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

Faraday's law states that a changing magnetic flux induces an electromotive force (emf). Lenz's law determines the direction of this induced current, which opposes the change in magnetic flux. For example, if a bar magnet moves downward, the induced magnetic field points upward, resulting in a counterclockwise current. Conversely, if a loop exits a magnetic field, the induced current is clockwise. The relationship can be expressed as $\epsilon =-n\times \frac{\Delta }{\Phi }/\Delta t$.

Lenz's Law

Hey, guys. So in the last couple of videos, we saw that a changing magnetic flux through a surface produced an EMF, and that was called Faraday's law. Well, in this video, we're gonna talk about the direction of those induced EMFs and currents. So let's check it out. Again, Faraday's law gives us the magnitude of the induced EMF and the currents. So in other words, we could relate the EMF to the current by using Ohm's law. But one of the things we haven't talked about yet is the direction. So we're gonna use another law called Lenz's law to find the direction of the induced currents. And basically, what Lenz's law says is that the direction of the induced current creates an induced B field that opposes any change in magnetic flux. So the keywords here are opposes the change. So whatever the change in the magnetic flux is, Lenz's law says that the direction of the induced current is gonna set up something that counteracts that change. So we're always gonna be looking for what the system is doing, and then we wanna reverse that change. Now we're gonna be using magnetic fields and we're also gonna be using directions. So we're gonna be using our right hand rule for circular currents. So if you use your right hand so if you take your right hand like this, our thumb is going to point in the direction of the B field because that's gonna be the straight line. And so that's gonna be our magnetic field here. And then what happens is our fingers are going to wrap around in the direction of the induced current. So that's the right hand rule we're going to be doing for this. We're going to take it really slow on each one of these examples because I really want you guys to understand this. Now, in fact, you might actually see Faraday's law represented as

ε = -n Δφ / Δt.

