Faraday's Law - Online Tutor, Practice Problems & Exam Prep

Faraday's law describes electromagnetic induction, stating that the induced electromotive force (emf) is proportional to the rate of change of magnetic flux through a coil. The equation is expressed as ε = n×|ΔΦΔt|, where n is the number of turns, ΔΦ is the change in magnetic flux, and Δt is the change in time. Changes in magnetic field, area, or angle can alter magnetic flux, leading to induced currents in circuits, as demonstrated through practical examples.

Faraday's Law

Hey, guys. So in this video, we're going to talk about Faraday's law, which is the mathematical equation for electromagnetic induction. This is a super important, very critical topic in electromagnetism, so pay attention and let's get to it. So we saw that changing a magnetic field through conducting loops was able to create an induced EMF. Now, what's actually happening here is when you're changing the magnetic field through a conducting loop, so we had some loops like this and you're changing the magnetic field, what you're actually changing is you're changing the magnetic flux. And so we saw that the higher that you change the magnetic flux, the larger the induced currents. So that's that delta φ_B. That's that magnetic flux right there. We also saw that faster changes produced higher induced currents and higher EMFs. So that means there's also a relationship of delta t, which is the amount of time that it takes. So there's clearly a relationship between these variables and induced EMF or the induced current in a coil of wire. And the mathematical relationship that describes those two variables is called Faraday's law.

Faraday's law tells us that the induced EMF is the rate at which the magnetic flux changes over time. Now this EMF is actually responsible for producing an induced current. And by the way, this is just v equals I r. This is just Ohm's law, which we've seen before. But this EMF, more specifically and more importantly, is related to n, which is the number of turns in a coil times the absolute value of the change in the magnetic flux over the change in time. This n | Δ φ φ t is Faraday's law and it's super important. We're going to be talking about this in the next couple of videos. Now the units for this are actually just volts because remember, at the end of the day, this is just an EMF. It's just a voltage. So basically what it tells us is that depending on the number of turns, which is n in a turn in a coil of wire, if you have a magnetic field that is going through this coil that produces a flux, if that flux changes with time, then it creates an EMF that is across these coils. So it induces some EMF, which produces an induced current.

