A square conducting loop of wire of side a, mass m and...

A square conducting loop of wire of side $a$, mass $m$ and resistance $R$ falls under gravity normally to a uniform, horizontal magnetic field $B$, as shown in Figure $7.22 .$ The area of the loop is small enough so that it can be contained totally within the region of the field. (a) While the loop is entering the magnetic field, find the magnitude and direction of the induced current. (b) Write down an expression for the acceleration of the loop as it is entering the field. (c) Write the acceleration as the time derivative of the velocity and hence show that after time $t$ the velocity of the loop is $$v=\frac{g m R}{B^{2} a^{2}}\left[1-\exp \left(-\frac{B^{2} a^{2}}{m R} t\right)\right]$$ where we have assumed for simplicity that the loop is dropped from rest from a position where the bottom leg is just about to enter the region of magnetic field. (d) Show that the quantity $\frac{k m R}{B^{2} a^{2}}$ has units of velocity and $\frac{m R}{B^{2} a^{2}}$ units of time.

A square conducting loop of wire of side $a$, mass $m$ and resistance $R$ falls under gravity normally to a uniform, horizontal magnetic field $B$, as shown in Figure $7.22 .$ The area of the loop is small enough so that it can be contained totally within the region of the field.
(a) While the loop is entering the magnetic field, find the magnitude and direction of the induced current.
(b) Write down an expression for the acceleration of the loop as it is entering the field.
(c) Write the acceleration as the time derivative of the velocity and hence show that after time $t$ the velocity of the loop is
$$v=\frac{g m R}{B^{2} a^{2}}\left[1-\exp \left(-\frac{B^{2} a^{2}}{m R} t\right)\right]$$ where we have assumed for simplicity that the loop is dropped from rest from a position where the bottom leg is just about to enter the region of magnetic field.
(d) Show that the quantity $\frac{k m R}{B^{2} a^{2}}$ has units of velocity and $\frac{m R}{B^{2} a^{2}}$ units of time.

Key Concepts

Faraday's Law of Induction

This fundamental principle states that a changing magnetic flux through a closed circuit induces an electromotive force (emf) in the circuit. In problems where a conducting loop moves in or out of a magnetic field, the rate of change of the magnetic flux—due to the motion of the loop—determines the magnitude of the induced emf according to Faraday’s Law.

Lenz's Law

Lenz's Law provides the direction of the induced current by stating that the induced current will flow in such a direction that its associated magnetic field opposes the change in the original magnetic flux. This concept is key in determining how induced currents act to oppose the motion of the loop, effectively providing a damping force.

Lorentz Force

The Lorentz force is the force that a current-carrying conductor experiences when placed in a magnetic field. It is fundamental in explaining how the induced current in the loop interacts with the magnetic field to create a force that opposes the motion, thereby linking the electromagnetic effects with mechanical motion.

Electromagnetic Damping

Electromagnetic damping refers to the resistive force that arises from the induced currents (eddy currents) in a conductor moving relative to a magnetic field. This effect, which is a direct consequence of Faraday’s and Lenz’s laws combined with the Lorentz force, acts to reduce the acceleration of the moving loop and can lead to a terminal velocity where gravitational pull and electromagnetic drag balance each other.

Newton's Second Law in the Context of Electromechanical Systems

Newton’s Second Law (F = ma) is applied to the moving loop, taking into account both the gravitational force and the opposing electromagnetic force resulting from the induced current. The formulation and solution of the resulting differential equation allow one to derive the velocity as a function of time, which is characteristic of systems exhibiting electromagnetic damping.

Dimensional Analysis

Dimensional analysis is used to verify that the derived expressions are consistent with the physical units expected for the quantities in question. For example, checking that certain combinations of parameters have the correct units for velocity or time ensures that the equations are physically meaningful and correctly formulated.

Transcript

00:01 Our goal is to determine the induced current and subsequent motion of a square loop of wire that is going to be dropped into a uniform horizontal magnetic field.

00:16 So there are a couple things that we are going to need.

00:20 In order to figure out subsequent motion, we're going to need that the force on that loop is coming from the magnetic field, as well as the induced current in the loop.

00:35 And that force is i -l -cross -b, with the current flowing perpendicular to the magnetic field.

00:45 The other thing that we need is faraday's law of induction.

00:54 And this says that there will be a voltage induced in the loop, or an emf, if we want to think about it, that is coming from the change in magnetic flux in time.

01:11 And that change in magnetic flux is basically the change in the magnetic field poking into the area of the loop.

01:23 That's usually a dot product, but here they are parallel.

01:29 So working with that faraday's law, we can, for the more, use oms law.

01:37 And equate the induced emf to the current times the resistance of the wire, which i've represented with a resistor.

01:48 The magnetic field is not changing, so we can pull that out.

01:53 And essentially, the area is a times y, with y being the amount of the loop that's going to extend down into the magnetic field.

02:11 So y is the dimension that is going to change as that loop drops.

02:17 So we have b times d -y by d -t, and that is simply b times the y component of the velocity of the wire.

02:30 So we've already accomplished one thing.

02:32 We can write down the, oops, and there's an a in there, a -y, a -v -y.

02:42 So we've already accomplished figuring out the magnitude of the current through that loop as it drops.

02:58 And it is proportional to the velocity, which is going to get bigger.

03:03 So seeing that the force is proportional to this current, we can see that there's going to be some sort of velocity -dependent force acting upon the loop.

03:18 Now let's figure out the direction of the current.

03:21 So to figure out the direction of the induced current, we are going to use lentz's law, which says that the current will try to oppose the change in the flux.

03:43 And the way i think about that is the flux is changing in a positive way.

03:49 As that loop drops, there'll be more and more area getting a cover.

03:53 By the magnetic field.

03:56 The loop wants to decrease the flux, so it will actually have an induced magnetic field opposite the external one.

04:08 And i can use my right -hand rule, stick my thumb into the plane of the loop.

04:14 The fingers on my right hand then wrap around in a clockwise direction.

04:22 And that's the direction of i.

04:24 I will be clockwise as viewed in the diagram i've shown.

04:36 Okay, now this is going to be nice because in order to find the subsequent motion, we'll have to drag out newton's second law with the sum of the forces on the loop, and these are going to be in the y direction, is mass times acceleration in the y direction.

05:05 And not too surprisingly, if we use the right -hand rule, we can figure out the direction of the magnetic force from i, l, which is along the segment of the lower part of the loop towards the left, l for left, cross b.

05:36 And that direction is upwards.

05:39 So if we draw the loop as a point, it has two simple forces on it.

05:50 It's got m -g downwards, pulling it downwards.

05:54 And it has the magnetic force i, l, b, pointing upwards.

06:04 And using newton's second law, we can then say, calling positive i down, m -g minus i -l -b, and actually the length is a, is equal to m -a -y.

06:30 And here's where we can see that the current is proportional to the velocity, and put that into our equation.

06:42 And we get b -a -squared, b -y over the resistance equals m -a -y.

06:53 So this is going to be our equation that we want to start with.

06:57 And there are a couple things to clean up here.

07:00 It looks fairly complicated.

07:03 But we know that the acceleration is the time rate of change of the velocity...