Where this negative sign, remember negative signs in physics just to do usually indicate a direction, this neg this negative sign is really what Lenz's law is all about. It just says that whatever the change in the magnetic flux is, the epsilon is gonna be the negative that in the opposite direction. So this sort of wraps up, Faraday's law and Lenz's law all into one neat equation. Okay. So let's check out a quick example here. We've got a bar magnet that is moving downwards, and this is something we saw in the first couple of videos. We saw that a moving bar magnet will generate an EMF. Well, now we're actually going to talk about the direction. So we have a bar magnet that's moving downwards. So remember that the north pole of a bar magnet means that the magnetic field lines go down here like this. So they actually form these little loops like that. So that's what a magnetic field looks like from a bar magnet. Now what happens is through the loop itself, the magnetic field is pointing downwards because these magnetic field lines sort of spread out like that, right? So that means that the magnetic field through the surface points in the downward direction. So these are gonna be the steps that we follow for every single one of these problems. So the magnetic field is gonna point downwards. Now what happens is the change in the magnetic flux comes from the fact that the bar magnet is moving with some velocity. So as the bar magnet is moving downwards, this magnetic field is getting stronger and stronger through the loop. So that means that the magnetic flux, which remember depends on B times A times cosine of theta, Which of the variables is changing? Well, that's actually going to be our magnetic field variable that is changing because it's getting stronger. So what happens is this magnetic field gets stronger and the change in the magnetic flux is positive. So Lenz's law says that there's going to be an induced current that wants to counteract that change. So that means that for the direction of B induced, what we have to do is we have to figure out what direction would counteract that change. What direction of the magnetic field induced will want to bring the system back to the way it was before? Well, the magnetic field changed from here, and then it got stronger in this direction. So that means that the induced magnetic field sort of wants to fight that change. So this is the direction of B induced. It points upwards to counteract that change in magnetic flux. So that means that the B induced is gonna be upwards. Now what does that tell us about the direction? Well, in order to get the direction from a magnetic I want you to put your hand over your page like this, and then we're gonna change it so that our thumb points in the direction of that magnetic field induced. And then I want you to wrap your fingers around like that. So what you should see, and I want you guys to do this on your page every single time, is that your fingers sort of curl away from you on the right and then towards you on the left. So what that means is that if we actually take a look at this magnetic field so what this magnetic field is doing, the induced current is gonna go in this direction and then around this way sort of on the front side. Right? So that means that if you were to look at this from the top, this actually is going to be a counterclockwise current. So this is going to be counterclockwise. Alright? And that's how Lenz's law works. We're going to check out another example in which we don't have a bar magnet that's moving, but we have a loop that's moving in or out of a magnetic field. Okay. So now what happens is we have a bunch of magnetic field lines that are passing through the surface, but the loop is sort of going outside of that magnetic field. So we're start off with the same two questions. Where is the magnetic field pointing through the surface? Well, clearly, it's pointing downwards, so that we're gonna do that. That's downwards. Now how about the change in the magnetic flux? What's the change in magnetic flux? Well, remember this depends on 3 variables. So we have B times A and then cosine of theta. Well, the magnetic field strength remains the same. We have the same amount of magnetic field lines. But the area is actually changing in this one because the amount of area that the magnetic field passes through is this, you know, big surface here in the initial case and then it's the smaller surface here in the final. So what happens is this is our changing variable, which is the A, and this area is getting smaller. So that means that the change in the magnetic flux is actually negative. So this is going to be negative right here. So the induced field, so the B field that's induced, wants to counteract that change. Now, the magnetic field points downwards, but the flux is getting smaller. So the magnetic field wants to do the opposite of that. So it actually wants to induce a field that is going to sort of strengthen that weakening magnetic field in this direction. So it sort of wants to bring it back to where it was before. So if it's downwards and getting smaller, the magnetic field wants to be downwards to bring it back to the way it was before. Okay. So that means that the induced field is going to point downwards. So what does that tell us about the induced currents? Well, now we have to use our right hand rule again. So go get your right hand. Now put it over the page like this. And what you should see is that when you point your thumb in the downwards direction, exactly like what I have here, and wrap your fingers around, you should see that they point away from you on the left and then curl towards you on the right. So they're gonna be doing this. Right? So go ahead and work that on your papers. You're gonna be doing this on your test, and it's gonna look a little silly. So point your thumb downwards, your fingers curl around away from you on the left and towards you on the right. So what that means is that this magnetic field is actually pointing in this direction on the left and this direction sort of on the right side. So if you were to view that from the top, this actually would look like a clockwise current, right? So if you view it from the top, it's clockwise. Alright, guys. That's basically it. So we're gonna do a couple more examples because really this is all just about practice. It's very straightforward when you actually get the hang of it. Cool. So in the following scenarios, we're gonna find the direction of the current induced on the conducting wires. Okay. So we got a bar magnet again. So we always wanna identify the north pole, and that's where the magnetic field is gonna be coming out of. Right? So now we have a magnetic field that points in this direction. So that means that through the surface, the magnetic field points up. So as this magnet is moving upwards, how is the magnetic flux changing? Well, we have B, and we have A, and we have cosine theta. What happens is as the bar magnet gets closer, the magnetic field is gonna get stronger and stronger. So it's gonna keep getting stronger like this. So that means that the B field is our changing variable and it's gonna be positive because it's getting stronger. So the induced field wants to do the opposite of that. It's upwards and getting stronger, so the magnetic field actually wants to point downwards to bring it back to the way it was before. So it bring back to the original state. So that means our downward magnetic field is going to be the or our downward field is gonna be induced. So we have this downward, and we just get out our right hands again, and we say, okay. Well, we have a downwards pointing magnetic field, so that means that the current is gonna go away from you on the left and then towards you on the right. So it's gonna be doing this. Okay? So that means that the magnetic field lines are gonna be going this way and this way. So that's I induced, and that is actually going to be a clockwise current when viewed from the top. And for this final example right here, we've got a bar magnet that's moving now away. The magnetic field is going to come out of the north pole like this. So that means that through the surface, it's going to be pointing downwards like that. So we have this. And as the bar magnet is now moving away from this loop, the magnetic flux is actually going to be decreasing. So in this case, when the bar magnet was moving toward, the magnetic flux was increasing. So if it's moving away, then that means the magnetic field gets weaker and the flux is smaller. So the magnetic field induced is going to want to counteract that change. So it's it's downwards and the flux is getting smaller. So the induced current or the induced B field actually wants to bring it back to the way it was by strengthening or sort of by reinforcing that existing magnetic field. So that means that the magnetic field induced points downward, and we already know what direction that that current actually goes in. A downwards induced magnetic field produces a current that goes in this direction using the same exact right hand rule that we just used for the other example. Okay? So that means that this is I induced and this is going to be clockwise. Alright. So hopefully you guys got the hang of this. There's actually a pattern here if you haven't noticed. So when we had a strengthening magnetic field, so in this case when the B field was downwards and the flux was increasing, the magnetic field did the opposite of that. Here, it did the exact same thing. When the magnetic field was upwards and the flux was positive, so in other words getting stronger, the magnetic field did the opposite. So that means that the induced magnetic field is always gonna do the opposite and be pointed opposite the increasing magnetic field. And here in these situations, when the the in other words, the So the in other words, the induced magnetic field is always going to be directed along a decreasing magnetic field to restore it back to the way it once was. Alright, guys. That wraps up our discussion on lenses law. We're gonna do a couple more practice problems. Let me know if you guys have any questions.

