Question

A sphere made of a linear magnetic material with X_m is placed in a uniform magnetic field B₀. Determine the field inside the sphere. 

Hint: The external field will magnetize the sphere. This magnetization will create another uniform magnetic field inside the sphere which will cause an additional magnetization. Thus, you need to find a series expression.

Answer 1

The mobile charges do tend to line up near the axis, but the resulting concentration of (negative) charges sets up an electric field that repels away further accumulation. Equilibrium is reached when the electric repulsion on a mobile charge q balances the magnetic attraction:

F = q[E + (v × B)] = 0

⇒ E = −(v × B)

Let us assume the current to be in the z direction: J = ρ−vzˆ (where ρ− and v are both negative). The magnetic field is then given by I B · dl = µ0Ienc ⇒ B2πs = µ0Jπs2 ⇒ B = µ0ρ−vs 2 φˆ

The electric field is given by

in equilibrium....

1/ 2²0 (ρ+ + ρ−)sˆs = − h (vzˆ) × ³µ0ρ−vs 2 φˆ ´i = µ0 /2 ρ−v 2 sˆs

⇒ ρ+ + ρ− = ρ−(²0µ0v 2 ) = ρ− µ( v 2 /c 2 )

⇒ ρ+ = −ρ− µ (1 − v 2 /c 2 )= −ρ−/γ2 , or ρ− = −γ 2 ρ+

In this naive model, the mobile negative charges fill a smaller inner cylinder, leaving a shell of positive (stationary) charge at the outside

. B0 magnetizes the sphere: M0 = χmH0 = χm /(µ0(1 + χm) )B0.

This magnetization sets up a field wihin the sphere given by

B1 = 2/3 µ0M0 = 2 /3 κB0 (where κ ≡ χm /(1 + χm) ).

Now B1 magnetizes the sphere an additional amount

This sets up an additional field in the sphere:

and so on. The total field is the sum of all these fields: