ respectively. The positively charged wall occupies the region 0 ≤ x ≤ a, 0 ≤ y ≤ b of the plane z = h, while the negatively charged wall occupies the region 0 ≤ x ≤ a, 0 ≤ y ≤ b of the x-y plane. Inside the box, there is a uniform magnetic field B = (0, B₀, 0). Assume that is much smaller than both a and b and that the charged walls are non-conducting (see Figure 1.1).

 Figure 1.1

a) Estimate the impulse experienced by the box when the magnetic field is switched off.

b) Show that it is equal to the initial momentum of the electromagnetic field.

SOLUTION

a) From the Maxwell equation

(1)

we can find the electric field induced in the box. We have for the and ẑ components

(2)

From these equations, we obtain

where E₀ is some constant. From the equation for the force

we obtain the impulse received by the box when the magnetic field goes to zero

(3)

b) The initial momentum may be found from the Poynting vector

(4)

(5)

which is the same result as (3)