Question

The instantaneous voltages at three terminals marked X,Y and Z are given by;
${V_X} = {V_0}\sin \omega t$
${V_Y} = {V_0}\sin (\omega t + \dfrac{{2\pi }}{3})$
${V_Z} = {V_0}\sin (\omega t + \dfrac{{4\pi }}{3})$
An ideal voltmeter is configured to read the rms value of potential difference between its terminals. It is connected between the points X and Y and then between Y and Z . the readings of the voltmeter will be:
A) ${V_{YZ}}^{rms} = {V_0}\sqrt {\dfrac{1}{2}} $
B) ${V_{XY}}^{rms} = {V_0}\sqrt {\dfrac{3}{2}} $
C) ${V_{XY}}^{rms} = {V_0}$
D) Independent of the choice of two terminals

Answer 1

Hint: Instantaneous voltages means the voltage at any instant of time present in the circuit.
In order to find the voltage between the two terminals we will apply the parallelogram law of vector addition which is given as:
$\sqrt {{V_1}^2 + {V_2}^2 + 2{V_1}{V_2}\cos \theta } $ ($\theta $ is the angle between two vectors and $V_1$ and $V_2$ are the voltages)
Using the above concept we will find out the reading shown by voltmeter.

Complete step by step solution:
Instantaneous voltage is the induced EMF in the coil at any instant of time depending upon the speed at which the coil cuts the magnetic lines of flux between the poles. An AC waveform keeps on changing at every instant, then the value of induced EMF will also change at every instant.
Now, we will come to the calculation part of the problem;
The potential difference between X and Y is calculated as;
$ \Rightarrow {V_{XY}} = {V_X} - {V_Y}$
$\Rightarrow {V_{XY}} = {({V_{XY}})_0}\sin (\omega t + {\theta _1})$
where ${({V_{XY}})_0} = \sqrt {{V_0}^2 + {V_0}^2 - 2{V_0}{V_0}\cos \theta } $
Then we can find the value of Maximum voltage ;
$ \Rightarrow \sqrt {2{V_0}^2 - 2{V_0}^2 \times -\dfrac{1}{2}} $ ( cos$\dfrac{{2\pi }}{3}$ is equal to $-\dfrac{1}{2}$)
$ \Rightarrow \sqrt 3 {V_0}$
The value of voltage which we have calculated is the maximum value of voltage, therefore the rms value is calculated as:
$ \Rightarrow {V_{RMS}} = \dfrac{{{V_{MAX}}}}{{\sqrt 2 }}$
$ \Rightarrow {V_{RMS}} = \dfrac{{{V_0}\sqrt 3 }}{{\sqrt 2 }}$ is the value of voltmeter reading.
Similarly, if we calculate the value of voltage for Y and Z points then we can have the same procedure to follow;
The potential difference between X and Y is calculated as;
$ \Rightarrow {V_{YZ}} = {V_Y} - {V_Z}$
$ \Rightarrow {V_{YZ}} = {({V_{YZ}})_0}\sin (\omega t + {\theta _2})$
where ${({V_{YZ}})_0} = \sqrt {{V_0}^2 + {V_0}^2 - 2{V_0}{V_0}\cos \theta } $
Then we can find the value of Maximum voltage ;
$ \Rightarrow \sqrt {2{V_0}^2 - 2{V_0}^2 \times -\dfrac{1}{2}} $ ( cos$\dfrac{{2\pi }}{3}$ is equal to $-\dfrac{1}{2}$)
$ \Rightarrow \sqrt 3 {V_0}$
The value of voltage which we have calculated is the maximum value of voltage, therefore the rms value is calculated as:
$ \Rightarrow {V_{RMS}} = \dfrac{{{V_{MAX}}}}{{\sqrt 2 }}$
$ \Rightarrow {V_{RMS}} = \dfrac{{{V_0}\sqrt 3 }}{{\sqrt 2 }}$ is the value of voltmeter reading.
$\Rightarrow {V_{YZ}} = \dfrac{{\sqrt 3 {V_0}}}{{\sqrt 2 }}$
Thus, we can say that the value of voltage read by the voltmeter does not depend upon the points across which it is calculated.

Thus, options B and D are correct.

Note: Voltmeter is a voltage measuring instrument and is always connected in parallel with the circuit. Voltmeter usually has high resistance, so that it takes almost negligible current from the circuit. Normally two types of voltmeters are available analog and digital voltmeters are available.