An N -turn square coil with side ℓand resistance R is pulled to the right at constant speed v in

An N -turn square coil with side ℓand resistance R is pulled...

An $N$ -turn square coil with side $\ell$ and resistance $R$ is pulled to the right at constant speed $v$ in the positive $x$ -direction in the presence of a uniform magnetic field $B$ acting perpendicular to the coil, as shown in Figure $\mathrm{P} 20.66$ . At $t=0$ , the right side of the coil is at the edge of the field. After a time $t$ has elapsed, the entire coil is in the region where $B=0 .$ In terms of the quantities $N, B, \ell, v,$ and $R,$ find symbolic expressions for $(a)$ the magnitude of the induced emf in the loop during the time interval $t,(\mathrm{b})$ the magnitude of the induced current in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the field. (e) What is the direction of the induced current in the loop? (f) What is the direction of the magnetic force on the loop while it is being pulled out of the field?

Key Concepts

Faraday's Law

This fundamental principle of electromagnetic induction states that a time-varying magnetic flux through a closed loop induces an electromotive force (emf) in the loop. It is the foundational reason why moving a coil in or out of a magnetic field generates an emf, and it quantitatively relates the rate of change of flux to the induced voltage.

Magnetic Flux

Magnetic flux represents the total magnetic field passing through an area and is defined as the product of the magnetic field strength and the perpendicular area it penetrates. It plays a central role in Faraday’s Law, as changes in the magnetic flux through the loop are what drive the generation of an induced emf.

Lenz's Law

Lenz's Law explains the direction of the induced current in a loop; it states that the induced current will flow in such a way that its own magnetic field opposes the change in the original magnetic flux. This concept ensures the conservation of energy and provides a method to predict the polarity of the induced emf.

Motional EMF

Motional emf refers to the voltage generated when a conductor moves through a magnetic field. This concept is derived from the Lorentz force acting on the charge carriers in the moving conductor, and it is particularly useful in understanding how the speed of a moving loop and its dimensions contribute to the induced emf.

Ohm's Law

Ohm's Law relates the voltage, current, and resistance in an electrical circuit by stating that the voltage is the product of the current and resistance (V = IR). This principle is crucial for determining the induced current once the emf is known, by taking into account the resistance of the coil.

Joule Heating (Resistive Power Loss)

Joule heating describes the process by which electrical energy is converted into heat in a resistor. The power dissipated can be calculated using the expression P = I²R, linking the induced current and the resistance to the rate of energy loss in the circuit.

Magnetic Force on a Current-Carrying Conductor (Lorentz Force)

A current-carrying conductor placed in a magnetic field experiences a force due to the interaction between the current and the magnetic field, as described by the Lorentz force law. This concept is essential for understanding the mechanical force required to move the coil against the magnetic forces that arise from the induced current.

Transcript

00:02 For part a of our question, we're asked to find a symbolic expression for the magnitude of the induced emf in the loop during the time interval t.

00:12 Okay.

00:15 So to do this, we're going to use faraday's law of induction, where the induced emf is defined as the rate of change of magnetic flux enclosed in the circuit.

00:27 So for a coil with in number of turns, the induced emf is going to be expressed as follows.

00:35 Epsilon is equal to minus the number of turns in the coil times the change in magnetic flux delta phi over the change in time delta t okay well the magnetic flux phi is equal to the magnetic field b multiplied by the area but the area here it's a square with uh with sides of length l so it's going to be magnetic field times l squared.

01:10 And where the velocity here, we'll call it v, of course, is just distance divided by time.

01:17 Distance here traveled is l divided by time, t.

01:21 So we can go ahead and solve for t because we want t in our expression of the induced dmf, where t is equal to l divided by velocity v.

01:33 Therefore, the induced dmf, epsilon, is equal to minus, the number of turns in the coil, multiplied by the magnetic field times l squared.

01:46 And the reason here that it's not delta phi is because we're told that the initial magnetic field is zero, initial time is zero.

01:56 So those values just go away.

01:58 So we're left with the expression for phi, which is b times l squared, divided by the expression for t, which is l over v.

02:07 So the l on the bottom cancels the squared l on top, so we end up with the number of turns of the magnetic field times b.

02:20 And this is, there's actually a minus here.

02:23 So those minus signs will cancel.

02:26 So number turns in the magnetic field times b, which is the magnetic field.

02:30 Let's make that expression look a little nicer here for b.

02:38 I can read it.

02:40 There we go.

02:42 And again, there's a minus sign here.

02:43 So let's rewrite that.

02:46 Okay.

02:46 So times the magnetic field times the length times the velocity v.

02:54 And we can go ahead and box that in as our answer for part a.

03:01 So then for part b, we are asked to find an expression for the magnitude of the induced current in the coil.

03:10 Okay, so let's get a new page going.

03:14 So this is part b.

03:16 The magnitude of the induced current.

03:18 So the current here, i using oams law, is equal to the induced dmf divided by, the resistance.

03:27 Well, we just found the induced emf.

03:28 The induced emf is the number of turns in the coil times the magnetic field, times the length, l, times the velocity, v, and then, of course, this is just divided by r.

03:42 And that there is our expression.

03:43 So that was simple enough.

03:44 We can box it in as our solution for part b.

03:49 Now part c.

03:51 For part c, we're asked to find an expression for the power...

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