A small, closely wound coil has N turns, area A, and...

A small, closely wound coil has $N$ turns, area $A$, and resistance $R$. The coil is initially in a uniform magnetic field that has magnitude $B$ and a direction perpendicular to the plane of the loop. The coil is then rapidly pulled out of the field so that the flux through the coil is reduced to zero in time $\Delta t$. (a) What are the magnitude of the average $\operatorname{emf} \mathcal{E}_{\text {av }}$ and average current $I_{\mathrm{av}}$ induced in the coil? (b) The total charge $Q$ that flows through the coil is given by $Q=I_{\mathrm{av}} \Delta t .$ Derive an expression for $Q$ in terms of $N, A, B,$ and $R .$ Note that $Q$ does not depend on $\Delta t .$ (c) What is $Q$ if $N=150$ turns, $A=4.50 \mathrm{~cm}^{2}, R=30.0 \Omega,$ and $B=0.200 \mathrm{~T} ?$

Key Concepts

Ohm's Law

Ohm's Law provides a relationship between voltage, current, and resistance in an electrical circuit. When an emf is induced in a coil, the resulting current is determined by dividing the emf by the coil's resistance, establishing how the electrical properties of the circuit influence the induced current.

Total Charge and Time Independence

Integrating the induced current over time provides the total charge that flows through the coil, with the intriguing result that this total charge depends solely on the total change in magnetic flux and the resistance of the coil, rather than on the duration of the flux change. This demonstrates a fundamental aspect of electromagnetic induction where the integrated effect remains constant despite variations in the time interval of the change.

Faraday's Law of Electromagnetic Induction

This principle states that a change in magnetic flux through a loop or coil induces an electromotive force (emf) in that coil, with the induced emf being proportional to the rate of change of the flux. It is foundational in understanding how varying magnetic environments can generate electric currents in conductors.

Magnetic Flux

Magnetic flux represents the quantity of magnetic field passing through a given area, typically calculated as the product of the magnetic field strength, area of the loop, and the cosine of the angle between the field and the normal to the surface. It is the change in this flux that directly relates to the induced emf via Faraday's Law.

Transcript

00:01 Hi.

00:02 In the given problem, the closely wound coil is having number of turns, the total turns as n.

00:26 Area of this coil is a resistance of the wire which is used to make this coil is r and initially the plane of coil is perpendicular to the magnetic field hence angle between the area vector of the coil and magnetic field is 0 degree.

00:52 Then in a time duration of delta t, the coil is pulled out of the magnetic field.

01:01 So the flux, the final flux, magnetic flux linked through the coil 5 -2 will become 0, while initial flux passing through the coil will be given as b.

01:21 Dot a n times of b .a for the total magnetic flux or we can say this is n b a cos theta or this is n b a cos zero degree and the value of cause zero degree is one so initial flux linked through the coil is 5 1 is equal to n b a now in the first part of the problem we have to find average current passing through this coil then it will be being pulled out of the magnetic field.

02:00 So emf, average emf induced in the coil will be given by using faraday's loss of electromagnetic induction, which says average emf induced is equal to the time rate of the negative of time rate of change of magnetic flux.

02:19 So it will become minus minus 5 to minus 5 1 by delta t the time.

02:30 So it becomes minus 0 minus n b a divided by delta t or we can say the average value of emf induced in this coil is n b a by delta t one of the answer of the first part of the problem.

02:54 Then we have to find the average current passing through this coil.

03:00 So that average current using omns law will be given by ratio of average emf induced with the resistance.

03:09 So it becomes i average is equal to nb a by delta t into r...

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