Question

In a series LCR circuit, the inductance L is 10 mH, Capacitance C is 1μF and resistance R is 100Ω. The frequency at which resonance occurs is

• A. 1.59 kHz
• B. 15.9 rad/s
• C. 15.9 kHz
• D. 1.59 rad/s

Answer 1

The Correct Option is A

To determine the frequency at which resonance occurs in a series LCR circuit, we use the formula for the resonant frequency \( f_0 \), given by:

\( f_0 = \frac{1}{2\pi\sqrt{LC}} \)

where \( L \) is the inductance, \( C \) is the capacitance, and \( \pi \) is a constant approximately equal to 3.14159.

Given:

• Inductance \( L = 10 \text{ mH} = 10 \times 10^{-3} \text{ H} \)
• Capacitance \( C = 1 \mu\text{F} = 1 \times 10^{-6} \text{ F} \)

Substituting the values into the formula:

\( f_0 = \frac{1}{2\pi\sqrt{10 \times 10^{-3} \times 1 \times 10^{-6}}} \)

\( f_0 = \frac{1}{2\pi\sqrt{10 \times 10^{-9}}} \)

\( f_0 = \frac{1}{2\pi\sqrt{10^{-8}}} \)

\( f_0 = \frac{1}{2\pi \times 10^{-4}} \)

\( f_0 = \frac{10^{4}}{2\pi} \)

\( f_0 \approx \frac{10000}{6.2832} \)

\( f_0 \approx 1591.55 \text{ Hz} \)

Thus, the resonant frequency is approximately 1.59 kHz. The correct answer is 1.59 kHz.

Answer 2

The correct option is (A): 1.59 kHz
The resonant frequency (fr) of a series LCR circuit is given by the formula:
fr = \(\frac{1}{2\pi(LC)}\)
where L is the inductance in Henries, C is the capacitance in farads, and π is the mathematical constant pi (approximately equal to 3.14).
Substituting the given values, we get:
\(f_r=\frac{1}{2\pi\sqrt{10\,mH\times1\mu F}}\)
\(= \frac{1}{(2π\sqrt{(10 x 10-3 H x 10-6 F)) }}\)
\(= \frac{1}{(2π\sqrt{(10-8))}}\) 
\(=\frac{1}{(2π \times 10^{-4})}\) 
\(= \frac{1}{(6.28 \times 10^{-5})} \)
= 1591.5 Hz (approx)
Therefore, the resonant frequency of the series LCR circuit is approximately 1591.5 Hz, = 1.59K Hz