Question

A basic one-loop dc generator is rotated at $90{\text{ rev }}{{\text{s}}^{ - 1}}$. How many times each second does the dc output voltage peak (reaches a maximum)?
A) 90
B) 180
C) 270
D) 360

Answer 1

The given speed at which the generator rotates is essentially the frequency of the generator. The instantaneous value of the induced emf in the generator due to its rotation has a sinusoidal nature. So if we were to obtain the number of times the voltage peaks in one rotation, then we can determine the number of times it peaks as the frequency is given.

Complete step by step answer:
Step 1: Express the instantaneous value of the induced emf in the generator.
Here the coil of the generator rotates with an angular velocity $\omega$. The magnetic field present is $B$ and the area of the coil is $A$ .
According to Faraday’s law, the emf induced in a coil with $N$ number of turns is given by, $\varepsilon = - NBA\dfrac{{d\left( {\cos \omega t} \right)}}{{dt}}$ ---------- (1)
Taking the derivative of equation (1), we get the instantaneous value of the induced emf as $\varepsilon = NBA\omega \sin \omega t$ -------- (2)
Step 2: Obtain the number of times the voltage peaks in one rotation to obtain the number of times it peaks in each second of its rotation.
The peaking of the voltage corresponds to reaching a maximum voltage.
The instantaneous voltage is expressed as $\varepsilon = NBA\omega \sin \omega t$.
The instantaneous voltage varies as the value of $\omega t$ varies from $0^\circ$ to $360^\circ$ i.e., as the coil makes one complete rotation.
It is maximum when $\sin \omega t = \pm 1$
$\Rightarrow$ $\omega t = 90^\circ$ and $\omega t = 270^\circ$.
So in one complete rotation, the voltage peaks twice.
So as the frequency of the generator is given to be $f = 90{\text{ rev }}{{\text{s}}^{ - 1}}$, the number of times the voltage peaks each second will be $n = 2f = 2 \times 90 = 180$

Hence, the correct option is B.

Note:
The value of the sine function in equation (2) varies from $+ 1$ to $- 1$ and so the polarity of the induced emf changes with time. The instantaneous value of the induced emf has a negative maximum $\varepsilon = - NBA\omega$ corresponding to $\omega t = 270^\circ$ and a positive maximum $\varepsilon = NBA\omega$ corresponding to $\omega t = 90^\circ$. Here we considered both maxima. The dc generator converts mechanical energy into electrical energy as the rotation of the coil changes the magnetic flux associated with it which then induces an emf.