So in other words, you have some current that will go around the coil like this. Okay? So this is very, very important, and you definitely need to know this. So how does this actually work? Well, remember that the flux will change depending on 3 variables. The flux is b a cosine of theta, which means that there are 3 ways to change the magnetic flux. You can either change the magnetic field, you can change the area, or you can change the angle of the magnetic field and the area. So the way that these problems are going to go is that in all of these problems, one variable will change, whether it's b, a, or theta, while the other 2 remain constant. So what we have to do is we just have to identify what is that changing variable in each of the problems, and then just go ahead and use that equation for the changing flux. Alright. Let's get to it. Let's get, let's take a look at a couple of really, really common examples that you might see. So in one example, you'll have a sort of loop given by this square like this. You'll have a magnetic field. In this case, the magnetic field is pointing outwards like this. So you have some magnetic field. And then at some later time, delta t, you're going to have more field lines, which means that the magnetic field has changed. So when this situation happens, is that the b field has changed, so this is a changing b field. But if you take a look here, the area of the loop has remained constant, and the angle at which the magnetic field and the area, the angle between those two has also remained constant. So this is the variable that changes. Let's take a look at a different example. So now what we have is we have the same amount of magnetic field lines given by those little circles. The area is still coming out at you. There the magnetic field is. But the thing that's different is that the loop has now changed area. So you have some initial area of the loop, and then at some later time, you have a final area of the loop. So what happens is the area in which the magnetic field line has are coming out is changing, while the other 2 are remaining constant. The magnetic field is the same and the angle between them is the same as well. So this is an example of changing area. Now let's take a look at the third example. Here, what happens is we have the same amount of magnetic field lines in the before and the after, and the size of the loop is the same, so you have the same amount of area. But what's different between these is that in this case, we have some angle initial that the magnetic field in the area are making, and then at some later time, delta t, you have some other angle, so that's theta. So what's happening between here is that the angle is changing. So this is an example of changing angles. And so now what happens is your cosine of theta is going to change, while the other two remain constant. So all of these things are constant here. So it's your job to kind of figure out which one of these three scenarios a problem falls under. Alright. So let's go, let's take a look at an actual example of this and see how this works. So in this example here, we've got an EMF in the following circuit. We're told what the area of this loop of the circuit is, which by the way is in that blue line. And we're told that the magnetic field is going to change from 3 tesla to 6 tesla in 5 seconds. So let's take a look at the first part of the problem. We're supposed to figure out what is the induced e m f in the circuit. So that induced e m f is our variable e induced. So we're going to relate this to n, which is the number of turns times the absolute value of the change in the magnetic flux divided by the change in time. And so the key thing here is Let's take a look at our variables. We have this loop here, or this circuit, and we're told that it just basically goes around once. What? We're not actually told that, but we can sort of, infer that because it doesn't tell us the amount of turns in the circuit, that n is just equal to 1. So n is 1 in this case, which just means that we can just replace this with a one. And we know the amount of time that it takes in order for some change to happen. That's going to be 5 seconds. So this is delta t right here. So we have that. So the key things in these problems is we have to figure out what the magnetic flux change is. So let's go ahead and figure that out. So delta φ_B is actually what we're trying to find. Now, we know that φ is equal to b a cosine theta, so that means delta φ is going to be delta b a cosine of theta. Right? So we have 3 variables, and we just have to figure out which is the one that's changing. Well, let's take a look. The magnetic field is changing from 3 Tesla to 6 Tesla. So that means that our b is actually the changing variable, so this changes. And then these guys, the area and the cosine of theta, these are actually going to be constants. So what we do is we actually just pull those things out of the delta. But first, we can also just figure out what the cosine of this theta angle is. Now let's see. The magnetic field points into the page. Right? So into the page like this. And the area or of this circuit here is like flat on the page. What that actually means is that the cosine of the angle is just equal to 1. So in other words, both of these things, the area vector and the magnetic field, they point in the same direction. So we can kinda just eliminate that and that just goes to 1. So that means that the change in the magnetic flux is going to be the area times the delta b. So in other words, this is going to be the area times the final magnetic field minus the initial magnetic field. So now what we do is we can just plug this back into our expression for the induced EMF. So our induced EMF is going to be the absolute value of the area, and we're told that the area is 50 centimeters squared. So we actually have to do point 0 5 meters squared. And now we have to do the in the final magnetic field, which is 6 Tesla, minus the initial magnetic field 3 Tesla, and then the delta t is equal to 5 seconds. So that is going to be our induced EMF, and what we get is we get an induced EMF that is equal to 0.03 volts. And by the way, that is a positive number because the absolute value takes care of that. Right? Even if it was negative or positive, it's always going to be positive. Alright. So that is the answer to part a. So part b now asks us, what is the induced current if the resistor has a resistance of 2 ohms? So let's take a look here. So remember that we're our induced EMF can always be related to an induced current using Ohm's law. E equals IR or epsilon equals IR, because really this is just v equals I r right here. Right? So all you have to do is just move the resistance over, and so the induced current is just going to be the induced voltage or the induced EMF of 0.03 divided by 2. So we just get an induced current of 0.06, and that's going to be in amps. So these are our two answers. Alright, guys. So we're going to get some more practice in the next couple of videos. Let me know if you guys have any questions.

A tightly-wound 200-turn rectangular loop has dimensions of 40cm by 70cm. A constant magnetic field of 3.5T points in the same direction as the normal of the loop. If the dimensions of the loop change to 20cm by 35cm over 0.5s, with the number of turns remaining the same, what is the induced EMF on the rectangular loop?

• A. 48 V
• B. 97 V
• C. 294 V
• D. 588 V
• E. 0 V

Current in a Circuit with a Changing Magnetic Field

All right, guys. We're going to get some more practice with Faraday's law, and we're going to work this one out together. So, we have a small circular loop of some radius with some resistance. It's centered inside of a larger loop with some bigger radius r equals 5. We're told the initial current. And then what happens is this large loop gets disconnected from its voltage source and the current steadily decreases. And we're supposed to figure out what is the change in the magnetic flux throughout this time. So for part a, what we're really trying to find is not the induced EMF. We're actually just looking for what is the change in the magnetic flux. So in other words, what is the change in B times A times cosine of theta? That's going to be the first part. But before we dive into the math, let's just go ahead and draw a diagram of what's going on. We have this larger circular loop that we're told it's going to look something like that, and then we have an inner circular loop that's going to be inside of it. So, obviously, it's not going to be to scale or anything like that. And we're told that the radius of this small loop is 5 centimeters. So in other words, I'm going to say this is r_s for r small, and this is equal to 0.05 m. And then the larger ring here, which is going to have r_big, and that's going to equal 5 meters.

