11. The rms current in an RLC circuit depends on the frequency of the power source as shown below. It reaches its maximum value at fo = 80 Hz. Find the rms voltage applied to the circuit if R = 10 92, C = 198 µF, and L = 20 mH.
11. The rms current in an RLC circuit depends on the frequency of the power source as shown below. It reaches its maximum value at fo = 80 Hz. Find the rms voltage applied to the circuit if R = 10 92, C = 198 µF, and L = 20 mH.
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ISBN:9781305952300
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Publisher:Raymond A. Serway, Chris Vuille
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![### Problem 11: RLC Circuit and RMS Voltage Calculation
**Problem Statement:**
The rms current in an RLC circuit depends on the frequency of the power source as shown below. It reaches its maximum value at \( f_0 = 80 \) Hz. Find the rms voltage applied to the circuit if:
\[ R = 10 \, \Omega, \; C = 198 \, \mu F, \; L = 20 \, mH \]
**Options:**
A. 25 V
B. 50 V
C. 75 V
D. 100 V
E. 125 V
**Graph Explanation:**
The provided graph plots Current (Amps) on the y-axis against Frequency (Hertz) on the x-axis. The graph depicts a curve that peaks sharply at \( f_0 = 80 \) Hz, indicating the resonance frequency where the rms current reaches its maximum value of approximately 2.5 Amps. Beyond this peak value, the current decreases on either side as the frequency moves away from 80 Hz.
**Solution Approach:**
Given:
- \( R = 10 \, \Omega \)
- \( C = 198 \, \mu F \)
- \( L = 20 \, mH \)
- \( f_0 = 80 \, Hz \) (Resonance Frequency)
- Maximum rms current (\( I_{rms} \)) at \( f_0 = 2.5 \, A \)
The resonance frequency \( f_0 \) in an RLC circuit is given by:
\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]
From this, we can verify the resonance condition:
\[ L = 20 \times 10^{-3} \, H \]
\[ C = 198 \times 10^{-6} \, F \]
\[
f_0 = \frac{1}{2\pi \sqrt{(20 \times 10^{-3})(198 \times 10^{-6})}} \approx 80 \, Hz
\]
(This verifies that the given components are correct for the resonance frequency.)
At resonance, the impedance \( Z \) of the circuit is purely resistive and equal to \( R \). Therefore, using Ohm's law:
\[ I_{rms} = \frac{V_{rms](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53b47ed7-4497-4fb9-9d0f-c41504626a91%2Fb79ea4ab-8f04-4969-8897-087d2b0593ee%2Fz82dlza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 11: RLC Circuit and RMS Voltage Calculation
**Problem Statement:**
The rms current in an RLC circuit depends on the frequency of the power source as shown below. It reaches its maximum value at \( f_0 = 80 \) Hz. Find the rms voltage applied to the circuit if:
\[ R = 10 \, \Omega, \; C = 198 \, \mu F, \; L = 20 \, mH \]
**Options:**
A. 25 V
B. 50 V
C. 75 V
D. 100 V
E. 125 V
**Graph Explanation:**
The provided graph plots Current (Amps) on the y-axis against Frequency (Hertz) on the x-axis. The graph depicts a curve that peaks sharply at \( f_0 = 80 \) Hz, indicating the resonance frequency where the rms current reaches its maximum value of approximately 2.5 Amps. Beyond this peak value, the current decreases on either side as the frequency moves away from 80 Hz.
**Solution Approach:**
Given:
- \( R = 10 \, \Omega \)
- \( C = 198 \, \mu F \)
- \( L = 20 \, mH \)
- \( f_0 = 80 \, Hz \) (Resonance Frequency)
- Maximum rms current (\( I_{rms} \)) at \( f_0 = 2.5 \, A \)
The resonance frequency \( f_0 \) in an RLC circuit is given by:
\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]
From this, we can verify the resonance condition:
\[ L = 20 \times 10^{-3} \, H \]
\[ C = 198 \times 10^{-6} \, F \]
\[
f_0 = \frac{1}{2\pi \sqrt{(20 \times 10^{-3})(198 \times 10^{-6})}} \approx 80 \, Hz
\]
(This verifies that the given components are correct for the resonance frequency.)
At resonance, the impedance \( Z \) of the circuit is purely resistive and equal to \( R \). Therefore, using Ohm's law:
\[ I_{rms} = \frac{V_{rms
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