Question

An ideal inductor, a resistor of resistance \(R\) Ohms and a capacitor with adjustable capacitance are connected in series to an alternating voltage with an effective value of \(V\) Volts and frequency of \(f\) Hz. The current flowing through the circuit when the capacitance of the capacitor is set to \(C_1\) is the same as when the capacitance of the capacitor is set to \(C_2\), \(C_2 > C_1\). The inductance of the inductor \(L\) is given by:

• A. \(\dfrac{1}{8\pi^2 f^2}\dfrac{C_1 + C_2}{C_1 C_2}\)
• B. \(\dfrac{1}{8\pi^2 f^2}\dfrac{C_1 C_2}{C_1 + C_2}\)
• C. \(\dfrac{1}{8\pi^2 f^2}\dfrac{C_1 - C_2}{C_1 C_2}\)
• D. \(\dfrac{1}{2\pi^2 f^2}\dfrac{1}{R(C_1 - C_2)}\dfrac{C_1 + C_2}{C_1 C_2}\)

Hint:

Equal current means the two settings are symmetric about resonance: \(X_L - X_{C_1} = -(X_L - X_{C_2})\), so \(2\cdot 2\pi f L = \tfrac{1}{2\pi f C_1} + \tfrac{1}{2\pi f C_2}\).

Answer 1

The Correct Option is A

Reasoning about equal current on both sides of resonance.

The current in a series RLC circuit depends on capacitance only through the net reactance $X = X_L - X_C$. Current is maximum at resonance, where $X = 0$, and falls off symmetrically as $X$ moves away from zero in either direction. So two different capacitor settings give the same current only if they sit symmetrically about resonance: one with a net inductive reactance $+x$ and the other with net capacitive reactance $-x$.

That symmetry condition is \[X_L - \frac{1}{2\pi f C_1} = -\left(X_L - \frac{1}{2\pi f C_2}\right).\]
Adding $X_L$ terms together: \[2X_L = \frac{1}{2\pi f C_1} + \frac{1}{2\pi f C_2}.\]
With $X_L = 2\pi f L$: \[2\cdot 2\pi f L = \frac{1}{2\pi f}\left(\frac{1}{C_1} + \frac{1}{C_2}\right).\]
Combine the fractions using $\dfrac{1}{C_1} + \dfrac{1}{C_2} = \dfrac{C_1 + C_2}{C_1 C_2}$: \[4\pi f L = \frac{1}{2\pi f}\cdot\frac{C_1 + C_2}{C_1 C_2}.\]
Dividing both sides by $4\pi f$ gives \[L = \frac{1}{8\pi^2 f^2}\cdot\frac{C_1 + C_2}{C_1 C_2}.\]
Note that $R$ drops out completely, which is why the option containing $R$ cannot be correct. \[\boxed{L = \frac{1}{8\pi^2 f^2}\,\frac{C_1 + C_2}{C_1 C_2}}\]