We have to figure out what the changing variable is in order to figure out the magnetic flux. So, let's start at the top or the left, I guess. What is the magnetic field? Where does the magnetic field even appear in this equation? We're never told anything about a magnetic field. So, we're told that the larger loop carries an initial current of 6 amps. In other words, it's going to be in either this direction or that direction. This is the current, and that's equal to 6 amps. So, where do we get our magnetic field from that? Well, remember that current-carrying wires always produce magnetic fields. So, in order to find out what direction that magnetic field is, we're going to use our right-hand rule. So, there is a B-field that is generated somewhere inside of this loop, and we're going to use the right-hand rule for this. If you take your right hand and you curl your fingers in the direction of the current, and by the way, the direction doesn't matter. So we're actually just going to go ahead and choose clockwise to be the direction. The truth is it doesn't matter because we're only looking for the change in the magnetic flux. So, I'm just going to curl my fingers in the direction of that current, and then my thumb should point away from me. It's going to look exactly how my hand looks on the screen right now. So, what happens is that this magnetic field actually points into the page like this. So, we have a magnetic field that points inwards, and that magnetic field is going to be constant inside of the wires that point inwards.

Now what about the formula for the magnetic field? That magnetic field is going to be μ₀ times I divided by 2 times r. But which radius are we going to use? We're talking about the larger loop, so we're actually just going to use this r_big equation right here. By the way, this equation we've used before for the center of a loop of current. So, you should have this in your notes somewhere.

Now that we have the magnitude of this B-field and the direction, now we just need to figure out how the magnetic flux changes. So, we need to identify which one is our changing variable. What's happening is that this larger loop is disconnected from its voltage source, and the current is going to decrease to 0 over some time. So, what's happening during these 20 microseconds? Is the area changing, is the angle changing, or is the B-field changing? The area is going to be the area of the smaller loop, but we're not told anything about that changing area. That area is just equal to radius small, which is 5 centimeters, and the angle doesn't change either. We're not told that this ring rotates or anything like that. So, what happens is our magnetic field is the variable that's going to be changing. Because if you take a look at this equation, the magnetic field strength is proportional to the current going through the larger loop. So, as this current I decreases, the magnetic field is also going to be decreasing. As the current around this loop starts to go to 0, the magnetic field strength will also start to decrease. So B is our changing variable. So, that means that our δφ is going to be. We can pull these out of the δ, so we can pull these to the outside like that. And we're going to have A times the cosine of theta times δB, which is equal to B_final minus B_initial.

As for the area, so we have δφ is equal to the area, and the area we're going to take is the area of the small loop because that is the flux that we're trying to evaluate. So, we're actually going to use π times rs squared. Now, how about the cosine of the angle? Well, the normal of this small loop actually points into the page as well. So if the normal points into the page and the magnetic field points into the page, then the cosine of this angle is equal to just 1 because θ is equal to 0.

And now what we have is the magnetic field final. Well, remember that that magnetic field has an equation. And because we have a final current and an initial current, we're going to substitute these two equations. So, I'm going to have μ₀ and this is going to be I_initial divided by 2 r_big minus μ₀ I_final divided by 2 r_big. So, this is actually the expression for the changing magnetic flux. We're going to have to evaluate the final magnetic field and the initial magnetic field. Well, we're told that the current will steadily decrease to 0 over some time. So that means that this whole entire thing will go to 0 because I_final is equal to 0. So that whole entire term drops away.

Alright. So that means that the change in the magnetic flux is equal to now we've got π times the radius, the small radius which is 0.05 m squared, and then we have times negative, μ₀ is just equal to 4π times 10 to the minus 7. That's the magnetic permeability. Now, the initial currents is equal to 6 amps. And the 2 times the r_big, the r_big is just 5 meters. So if you work this out, you're going to get that the change in the magnetic flux is equal to negative 5.92 times 10 to the minus 9 and that's Webers. So that's the change in the magnetic flux. And notice how there's no absolute value that we have to take into account because it's actually just asking for the change in the magnetic flux.

