Figure P 31.50 is a graph of the induced emf versus time for...

Figure $\mathrm{P} 31.50$ is a graph of the induced emf versus time for a coil of $N$ turns rotating with angular velocity $\omega$ in a uniform magnetic field directed perpendicular to the axis of rotation of the coil. Copy this graph (on a larger scale), and on the same set of axes show the graph of emf versus $t$ (a) if the number of turns in the coil is doubled, (b) if instead the angular velocity is doubled, and (c) if the angular velocity is doubled while the number of turns in the coil is halved.

Key Concepts

Influence of Angular Velocity

The emf induced in a rotating coil is also affected by the angular velocity (?) of the coil. A higher angular velocity not only increases the rate at which the magnetic flux changes but also decreases the period of the sine function, leading to a higher frequency in the sinusoidal emf wave. Doubling the angular velocity doubles both the frequency and the amplitude (provided other factors remain constant).

Faraday's Law of Electromagnetic Induction

Faraday's Law states that an electromotive force (emf) is induced in a circuit when there is a change in magnetic flux through the circuit. This law is fundamental to understanding how variations in parameters like the number of turns in a coil or its angular velocity in a magnetic field affect the magnitude of the induced emf.

Impact of the Number of Turns

The induced emf in a coil is directly proportional to the number of turns (N) in the coil. Doubling the number of turns doubles the emf amplitude because each loop contributes equally to the total induced emf, thereby directly scaling the overall effect as the number of turns increases.

Sinusoidal Variation and Graphical Transformation

The emf versus time graph for a rotating coil in a uniform magnetic field is sinusoidal due to the periodic nature of the magnetic flux change. Variations in parameters like the number of turns and angular velocity result in changes in the amplitude and frequency of the sine wave. Graphically, modifying these parameters involves scaling the y-axis (amplitude changes) and x-axis (period/frequency changes) accordingly to accurately reflect the new functional form of the induced emf.

Transcript

00:02 Hi, in the given problem the graph between induced emf and the time for a coil having n turns and rotating with an angular velocity omega is given as here.

00:30 This is the emf induced over the y -axis and for the x -axis this is time.

00:39 Time is in middle.

00:41 Second and this emf is in millivolt.

00:47 The peak value of emf is given as 5 millivolt and the time period here has been given as 2 milliseconds.

01:01 So this is 1 this is 2.

01:03 We have to extend the curve in the same system which will include the rotation of the same coil in the same magnetic field but with some changed parameter and we know the expression for emf induced in this coil at any instant is given as n omega the rate of rotation omega a is the area of the coil b is the magnetic field into sign omega t where if we compare this equation with the standard equation equals to e not sine omega t this is the peak mf induced which is given as a product of number of turns with the rate of rotation with the area of the coil and with the magnetic field and here the value of this e not is given as 5 millivolt in the first part of the problem they are saying if the number of turns is made twice means if we increase the number of turns to double of its then their initial value it is clear that the peak mf induced is depending directly on the number of turns so it is clear that on increasing the number of turns to twice only the amplitude of this emf induced will be increased and it will become twice so if initially it was 5 milovolts now it will be increased to 10 milovol so we will make a twice of the initial value on both the sides and the time period we have to keep the same with the help of scale we can draw the curves the time period should remain same so we make some index lines in order to get the same time period like this the new shape of the curve will become something like this this is wrong here we can change its value to some lower one this is and here this emf induced means the magnet means the amplitude of the smf will be 10 really hold now in the second part of the problem they are increasing the rate of rotation to twice means omega dash equals to twice of omega in doing so the frequency and the amplitude both will be increased to double in doing this the frequency means the rate of rotation and amplitude both will be increased to twice so again repeating the same the curve, amplitude we have to keep same as 10 millivolt, but the frequency will be double means the time period will be reduced to half...

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