Question

A series resonant ac circuit contains a capacitance \(10^4\) F and an inductor of \(10^-4\) H. The frequency of electrical oscillations will be

• A. \(\frac{{10^5}}{{2\pi}}\)Hz
• B. \(10^3\) Hz
• C. \(\frac{{10}}{{2\pi}}\) Hz
• D. 10 Hz

Answer 1

The Correct Option is A

The resonance frequency (\( v \)) of an LC circuit is given by the formula:

\( v = \frac{1}{2 \pi \sqrt{L C}}\)

Substituting the values for the capacitance (\( C = 10^{-4} \)) and inductance (\( L = 10^{-6} \)):

\( v = \frac{1}{2 \pi \sqrt{10^{-6} \times 10^{-4}}} = \frac{1}{2 \pi \sqrt{10^{-10}}}\)

\(v = \frac{10^5}{2 \pi} \, \text{Hz}\)

Therefore, the resonance frequency of the circuit is: \( \frac{10^5}{2 \pi} \, \text{Hz} \).

Answer 2

The resonant frequency \(f\) of a series resonant AC circuit containing an inductor \(L\) and a capacitor \(C\) is given by the formula:

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

Where:

• \(f\) is the resonant frequency in Hertz (Hz)
• \(L\) is the inductance in Henrys (H)
• \(C\) is the capacitance in Farads (F)

Given:

• \(C = 10^{-4} \, \text{F}\)
• \(L = 10^{-4} \, \text{H}\)

Substitute the given values into the formula:

\(f = \frac{1}{2\pi\sqrt{(10^{-4})(10^{-4})}}\)

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

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

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

\(f = \frac{10000}{2\pi} \, \text{Hz}\)

It seems there might be a typo in the capacitance value given in the question. If the capacitance was \(10^{-4} F\) instead of \(10^4 F\), the correct resonant frequency would be \(\frac{10000}{2\pi}Hz\), but that is not among the options. Therefore, let's proceed with the assumption that the capacitance given is incorrect and intended to be \(10^{-4} F\).

If the capacitance is \(10^{-6} F\),

\(f = \frac{1}{2\pi\sqrt{(10^{-4})(10^{-6})}}\)

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

\(f = \frac{1}{2\pi \times 10^{-5}}\)

\(f = \frac{10^5}{2\pi} \, \text{Hz}\)

Thus, assuming a typo in the question, the closest answer would be: \(\frac{10^5}{2\pi} Hz\) is more correct if the capacitor is \(10^{-6}\)