Alright, so this is the answer to part a, what's the change in the magnetic flux? So, let's move on to part b. Part b is now asking us, what is the magnitude of the induced EMF? So, now we actually are going to take this EMF and use Faraday's law, and that's going to be the n, which is the number of turns in this loop, times the absolute value of the change in the magnetic flux divided by the change in time. Now, this circular loop here, we're not told that it has any turns, so we're just going to assume that the amount of turns, this n, is just equal to 1. So, this is just a 1. And the induced EMF is just going to be the absolute value of the change in the magnetic flux, which is 5.92 times 10 to the minus 9. And we're going to divide that by the change in time, which we're told is 20 microseconds. So, that's 20 times 10 to the minus 6, and you have to take the absolute value. So when you do that, when you work this out, you're going to get 2.96 times 10 to the minus 4 and that's in volts. You're going to get a negative number, but that has to be positive because of the absolute value. So, that's the answer to part b. Now we're almost done here. For the last part, we just have to figure out what is the induced current on the smaller loop. So, remember, when you're trying to find out what an induced current is, you have to relate that back to the induced EMF divided by by the resistance, which comes from Ohm's law.

So, the induced EMF is 2.96 times 10 to the minus 4, and the resistance is equal to, let's see. We've got 10 milliohms, so that's actually 0.010 and that's in ohms. By the way, this is a 4 that's volts. So, this should give us a current of 0.0296 amps. So, that's just kind of proportional or just roughly equal to 0.03 amps. Okay, guys. There's kind of a long problem. There were a lot of steps, but if you work the steps out and you kind of just work backward in the magnetic field and the currents and use Faraday's law, you should be able to figure it out. Let me know if you guys have any questions.

A square conducting wire of side length 4 cm is in a 2 T magnetic field. It rotates such that the angle of the magnetic field to the normal of the square increases from 30° to 60° in 2 s. What is the induced current on the wire if its resistance is 5 mΩ?

• A. 0.85 A
• B. 0.12 A
• C. 1.2 A
• D. 1.39 mA

Here’s what students ask on this topic:

What is Faraday's Law of Electromagnetic Induction?

Faraday's Law of Electromagnetic Induction states that the induced electromotive force (emf) in a coil is proportional to the rate of change of magnetic flux through the coil. Mathematically, it is expressed as:

$\epsilon =n\times \frac{\left|\mathrm{\Delta \Phi }\right|}{\mathrm{\Delta t}}$

where $\epsilon$ is the induced emf, $n$ is the number of turns in the coil, $\mathrm{\Delta \Phi }$ is the change in magnetic flux, and $\mathrm{\Delta t}$ is the change in time. This law is fundamental in understanding how electric generators and transformers work.

How does changing the magnetic field affect the induced emf according to Faraday's Law?

According to Faraday's Law, changing the magnetic field through a coil induces an emf. The induced emf is directly proportional to the rate of change of the magnetic flux. If the magnetic field $B$ changes, the magnetic flux $\Phi$ (which is the product of the magnetic field, the area of the coil, and the cosine of the angle between them) also changes. This change in flux induces an emf in the coil. Faster changes in the magnetic field result in a higher induced emf.

What are the three ways to change the magnetic flux in a coil?

There are three ways to change the magnetic flux $\Phi$ in a coil:

1. Change the Magnetic Field ($B$): Varying the strength of the magnetic field passing through the coil.
2. Change the Area ($A$): Altering the area of the coil that is exposed to the magnetic field.
3. Change the Angle ($\theta$): Modifying the angle between the magnetic field and the normal to the area of the coil.

Any of these changes will result in a change in magnetic flux, thereby inducing an emf in the coil.

How is the induced current in a circuit related to the induced emf and resistance?

The induced current $I$ in a circuit is related to the induced emf $\epsilon$ and the resistance $R$ of the circuit by Ohm's Law, which is expressed as:

$\epsilon =IR$

Rearranging this equation to solve for the current, we get:

$I=\frac{\epsilon }{R}$

This means that the induced current is equal to the induced emf divided by the resistance of the circuit. Higher resistance results in a lower induced current for a given emf.

What is the significance of the number of turns in a coil in Faraday's Law?

In Faraday's Law, the number of turns $n$ in a coil is significant because it directly affects the induced emf. The law is expressed as:

$\epsilon =n\times \frac{\left|\mathrm{\Delta \Phi }\right|}{\mathrm{\Delta t}}$

Here, $n$ represents the number of turns in the coil. A higher number of turns increases the induced emf for a given rate of change of magnetic flux. This is because each turn of the coil experiences the same change in flux, and the effects are cumulative.