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Question: In the figure which voltmeter reads zero, when \( \omega \) is equal to the resonant frequency of th...

In the figure which voltmeter reads zero, when ω\omega is equal to the resonant frequency of the series LCR circuit?

A). V1{V_1}
B). V2{V_2}
C). V3{V_3}

Explanation

Solution

An LCR circuit has three major components: resistor, inductor, capacitor.
These components can be connected in either a series or a parallel configuration.
Impedance is lowest at resonance in the LCR series. As a result, Zmin=R{Z_{\min }} = R . ω\omega is the angular frequency.

Complete Step By Step Answer:
A voltmeter will read zero when the points being measured are of the same potential.
Let the voltage across the inductor be VL{V_L} and voltage across the capacitor be VC{V_C} .
Voltage across the resistor is given by V1=IR{V_1} = IR , where I is the current and R is the resistance.
Voltage across inductor and capacitor is given by V2=VLVC{V_2} = {V_L} - {V_C}
Therefore voltage across V3{V_3} is given by V32=V12+(VLVC)2{V_3}^2 = {V_1}^2 + {({V_L} - {V_C})^2}
Only the presence of resistor affect the voltage through the voltmeter V1{V_1} , the inductor and the capacitor affect the voltage through the voltmeter V2{V_2} , all three components affects the voltage through the voltmeter V3{V_3}
Since the resonance voltage across the inductor and capacitor will be the same.
That is VL=VC{V_L} = {V_C}
VLVC=0\Rightarrow {V_L} - {V_C} = 0
\Rightarrow voltage across the voltmeter V2{V_2} will be zero.
The correct answer is option B, V2{V_2} .

Note:
The frequency at which the impedance of the LCR circuit becomes minimal or the current in the circuit becomes maximal is known as the resonance frequency.
Resonant frequency ωr==1LC{\omega _r} = = \dfrac{1}{{\sqrt {LC} }}
LCR circuits have a significant amount of resonance. Energy can be stored in LCR circuits in two ways: as an electric field in a capacitor when it is charged, or as a magnetic field in an inductor when current runs through it.