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Chapter 30: Problem 21

Is it possible for the voltage amplitude across the inductor in a series RLC circuit to exceed the voltage amplitude of the voltage supply? Why or why not?

Short Answer

Expert verified
Answer: No, the voltage amplitude across the inductor (VL) of a series RLC circuit cannot exceed the voltage amplitude of the voltage supply (Vs). The supplied voltage will always be equal to the sum of the voltages across the resistor, inductor, and capacitor, and it must be distributed between these components. Therefore, VL cannot be greater than Vs.

Step by step solution

01

Recall the Relationship between Voltages

A series RLC circuit has three components: a resistor (R), an inductor (L), and a capacitor (C). The voltage across each component must add up to the total supplied voltage (Vs) in a series circuit. In other words, Vs = VR + VL + VC where VR is the voltage across the resistor, VL is the voltage across the inductor, and VC is the voltage across the capacitor.
02

Consider the Resonance Condition

The voltage across the inductor and capacitor is related to the frequency of the AC source. When the circuit is at resonance, the capacitive reactance (XC) and inductive reactance (XL) are equal: XC = XL Remember that voltage across the capacitor (VC) = IX_C and voltage across the inductor (VL) = IX_L. At resonance, the voltages across the inductor and capacitor are equal in magnitude, but they differ in phase by 180 degrees. This means that the voltages across the inductor and capacitor cancel each other out, and the total voltage in the circuit becomes: Vs = VR
03

Determine if VL can be Greater than Vs

Since the voltage across the inductor and capacitor cancel each other out at resonance, we can conclude that the voltage across the inductor (VL) cannot exceed the voltage supply (Vs). The supplied voltage will always be equal to the sum of the voltages across the resistor, inductor, and capacitor, and it must be distributed between these components. Therefore, VL cannot be greater than Vs. So, it is not possible for the voltage amplitude across the inductor in a series RLC circuit to exceed the voltage amplitude of the voltage supply.

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Most popular questions from this chapter

If you use a parallel plate capacitor with air in the gap between the plates as part of a series RLC circuit in a generator, you can measure current flowing through the generator. Why is it that the air gap in the capacitor does not act like an open switch, blocking all current flow in the circuit?

Why can't a transformer be used to step up or step down the voltage in a DC circuit?

Laboratory experiments with series RLC circuits require some care, as these circuits can produce large voltages at resonance. Suppose you have a 1.00 - \(\mathrm{H}\) inductor (not difficult to obtain) and a variety of resistors and capacitors. Design a series RLC circuit that will resonate at a frequency (not an angular frequency) of \(60.0 \mathrm{~Hz}\) and will produce at resonance a magnification of the voltage across the capacitor or the inductor by a factor of 20.0 times the input voltage or the voltage across the resistor.

A capacitor with capacitance \(C=5.00 \cdot 10^{-6} \mathrm{~F}\) is connected to an AC power source having a peak value of \(10.0 \mathrm{~V}\) and \(f=100 . \mathrm{Hz} .\) Find the reactance of the capacitor and the maximum current in the circuit.

An AC power source with \(V_{\mathrm{m}}=220 \mathrm{~V}\) and \(f=60.0 \mathrm{~Hz}\) is connected in a series RLC circuit. The resistance, \(R\), inductance, \(L\), and capacitance, \(C\), of this circuit are, respectively, \(50.0 \Omega, 0.200 \mathrm{H},\) and \(0.040 \mathrm{mF}\). Find each of the following quantities: a) the inductive reactance b) the capacitive reactance c) the impedance of the circuit d) the maximum current through the circuit e) the maximum potential difference across each circuit element
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