An outer ring is connected to a variable voltage source. If the battery's voltage is continuously INCREASING, what is the direction of the induced current in the inner ring, centered inside of the outer ring?

• A. Clockwise
• B. Counterclockwise
• C. Into the page
• D. Out of the page

Lenz's Law for a Long Straight Wire

Alright, guys. Let's work this one out together. We have a long straight wire on a horizontal surface in the xy plane, and it carries a constantly increasing current in the plus y direction. So now we have a square loop, and we need to figure out what the direction is of the induced current. So that should automatically tell you this is going to be a Lenz's law problem. So the first thing I like to do in these kinds of problems where I'm given sort of like a 3D perspective is just draw out sort of what's going on here. So we have this axis right here, which we're told is the plus y direction, and we're told directly on the side of it, on the right, that is the plus x direction. So in other words, sort of I have a straight line like this, and this is going to be my x direction. Alright? So it's kind of weird. So that means that this direction was sort of like the z-axis. So I've got some three-dimensional stuff going on here. So what I always like to do in these kinds of situations is see if I can change my perspective a little bit and sort of draw, like, a parallel or a different diagram. So if I were to view this current from the side, then basically what would happen is, if I were to view along the axis of the current, then that means that the current would be going away from me like this, and then that means the square loop would be sort of on the right like that. So I'm looking sort of, like, perfectly on the side on the axis like that. Okay. So we need to find out what the direction of the induced current is. We need to use Lenz's law, but there are a couple of things we need to figure out first. So we need to figure out what the direction is of the magnetic field through the square loop, and then we need to figure out how the magnetic flux is changing. So in other words, what's delta phi b? That'll tell us what the direction is of the induced magnetic field. Okay?

