A semicircular conductor of radius R=0.250 m is rotated...

A semicircular conductor of radius $R=0.250 \mathrm{~m}$ is rotated about the axis $A C$ at a constant rate of 120 rev/min (Fig. P31.42). A uniform magnetic field in all of the lower half of the figure is directed out of the plane of rotation and has a magnitude of $1.30 \mathrm{~T}$. (a) Calculate the maximum value of the emf induced in the conductor. (b) What is the value of the average induced emf for each complete rotation? (c) How would the answers to parts (a) and (b) change if $\mathbf{B}$ were allowed to extend a distance $R$ above the axis of rotation? Sketch the emf versus time (d) when the field is as drawn in Figure $\mathrm{P} 31.42$ and $(\mathrm{e})$ when the field is extended as described in part (c).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic environment generates an electromotive force (emf) in a conductor. This principle is fundamental in understanding how moving conductors, or conductors in time-varying magnetic fields, convert mechanical energy into electrical energy according to the laws of electromagnetism.

Faraday’s Law of Induction

Faraday’s law quantitatively relates the rate of change of magnetic flux through a circuit to the magnitude of the induced emf. It states that the induced emf is equal to the negative of the time rate of change of magnetic flux, illustrating how variations in either magnetic field strength or the area through which the field penetrates can produce an electrical voltage.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetic field passing perpendicularly through a given area. It is calculated as the product of the magnetic field strength, the area of the surface, and the cosine of the angle between the field and the area vector. Changes in any of these factors over time result in a changing flux and, thereby, induce an emf.

Rotational Kinematics in Electromagnetism

When dealing with rotating conductors in magnetic fields, it is crucial to convert rotation rates (in revolutions per minute, for example) into angular velocity (in radians per second). This angular velocity is then used to determine how quickly the conductor’s orientation relative to the magnetic field changes, directly affecting the rate of change of magnetic flux.

Waveform Analysis of Induced EMF

The analysis of the induced emf waveform involves understanding how the voltage varies with time due to the cyclic nature of the conductor’s motion through the magnetic field. It includes the determination of the maximum instantaneous emf and the average emf over a full cycle, which are influenced by both the geometry of the conductor and the spatial distribution of the magnetic field.

Transcript

00:02 Hi, in the given problem the coil is half in the shape of a semi -circle and magnetic field which is coming out of the plane of paper is also extended to just half the region.

00:22 So when this coil will be rotated about its axis in the given manner the flux will be changed.

00:32 Only when the coil will be passing through this magnetic field in the lower segment.

00:39 In the upper segment there will be no change in the magnetic field.

00:44 So the radius of the coil has been given as 0 .250 meter.

00:50 The rate of rotation of the coil is 120 revolutions per minute or we can say this is 120 by 2 per second 120 by 60 revolutions per second or this is simply two revolutions per second and the magnetic field in the region has been given as 1 .30 tesla in outward direction.

01:23 So in the first part of the problem we have to find the maximum emf induced in the coil when it is being rotated.

01:31 So the expression for the maximum emf induced is n into omega the angular velocity of this coil into area of the coil into magnetic field but as the area of the coil will be taken just the half so here it becomes and the number of turn is just the one so this is one into two pi f the expression for angular velocity into half of the area into half of the magnetic field also as the magnetic field is only in the half of the region so when we put all these values this is 2 multiplied by 3 .14 multiplied by 2 the frequency then half of the area means pi r square this is capital r by r square for the area and it's half by 2 and half of the magnetic field also so here this is cancelled, so it will become 6 .28 times 3 .14 square of the radius means the square of 0 .25 multiplied by the magnetic field, half of magnetic field, 1 .30 by 2.

02:50 So this finally comes out to be 0 .801 volt...

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