25. Decay of Magnetic Field in a Cylinder
Step 1: Resistance and Inductance
The cylinder acts as a single-turn solenoid. Current flows circumferentially.
- Path Length: $2\pi r$
- Cross-section Area: $l \times d$
$$ R = \rho \frac{\text{Length}}{\text{Area}} = \rho \frac{2\pi r}{ld} $$
$$ L = \frac{\mu_0 N^2 A_{\text{loop}}}{l} = \frac{\mu_0 (1)^2 (\pi r^2)}{l} = \frac{\mu_0 \pi r^2}{l} $$
Step 2: Time Constant Calculation
The decay of the B-field follows $B = B_0 e^{-t/\tau}$, where $\tau = L/R$.
$$ \tau = \frac{L}{R} = \frac{\frac{\mu_0 \pi r^2}{l}}{\frac{2\pi r \rho}{ld}} $$
$$ \tau = \frac{\mu_0 \pi r^2}{l} \times \frac{ld}{2\pi r \rho} = \frac{\mu_0 r d}{2\rho} $$
$$ B(t) = B_0 \exp\left({-\frac{2\rho t}{\mu_0 r d}}\right) $$