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Ch 32: AC Circuits
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 32: AC CircuitsProblem 28
Chapter 32, Problem 28

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

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Identify the formula for the resonance frequency of an RLC circuit, which is given by \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance, \( C \) is the capacitance, and \( f \) is the resonance frequency.
Note the changes in the circuit components: the capacitor value is doubled (\( C' = 2C \)) and the inductor value is halved (\( L' = \frac{L}{2} \)).
Substitute the new values of \( L' \) and \( C' \) into the resonance frequency formula to find the new resonance frequency \( f' \). Use \( f' = \frac{1}{2\pi\sqrt{L' \cdot C'}} \).
Simplify the expression by substituting \( L' = \frac{L}{2} \) and \( C' = 2C \) into the formula, resulting in \( f' = \frac{1}{2\pi\sqrt{\frac{L}{2} \cdot 2C}} \).
Further simplify the expression to find that the new resonance frequency \( f' \) is equal to the original resonance frequency \( f \), because the changes in \( L \) and \( C \) counteract each other.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Resonance Frequency in RLC Circuits

The resonance frequency of a series RLC circuit is the frequency at which the inductive reactance equals the capacitive reactance, resulting in maximum current flow. It is given by the formula f₀ = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. At resonance, the impedance of the circuit is minimized, and the circuit can oscillate freely.
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Effect of Capacitance on Resonance Frequency

Doubling the capacitance in an RLC circuit affects the resonance frequency inversely. According to the resonance frequency formula, increasing capacitance decreases the resonance frequency, as it is in the denominator of the square root. This means that if the capacitance is doubled, the resonance frequency will decrease, leading to a lower frequency of oscillation.
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Effect of Inductance on Resonance Frequency

Halving the inductance in an RLC circuit also influences the resonance frequency, but in the opposite direction. Since inductance is also in the denominator of the square root in the resonance frequency formula, reducing the inductance increases the resonance frequency. Therefore, if the inductance is halved, the resonance frequency will increase, resulting in a higher frequency of oscillation.
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Related Practice
Textbook Question

For the circuit of FIGURE EX32.32, What is the resonance frequency, in both rad/s and Hz?

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Textbook Question

For the circuit of FIGURE EX32.32, Find VR and VL at resonance.

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Textbook Question

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?

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Textbook Question

An inductor is connected to a 15 kHz oscillator. The peak current is 65 mA when the rms voltage is 6.0 V. What is the value of the inductance L?

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For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?

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Textbook Question

A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?

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