Suppose the loopin Fig. 𝐏 29.48 is (a) rotated about the y -axis; (b) rotated about the x -axis;

Suppose the loopin Fig. 𝐏 29.48 is (a) rotated about the y...

Suppose the loopin Fig. $\mathbf{P} 29.48$ is (a) rotated about the $y$ -axis; (b) rotated about the $x$ -axis; (c) rotated about an edge parallel to the $z$ -axis. What is the maximum induced emf in each case if $A=600 \mathrm{~cm}^{2}, \omega=35.0 \mathrm{rad} / \mathrm{s},$ and $B=0.320 \mathrm{~T} ?$

Key Concepts

Faraday's Law of Electromagnetic Induction

This principle states that an emf is induced in a circuit when there is a change in magnetic flux through the circuit. It is quantitatively expressed as the negative rate of change of magnetic flux, emphasizing the connection between time-varying magnetic environments and electric potentials.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the field and the normal to the area. Changes in the orientation of a loop relative to a steady magnetic field cause variations in the flux, leading to induction effects.

Rotational Motion in Magnetic Fields

Rotating a loop in a magnetic field alters the angle between the loop’s normal and the magnetic field vector. The angular velocity of rotation plays a key role in determining how quickly the magnetic flux changes, which in turn influences the magnitude of the induced emf.

Maximum Induced emf

The maximum induced emf occurs at the point where the rate of change of the magnetic flux is greatest. This condition is reached during specific phases of the rotation, depending on the axis of rotation, the loop's orientation, and the angular velocity.

Transcript

00:01 The given problem here, this is the xx, this is the y -axis, and perpendicular to both of these.

00:24 This is the x -axis.

00:30 Then there is a rectangular coil which has been kept in such a manner that two or two or a its sides are parallel to y -axis and other two sides are parallel to the x -axis and x -axis is perpendicular the plane of this soil then magnetic field is along the x -axis here this is the magnetic field.

01:20 Area of the loop is given as a is equal to 600 centimeter square or we can say this is a 600 into 10 dash bar minus 4 meter square angular velocity of the rotation of the coil is given as 35 .10 radiant per second and this magnet which is along the x -axis is given as 0 .3 to 0 as law.

02:04 In the first part of the problem it is said that the coil, the loop is rotating about y -axis is here like this.

02:29 So then the loop will be rotating about y -axis flux linked through this loop will be given by b .a or we can say this is b a cost theta theta with the angle between the area vector and magnetic field and as the angular speed of this rotation is omega so at any instant t this angular displacement theta will be given as distance equals to speed into time.

03:08 So angular distance equals to angular speed into time means omega t.

03:14 So the emf induced across this coil will be given as minus d5 by dt as per faraday's laws of electromagnetic induction.

03:29 So it will be given as d by dt means differentiation of b8 cost omega t.

03:38 Here, b into a, it means as it is being constant, then differentiation of cosine will be minus sine omega -t, and then differentiation of omega -t with respect to time, that will give us omega.

03:54 So finally, it comes out to be omega into a into b, into sine omega -t.

04:03 And then for maximum emf induced, the sign should have a maximum value which is just one.

04:21 Therefore, this maximum emf induced will be given by omega into a into b.

04:30 So for omega, this is 35 radian per second...

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