Question

By the use of an A.C voltmeter the potential difference in the electrical line in a house is found to be $234V$. When the line frequency is found to be \[50\dfrac{\text{cycles}}{\text{second}}\], then what will be the equation for the line voltage?
\[\begin{align}
  & A.V=165\sin \left( 100\pi t \right) \\
 & B.V=331\sin \left( 100\pi t \right) \\
 & C.V=220\sin \left( 100\pi t \right) \\
 & D.V=440\sin \left( 100\pi t \right) \\
\end{align}\]

Answer 1

Hint: The maximum voltage will be equivalent to the product of the square root of two and the root mean square velocity. The product of the angular velocity and the time taken will be equivalent to the product of the twice the pi, frequency and the time taken. This will help you in answering this question.

Complete step by step solution:
 The equation of the voltage can be written as,
\[V={{V}_{0}}\sin \left( \omega t \right)\]
The root mean square voltage can be written as,
The maximum voltage will be equivalent to the product of the square root of two and the root mean square velocity. This can be written as,
\[{{V}_{rms}}=234V\]
Substituting the values in the equation,
\[{{V}_{0}}=\sqrt{2}\times 234V\]
Simplifying the equation can be written as,
\[{{V}_{0}}=331V\]
The product of the angular velocity and the time taken will be equivalent to the product of the twice the pi, frequency and the time taken. This can be written as,
\[\omega t=2\pi nt\]
Substituting the values in the equation,
\[\omega t=2\pi \times 50\times t=100\pi t\]
Thus, the equation of the line voltage can be given as,
\[V=331\sin \left( 100\pi t \right)\]

So, the correct answer is “Option B”.

Note: The rms voltage can be found by taking the square root of the mean of the squares of the velocity. Maximum voltage can be defined as the voltage at which an equipment that can be operated safely, without making any damage to the device itself.