Question

Which of the following statement(s) is/are true for a LC circuit with 𝐿=25 mH and 𝐶=4 µF?

• A. Resonance frequency is close to 503 Hz
• B. The impedance at 1 kHz is 15 Ω
• C. At a frequency of 200 Hz, the voltage lags the current in the circuit
• D. At a frequency of 700 Hz, the voltage lags the current in the circuit

Answer 1

The Correct Option is A, C

The question involves understanding the dynamics of an LC circuit, which is a fundamental topic in electromagnetism and electronics. Here, we need to verify which statements are true based on given values of inductance \(L\) and capacitance \(C\).

Given: \(L = 25 \, \text{mH} = 25 \times 10^{-3} \, \text{H}\) and \(C = 4 \, \mu\text{F} = 4 \times 10^{-6} \, \text{F}\)

Step-by-Step Solution: 

1. Calculate the Resonance Frequency:

The resonance frequency (\(f_0\)) of an LC circuit is given by the formula:

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

Substituting the given values:

\(f_0 = \frac{1}{2\pi \sqrt{(25 \times 10^{-3})(4 \times 10^{-6})}}\)

Calculating the denominator inside the square root:

\(\sqrt{(25 \times 10^{-3})(4 \times 10^{-6})} = \sqrt{1 \times 10^{-7}} = 1 \times 10^{-3.5}\)

So the resonance frequency is:

\(f_0 \approx \frac{1}{2\pi \times 10^{-3.5}} \approx 503 \, \text{Hz}\)

This confirms that the statement "Resonance frequency is close to 503 Hz" is true.

2. Calculate the Impedance at 1 kHz:

The impedance (\(Z\)) of an LC circuit at a particular frequency (\(f\)) is given by:

\(Z = \sqrt{(X_L - X_C)^2}\)

 

Where \(X_L = 2\pi fL\) and \(X_C = \frac{1}{2\pi fC}\).

For \(f = 1000 \, \text{Hz}\):

• \(X_L = 2\pi \times 1000 \times 25 \times 10^{-3} = 157 \, \Omega\)
• \(X_C = \frac{1}{2\pi \times 1000 \times 4 \times 10^{-6}} = 40 \, \Omega\)
• \(Z = \sqrt{(157 - 40)^2} = \sqrt{117^2} = 117 \, \Omega\)

The calculated impedance is 117 Ω, not 15 Ω. Therefore, the statement "The impedance at 1 kHz is 15 Ω" is false.

3. Phase Relationship at 200 Hz and 700 Hz:

To determine whether the voltage leads or lags, we examine the relative sizes of \(X_L\) and \(X_C\):

For \(f = 200 \, \text{Hz}\):

• \(X_L = 2\pi \times 200 \times 25 \times 10^{-3} = 31.4 \, \Omega\)
• \(X_C = \frac{1}{2\pi \times 200 \times 4 \times 10^{-6}} = 199 \, \Omega\)

Since \(X_L < X_C\), the voltage lags the current. Therefore, the statement "At a frequency of 200 Hz, the voltage lags the current in the circuit" is true.

For \(f = 700 \, \text{Hz}\):

• \(X_L = 2\pi \times 700 \times 25 \times 10^{-3} = 110 \, \Omega\)
• \(X_C = \frac{1}{2\pi \times 700 \times 4 \times 10^{-6}} = 57 \, \Omega\)

Since \(X_L > X_C\), the voltage leads the current. Therefore, the statement "At a frequency of 700 Hz, the voltage lags the current in the circuit" is false.

Conclusion:

The true statements are:

• "Resonance frequency is close to 503 Hz"
• "At a frequency of 200 Hz, the voltage lags the current in the circuit"