8. Current in Self-Exciting Dynamo
Step 1: Induced EMF
The rotating disc generates a motional EMF between the center and the rim. This EMF drives current through the solenoid. The solenoid, in turn, modifies the magnetic field.
Let $I$ be the current. The solenoid produces field $B_{sol} = \mu_0 n I$. The external field is $B$.
Total Field: $B_{total} = B \pm \mu_0 n I$ (depending on direction).
Induced EMF in disc: $\mathcal{E} = \frac{1}{2} B_{total} \omega r^2$.
Step 2: Circuit Equation
The EMF drives current $I$ through the total resistance $R$.
$$ I = \frac{\mathcal{E}}{R} = \frac{(B \pm \mu_0 n I) \omega r^2}{2R} $$ $$ 2IR = B \omega r^2 \pm \mu_0 n \omega r^2 I $$ $$ I (2R \mp \mu_0 n \omega r^2) = B \omega r^2 $$Step 3: Solving for Current
For the two cases (aiding or opposing fields):
$$ I = \frac{\omega r^2 B}{2R \mp \omega \mu_0 r^2 n} $$
$$ I = \frac{\omega r^2 B}{2R \pm \omega \mu_0 r^2 n} $$