Arectangular circuit is moved at a constant velocity of 3.0 m / s into, through, and then out o

Arectangular circuit is moved at a constant velocity of 3.0...

Arectangular circuit is moved at a constant velocity of $3.0 \mathrm{~m} / \mathrm{s}$ into, through, and then out of a uniform $1.25 \mathrm{~T}$ magnetic field, as shown in Fig. E29.33. The magnetic-field region is considerably wider than $50.0 \mathrm{~cm} .$ Find the magnitude and direction (clockwise or counterclockwise) of the current induced in the circuit as it is (a) going into the magnetic field; (b) totally within the magnetic field, but still moving; and (c) moving out of the field. (d) Sketch a graph of the current in this circuit as a function of time, including the preceding three cases.

Key Concepts

Induced Current Direction

The direction of the induced current is determined by the combination of Faraday's Law and Lenz's Law, and can be analyzed using the right-hand rule. It involves understanding how the circuit's motion relative to the magnetic field causes the current to flow in a way that opposes the change in the magnetic flux, thus resulting in either a clockwise or counterclockwise current depending on whether the circuit is entering or leaving the magnetic field.

Motional EMF

Motional emf refers to the voltage generated when a conductor moves through a magnetic field, owing to the Lorentz force exerted on the charges within the conductor. This phenomenon explains how a current is induced in the circuit when it moves with constant velocity into, through, or out of a magnetic field, even when the magnetic field itself is uniform.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field and the perpendicular area it penetrates. When a circuit moves into, within, or out of a magnetic field, the magnetic flux through its area changes, which is crucial for calculating the induced emf.

Faraday's Law of Electromagnetic Induction

This principle states that any change in the magnetic environment of a circuit will cause a voltage (emf) to be induced in the circuit. It emphasizes that the induced emf is directly proportional to the rate of change of magnetic flux through the circuit, which forms the foundation for understanding how moving a circuit into or out of a magnetic field generates an electromotive force.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from a change in magnetic flux. It states that the induced current flows in such a direction that its magnetic field opposes the change in the original magnetic flux that produced it. This law ensures the conservation of energy and is key to determining whether the current will be clockwise or counterclockwise in practical scenarios.

Transcript

00:01 In the given problem here, this is a uniform magnetic field which is perpendicular to the plane of paper and coming out of it.

00:17 In this magnetic field there is a circuit, a rectangular circuit which is having a resistance r, having the value of 12.

00:38 0 .5 om.

00:41 It is moving towards right with the velocity v having a magnitude of 3 .0 meter per second means it is being entered into this magnetic field.

00:59 L length of this circuit here is given as l is equal to 75 .0 centimeter or we can say this 0 .75 meter and here its width is b is equal to 50 .0 centimeter or we can say this is 0 .5 meter.

01:34 The magnetic field, magnitude of this magnetic field, which is coming out perpendicularly to the plane of paper, is given as b is equal to 1 .25 tesla.

01:47 Now, in the first part of the problem, we have to find the current induced in this circuit when it is being entered into the magnetic field.

02:02 So when the circuit is entering into the magnetic field, its right arm having this length l will be intercepting the magnetic field.

02:32 So, motional emf induced across the ends of this length will be epsilon is equal to b into v into l.

02:50 Hence, current induced in this circuit will be given by oms law as emf induced means epsilon divided by r.

03:02 Or we can say this is bvl by r.

03:07 Plugging in all known values for b, this is 1 .25 tesla multiplied by speed, which is 3 .0 meter per second multiplied by length, which is 0 .75 meter.

03:23 And divided by resistance of this circuit, which is 12 .5 oom.

03:29 So in numerator we get 2 .625 volt, the emf induced.

03:41 Denominator is 12 .5 om.

03:45 So finally, the magnitude of this current comes out to be 0 .21 ampere.

03:55 Using fleming's right hand rule, the direction of this current induced in the right arm of the circuit is downward.

04:34 So in the circuit it will become clockwise...

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