Can the Voltage Across a Capacitor Be Greater Than the Source Voltage in a Series RLC Circuit?

[Meow12]

Homework Statement

You have a ##200\ \Omega## resistor, a ##0.400\ H## inductor, a ##6.00\ \mu F## capacitor, and a voltage source that has a voltage amplitude ##30.0\ V## and an angular frequency of 250 rad/s. They are connected to form an L-R-C series circuit.

(a) What are the voltage amplitudes across the resistor, inductor, and capacitor?
(b) Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source.

Relevant Equations

##X_L=\omega L##, ##\displaystyle X_C=\frac{1}{\omega C}##, ##Z=\sqrt{R^2+(X_L-X_C)^2}##
##\displaystyle I=\frac{V}{Z}##
## V_R=IR##, ##V_L=IX_L##, ##V_C=IX_C##

(a) Substituting the values, I get ##X_L=100\ \Omega##, ##X_C=666.67\ \Omega##.

From this, I get ##Z=601\ \Omega##, ##I=49.9\ mA##

##V_R=9.98\ V##, ##V_L=4.99\ V##, ##V_C=33.3\ V##

(b) It's possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source because ##V_R##, ##V_L-V_C##, and ##V## constitute a right triangle where ##V^2=V_R^2+(V_L-V_C)^2##.

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My answer to (a) matches with the one given in the back of the textbook.
But is my answer to (b) correct?
Thanks.

 

[osilmag]

Your math checks out. Were they wanting a conceptual answer? If so, something like the frequency of the signal is high enough that it is storing energy in the capacitor.

 

[DaveE]

(b) It's possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source because VR, VL−VC, and V constitute a right triangle where V2=VR2+(VL−VC)2.

But is my answer to (b) correct?

Yes, but if I was grading, I'd want you to continue your explanation a bit more. It would be customary to express this with complex numbers or magnitudes, since the phase of stuff is critical to understanding it. But that stuff is also pretty clearly implied in your statement.

 

[Gavran]

The equation $$ V ^ 2 = V _ R ^ 2 + ( V _ L – V _ C ) ^ 2 $$ is a correct and not complete answer to the question how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source in a series RLC circuit.

The second addend in the equation must be kept constant while the voltage across the inductance and the voltage across the capacitor can be changed, can be increased, so...