A rectangular circuit is moved at a constant velocity of 3.0...

A rectangular circuit is moved at a constant velocity of 3.0 m/s into, through, and then out of a uniform 1.25-T magnetic field, as shown in $\textbf{Fig. E29.35}$. The magnetic-field region is considerably wider than 50.0 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

Circuit Analysis of Induced Current

This concept involves applying fundamental circuit laws, such as Ohm’s Law and Kirchhoff’s Laws, in the presence of an induced emf. It helps in determining both the magnitude and the direction of the current that flows in a circuit when it experiences a change in magnetic flux, such as when moving into, within, or out of a magnetic field.

Lenz’s Law

Lenz’s Law provides the direction of the induced current such that its magnetic effect opposes the change in the original magnetic flux. This principle, essential when analyzing induced currents, ensures the conservation of energy in electromagnetic systems.

Motional EMF

Motional emf refers to the voltage induced in a conductor moving through a magnetic field, resulting from the Lorentz force acting on the charge carriers. It is quantified by the product of the magnetic field strength, the speed of the conductor, and the effective length of the conductor in the magnetic field.

Magnetic Flux

Magnetic flux is the measure of the magnetic field passing through a given area, defined as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field and the area's normal. It is key in determining the amount of field interacting with a circuit as it moves relative to the magnetic field.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that a change in magnetic flux through a loop induces an electromotive force (emf) in that loop. This law is fundamental in understanding how a moving circuit, which changes the amount of magnetic flux it encloses, generates an induced voltage.

Transcript

00:01 For this problem on the topic of electromagnetic induction, we are given a retangular circuit that is moved at a constant velocity into, through, and then out of a uniform magnetic field as shown in the figure.

00:14 We want to find the magnitude and erection of the current that is induced in the circuit as it is firstly going into the magnetic field, and then secondly totally within the magnetic field, and still moving, and finally moving out of the magnetic field.

00:29 We finally want to sketch a graph of the current in the circuit as a function of time for the three cases.

00:36 Now when the loop is entering or leaving the region of the magnetic field, the flux rate is changing and there's an induced emf.

00:43 The magnitude of this induced emf epsilon is equal to the magnetic field strength times the length l times the speed of the circuit v.

00:55 Now the length l we know is 75 centimeters or 0 .75 meters.

01:03 Now when the loop is totally within the field the flux to the loop is not changing so there's no induced emf.

01:11 Now let's look at part a we know that the current that is induced is equal to the induced emf over the resistance and this is blv over r.

01:26 So we know the linear field strength to be 1 .25 tesla's.

01:34 The length of the circuit is 0 .75 meters and it's moving at a constant speed of 3 meters per second.

01:44 All of this divided by the resistance of 12 .5 oms.

01:51 So we get the induced current to be 0 .225 ampers.

01:59 So we have the magnitude of the current that is induced.

02:05 Now the magnetic field through the loop is directed out of the page and it's increasing...

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