1. Circuit Structure

The circuit consists of a lamp (resistance) in series with a reactive network. The network has an inductor $L$ in series with a parallel combination of another inductor $L$ and a capacitor $C$.

Conditions given:
1. Minimum brightness (Minimum current) at angular frequency $\omega$.
2. We need to find the frequency for Maximum brightness (Maximum current).

2. Minimum Brightness (Parallel Resonance)

Minimum brightness occurs when the impedance of the circuit is maximum. This happens when the parallel $L-C$ branch resonates (Parallel Resonance), creating an open circuit (infinite impedance).

$$ X_L = X_C \implies \omega L = \frac{1}{\omega C} $$ $$ \omega^2 = \frac{1}{LC} \implies \omega = \frac{1}{\sqrt{LC}} $$

3. Maximum Brightness (Series Resonance)

Maximum brightness occurs when the total impedance is minimum. This happens when the net reactance of the series inductor cancels the reactance of the parallel combination (Series Resonance).

Impedance of parallel part ($Z_p$):

$$ Z_p = \frac{(j\omega’ L)(-j/\omega’ C)}{j\omega’ L – j/\omega’ C} = \frac{L/C}{j(\omega’ L – 1/\omega’ C)} $$

For series resonance, $j\omega’ L + Z_p = 0$:

$$ j\omega’ L + \frac{L/C}{j(\omega’ L – 1/\omega’ C)} = 0 $$ $$ \omega’ L = \frac{L/C}{\omega’ L – 1/\omega’ C} $$ $$ \omega’ L (\omega’ L – \frac{1}{\omega’ C}) = \frac{L}{C} $$ $$ (\omega’)^2 LC – 1 = 1 \implies (\omega’)^2 LC = 2 $$ $$ \omega’ = \sqrt{\frac{2}{LC}} = \sqrt{2} \left( \frac{1}{\sqrt{LC}} \right) $$

4. Comparing Frequencies

Since the minimum brightness frequency is $\omega = \frac{1}{\sqrt{LC}}$, the maximum brightness frequency $\omega’$ is:

$$ \omega’ = \omega \sqrt{2} $$