A very long, rectangular loop of wire can slide without...

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude $B =$ 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance $R =$ 5.00 $\times$ 10$^{-3} \Omega$. The loop is initially at rest; then a constant force $F_{ext}$ = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when $v =$ 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

Key Concepts

Magnetic Damping

Magnetic damping refers to the resistive force that arises due to the induced currents (and the associated magnetic fields) opposing the motion that generated them, as predicted by Lenz’s Law. This phenomenon is important in analyzing systems where moving conductors experience a reduction in acceleration due to electromagnetic forces counteracting the applied force.

Faraday's Law of Electromagnetic Induction

This law states that a change in the magnetic flux through a circuit induces an electromotive force (emf) in the circuit. The magnitude of this induced emf is proportional to the rate of change of the flux, which can occur when either the magnetic field strength changes or the area encompassed by the circuit changes, as in the case of a moving loop in a magnetic field.

Lenz's Law

Lenz’s Law provides the direction of the induced current by stating that the induced emf always acts in such a way as to oppose the change in magnetic flux that produced it. This means that when the flux through a circuit changes, the resulting induced current and magnetic field will work to counteract the initial change, a critical principle in understanding electromagnetic braking phenomena.

Motional EMF

Motional emf arises when a conductor moves through a magnetic field, generating an emf along its length that is proportional to its velocity, the magnetic field strength, and the effective length of the conductor within the field. This concept is fundamental in problems where moving loops or rods experience induced currents due to their motion relative to the magnetic field.

Newton's Second Law and Terminal Velocity

Newton's Second Law relates the net force acting on an object to its mass and acceleration. In the context of electromagnetic induction, the balance of applied forces and the opposing magnetic force due to induced currents can lead to a terminal velocity where the net force (and thus the acceleration) becomes zero. Understanding how these dynamic forces interact is essential when analyzing the motion of objects in magnetic fields.

Transcript

00:01 For part a of our question, we're asked to find the acceleration of the loop when the velocity is equal to 3 times 10 of the minus 2 meters per second.

00:10 So an enf induced in the left side of the bar because of the motion in the magnetic field.

00:15 So this is the potential on the left side of the bar, which is given by the potential, we're just going to call it epsilon, is equal to the velocity of the bar times the strength of the magnetic field times the length of the bar.

00:29 Okay? so from that, we can find the current i, since it's equal to, according to oams law, potential divided by resistance.

00:47 Therefore, the force from the magnetic field b is equal to i times b times l, which is equal to v times b squared times l squared divided by are.

01:17 Okay.

01:18 So now that we know the force of the magnetic field, we can just use newton's second law here.

01:24 And we're going to have, let's start a new page for this.

01:29 We're going to have f external, which we were given, minus force of the magnetic field, f sub b, which we just found, is equal to mass times acceleration.

01:43 Acceleration of course being what we're trying to find here acceleration then is equal to f external divided by the mass minus v times b squared times l squared divided by the mass times the resistance so plugging all those values into the equation we find that the acceleration is equal to 4 .14 meters per second squared.

02:31 We can box that in as our solution for part a.

02:35 Part b asks us to find the terminal velocity and the acceleration when the velocity is terminal.

02:42 Well, since the terminal velocity is the velocity at which it can accelerate no more, the acceleration there a priori is known to be equal to zero...

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