A sphere of radius R carries a surface-charge distribution σ in the region of θ ∈ [π/2−α, π/2 +α], α is a fixed parameter. The rest of the sphere has no charge. The sphere is rotated about the z axis with constant angular velocity ω. (a) (20 pts) Find the magnetic dipole moment ~m. (b) (10 pts) Demonstrate that your solution is correct by considering two limits: α → π/2, α → 0. In the second limit assume that the total current is held fixed.

One hazard of space travel is debris left by previous missions. there are several thousand masses large enough to detect by radar orbiting the earth, but there are far greater numbers of very small masses such as flakes of paint. calculate the force exerted by a 0.110 mg chip of paint that strikes a space shuttle window at a relative speed of 5.00 ✕ 103 m/s and sticks, given the collision lasts 6.00 ✕ 10-8 s. such a collision chipped the window of the ill-fated challenger in june 1983, causing $50,000 of damage.