Answer to Question #192638 in Electricity and Magnetism for yosef

Question

In your electricity and magnetism lab, suppose you were enthusiastic to apply what you learned about Faraday’s work and electromagnetic Induction. You decided to uniformly charge the surface of a sphere of radius R with chare Q. Then you let the sphere to revolve around the vertical axis with rotation velocity ω. With some calculations you figured out that the magnetic field of such system should be as follows: Inside the sphere, 𝑩⃗⃗ 𝒊𝒏 = 𝟐 𝝁𝟎 𝝎 𝑸 𝟏𝟒 𝝅 𝑹 𝒆⃗ 𝒛 , and Outside the sphere, 𝑩⃗⃗ 𝒐𝒖𝒕 = 𝟐 𝝁𝟎 𝝎 𝑸 𝑅 2 𝟏𝟒 𝝅 [𝟐 𝒄𝒐𝒔(𝜽)𝒆⃗ 𝒓 + 𝒔𝒊𝒏(𝜽)𝒆⃗ 𝜽] 1 If the angular velocity is constant of time, figure out a formula for Faraday induced E field at the surface of the sphere. 2 Find out the torque of the charged sphere. 3 Find out the energy stored inside the magnetic field of the system. 4 Prove that 2/3 of the stored energy lies within the sphere 5 Express the total magnetic field as 0.5 Imag ω2. 6 Find out the angular momentum in the fields. 

Answer 1

Yes, the current will flow along the 'equator' of the sphere. and the magnetic field will induce inside the sphere.

The torque is calculated by the volume integral $τ=∫r×F.$ The torque is calculated by the volume integral τ=∫r×F. Thus, the torque on the object is (very) approximately

$τ≈ωR^2B^2V/ρ$

$E = \frac{1}{2}1LI^2$