Solution: Coaxial Rotating Cylindrical Shells
Figure 3: Cross-section of coaxial shells. The magnetic field behaves like superposed solenoids.
Each rotating charged shell behaves like a long solenoid. The effective surface current density (current per unit length) for a shell of radius $R$, charge density $\sigma$, and angular velocity $\omega$ is:
$$ K = \sigma R \omega $$The magnetic field produced by such a shell is:
- Inside the shell: $B = \mu_0 K = \mu_0 \sigma R \omega$
- Outside the shell: $B = 0$
Using the principle of superposition:
1. Region $r < R_1$ (Inside both shells):
Both shells contribute to the magnetic field.
2. Region $R_1 < r < R_2$ (Between shells):
We are outside the inner shell (so $B_{inner} = 0$) but inside the outer shell.
3. Region $r > R_2$ (Outside both):
$$ B = 0 $$
Magnetic pressure is the force per unit area exerted on the current-carrying surface by the magnetic field. The force on a surface current $K$ is $d\vec{F} = \vec{K} \times \vec{B}_{local} dA$.
Crucially, the field $\vec{B}_{local}$ acting on the surface itself is the average of the field just inside and just outside that surface.
Pressure on Outer Shell ($p_2$):
- Field inside: $B_{in} = \mu_0 \sigma_2 \omega_2 R_2$ (contribution from shell 2 only, as shell 1 field is zero here).
- Field outside: $B_{out} = 0$.
- Average Field: $B_{avg} = \frac{1}{2} \mu_0 \sigma_2 \omega_2 R_2$.
- Current density: $K_2 = \sigma_2 R_2 \omega_2$.
Pressure on Inner Shell ($p_1$):
- Field just inside ($r < R_1$): $B_{total} = \mu_0 K_1 + \mu_0 K_2$.
- Field just outside ($r > R_1$): $B_{total} = 0 + \mu_0 K_2$.
- Note: The field due to shell 2 ($B_2 = \mu_0 K_2$) is constant across the boundary of shell 1. The field due to shell 1 jumps from $\mu_0 K_1$ to 0.
- Average Field acting on Shell 1: $B_{avg} = B_2 + \frac{B_1}{2} = \mu_0 K_2 + \frac{\mu_0 K_1}{2}$.
Substituting $K_1 = \sigma_1 \omega_1 R_1$ and $K_2 = \sigma_2 \omega_2 R_2$: