Question

The resonant frequency of a series LCR circuit is $f$. The circuit is now connected to the sinusoidally alternating e.m.f. of frequency $2f$. The new reactances $X'_L$ and $X'_C$ are related as

• A. $X'_C = \dfrac{1}{4} X'_L$
• B. $X'_C = 2X'_L$
• C. $X'_C = X'_L$
• D. $X'_C = \dfrac{1}{2} X'_L$

Hint:

Doubling frequency doubles inductive reactance but halves capacitive reactance.

Answer 1

The Correct Option is A

Step 1: Write expressions for reactances.
Inductive reactance:
\[ X_L = 2\pi f L \] Capacitive reactance:
\[ X_C = \frac{1}{2\pi f C} \]

Step 2: At resonance.
At frequency $f$:
\[ X_L = X_C \]

Step 3: Change frequency to $2f$.
\[ X'_L = 2\pi (2f)L = 2X_L \] \[ X'_C = \frac{1}{2\pi (2f)C} = \frac{1}{2}X_C \]

Step 4: Find relation.
Since $X_L = X_C$:
\[ X'_C = \frac{1}{2}X_L = \frac{1}{4}X'_L \]

Step 5: Conclusion.
The correct relation is $X'_C = \dfrac{1}{4} X'_L$.