A flat, circular, steel loop of radius 75 cm is at rest in a uniform magnetic field, as shown i

A flat, circular, steel loop of radius 75 cm is at rest in...

A flat, circular, steel loop of radius $75 \mathrm{~cm}$ is at rest in a uniform magnetic field, as shown in an edge-on view in Fig. E29.8. The field is changing with time, according to $B(t)=(1.4 \mathrm{~T}) e^{-\left(0.057 \mathrm{~s}^{-1}\right) t}$ (a) Find the emf induced in the loop as a function of time. (b) When is the induced emf equal to $\frac{1}{10}$ of its initial value? (c) Find the direction of the current induced in the loop, as viewed from above the loop.

Key Concepts

Faraday's Law of Induction

This law states that an electromotive force (emf) is induced in a loop when the magnetic flux through the loop changes with time. The induced emf is proportional to the negative rate of change of the magnetic flux, which underpins the working principles of many electromagnetic devices and circuits.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is calculated by taking the product of the magnetic field strength and the area perpendicular to the field. Understanding magnetic flux is crucial as it directly relates to the induced emf when the field or the area changes.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from a changing magnetic flux. It states that the induced current will flow in such a direction that its magnetic field opposes the change in the original magnetic flux, ensuring energy conservation in the system.

Right-Hand Rule

The right-hand rule is a convention used to determine the direction of the induced current in a loop. Based on Lenz's Law, this rule helps correlate the vector directions of the magnetic field, the change in flux, and the current, allowing one to understand the spatial orientation of the induced effects.

Transcript

00:01 Hi, in the given problem here this is the direction of magnetic field like this and in this magnetic field there is a loop whose end side view is visible to us and the plane of the loop is making an angle of 60 degree with the direction of this magnetic field.

00:42 So if we draw a normal and the plane of this loop, this normal drawn at the plane, will be making an angle of 30 degree with the direction of magnetic field.

00:57 The radius of this loop has been given as 75 centimeter or we can say this is 0 .75 meter.

01:08 And the magnetic field variable, which is variable with the time, has been given as 1 .4 into e to the power minus 0 .057 t where this t is the time tesla this is the time varying magnetic field given to us so in the first part of the problem first of all we will find the total magnetic flux linked with the loop so using expression for the flux it becomes b dot a the dot product does scalar product of magnetic field with the area of the coil.

01:52 So it comes out to be b times a into cosine of angle between area vector and magnet field which is 30 degree.

02:02 So we put the values of magnet field and area and the magnetic field is 1 .4 times e dash the bar minus 0 .057 t as the magnet field is variable with the time and the area is pi 0 .75 to the whole square multiplied by cost 30 degree so when we calculate all these values and multiply them with the 1 .4 we get this to be 2 .14 multiplied by e to the power minus 0 .057 t so this is the expression for this magnetic flux linked with the loop in terms of time.

02:51 So the emf induced at any instant in this coil will be given by using paradis laws of electromagnetic induction as negative of the time rate of change of this magnetic flux, d5 by dt...

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