Question

Using an A.C voltmeter the potential difference in the electrical line in a house is read to be $234V$. If the line frequency is known to be $50\dfrac{\text{cycles}}{\text{second}}$, the equation for the line voltage is given as,
$\begin{aligned} & A.V=165\sin \left( 100\pi t \right) \\\ & B.V=331\sin \left( 100\pi t \right) \\\ & C.V=220\sin \left( 100\pi t \right) \\\ & D.V=440\sin \left( 100\pi t \right) \\\ \end{aligned}$

Answer 1

The voltage should be converted into root mean square velocity. The rms voltage is equivalent to the product of square root of two and the peak voltage level. And also angular velocity is given as the product of twice the pi and frequency. Substitute the values calculated in the equation of voltage in an AC circuit. These all may help you to solve this question.

Complete answer:
As we all know, the general equation for the voltage in AC circuit is given as the equation,
$V={{V}_{0}}\sin \omega t$
The voltage in the root mean square value can be determined as,
${{V}_{0}}=\sqrt{2}{V}'$
Where ${V}'$ be the velocity mentioned in the question. The value is given as,
${V}'=234V$
Substituting this in the equation will give,
${{V}_{0}}=\sqrt{2}\times 234=331V$
The angular velocity is given as twice the pi multiplied with frequency. This can be written as,
$\omega =2\pi n$
Where $n$ be the frequency
The frequency has been given as,
$n=50\dfrac{\text{cycles}}{\text{second}}$
Substitute this in the equation which will give,
$\omega =2\pi \times 50=100\pi$
Hence we can substitute all these terms in the equation for voltage in AC circuits. The equation will become,
$V=331\sin 100\pi t$

Therefore the answer has been obtained which is given as option B.

Note:
The rms voltage can be found out by taking the root of the mean of the square of the voltages. This rms value is used to denote the sine form of the wave in the alternating current circuits which is representing the equivalent heating produced in the DC circuits. It is also known as the quadratic mean.