Conducting Sphere in Constant Magnetic Field

A perfectly conducting sphere of radius R moves with constant velocity  through a uniform magnetic field B = B₀ŷ (see Figure 1.1). Find the surface charge density induced on the sphere to lowest order in v/c.

Figure 1.1

SOLUTION

In the frame K' moving with velocity v, we have, to lowest order in v/c

(1)

In the lab frame K, we have E = 0 and B = B₀ŷ. Using (1) for the frame K', we have an electric field in this frame

(2)

where E'₀ = vB₀/c. Now, we have a perfectly conducting sphere in a constant electric field, so we may write the potential outside the sphere in the form

(3)

where θ is the angle between r and E', or, in this case, between r and ẑ (the origin of the spherical coordinates is at the center of the sphere). The surface charge density a is given by

Finally, substituting E'₀ from (2), we have

(4)