So we need to do is first we need to figure out what the magnetic field is. So for a straight current-carrying wire, we know that there is some relationship between the current in that wire and the magnetic field that it generates. Now you don't have to necessarily remember this, but remember that the magnetic field for a straight current-carrying wire is μ₀I/2πr. Okay. So the direction of our magnetic field is going to be given by our right-hand rule. So now what happens is our thumb points in the direction of the straight thing, in this case the current, and our fingers will curl in the direction of the magnetic field. Alright. So get out your right hands, and what we're going to do is we're going to point our thumbs in the direction of that current. So in other words, our thumbs are going to point in this direction. But remember that we're sort of viewing this from a weird perspective. So what I like to do in this situation is we're going to use the side view. So in other words, we're going to be looking at what's going on in this diagram right here. So what happens is we need to point our thumbs in the direction of the current, in this case, and what you should see is that you should be pointing your thumb into the page away from you, and your fingers will be curling in the direction of the magnetic field from that wire. So what that means is that our magnetic field lines actually curl clockwise. So what that means in this sort of diagram is that in the first diagram on the right, I have magnetic field lines that are going like this. But on the side view, now what happens magnetic field lines that are sort of curling around like this. And so we know that they're given by our right-hand rule, they're going to go clockwise or sorry, they're going to go clockwise like, wait, no. That's right. Yeah. So clockwise like that. So what that means is that the magnetic field is going to be pointing in this direction through the square loop. So in other words, the magnetic field through the square loop points downwards. So that is the direction of b. So that's our first question. The magnetic field through the square loop points downwards. So now what we need to do is to figure out what the change in the magnetic flux is. So remember that the change in the magnetic flux is going to be given by three variables, b, a, and the cosine of theta. So what is changing as this current is constantly increasing in the wire? Well, remember that we said that the relationship between I and b is that as the current increases, the magnetic field also increases. So what happens is the changing variable in this case and our magnetic flux equation is going to be b. Right? Because the area of the square loop is not changing. It's always just a constant right there. And the direction of it, so there's the cosine of theta, is always going to be, again, straight and not changing. So what happens is our b field is changing. So as the current knows positive magnetic flux change and a downward-pointing b, magnetic field, then we can figure out the direction of the induced magnetic field using Lenz's law. Lenz's law says that it's going to do whatever the opposite is of that change in magnetic flux. So if it's downwards and it's increasing, then that means that Lenz's law wants to counteract that change by producing a magnetic field that sort of fights that. So in other words, our magnetic field is going to point upwards like that. So now we find out what the direction is of the induced currents by using our right-hand rule again. But in this case, we have to use a different right-hand rule. So before, we had our thumb pointing in our direction of the current because that was the straight thing, but now the straight thing here is our magnetic field. So now what happens is our right thumb is going to point in the direction of the magnetic field, and our fingers will give us the direction of the induced current. So be careful when you're using these right-hand rules because sometimes in the same problem, you might be using two different right-hand rules to solve it. Okay? So now what we're going to do is on our page, right? So just follow exactly what we are doing. We're going to take our right hands, and we're going to point in the direction of the magnetic field. So what happens is if we have a magnetic flux, or sorry, if we have an induced magnetic field that points upwards like this, then that means that our fingers will curl in the direction of that induced current. So if we're looking at it from the side view, so we're still looking at the side view right here, then what happens is our induced currents is going to point sort of in the counterclockwise direction. So Or sorry, that's going to be counterclockwise. So in other words, if we're looking at it from this perspective in the first diagram like this, it might be helpful to visualize this as your thumb pointing, so like it's quite like upwards, and our induced magnetic, or sorry, our induced current will point like this. So that means that in our first diagram, our induced current is going to be pointing in this direction. So that's the direction of our induced current, and that actually happens to be the counterclockwise direction. So that's going to be counterclockwise. Okay? So let me know if you guys have any questions with this, and I'll see you guys in the next one.

Here’s what students ask on this topic:

What is Lenz's Law and how does it relate to Faraday's Law?

Lenz's Law states that the direction of the induced current in a conductor due to a changing magnetic flux will be such that it creates a magnetic field opposing the change in flux. This is a direct consequence of the conservation of energy. Faraday's Law quantifies the induced electromotive force (emf) as $\epsilon =\; -n\frac{\mathrm{\Delta \Phi }}{\mathrm{\Delta t}}$, where $\epsilon$ is the emf, $n$ is the number of turns, and $\mathrm{\Delta \Phi }$ is the change in magnetic flux. The negative sign in Faraday's Law is a representation of Lenz's Law, indicating the opposition to the change in magnetic flux.

How do you determine the direction of the induced current using Lenz's Law?

To determine the direction of the induced current using Lenz's Law, follow these steps: First, identify the direction of the changing magnetic flux. If the magnetic flux is increasing, the induced current will create a magnetic field opposing this increase. If the flux is decreasing, the induced current will create a magnetic field to oppose the decrease. Use the right-hand rule: point your thumb in the direction of the induced magnetic field, and your fingers will curl in the direction of the induced current. This method ensures that the induced current always opposes the change in magnetic flux.

What is the right-hand rule in the context of Lenz's Law?

The right-hand rule is a mnemonic used to determine the direction of the induced current in a loop due to a changing magnetic field. To apply it, point your thumb in the direction of the induced magnetic field (which opposes the change in flux). Your fingers will naturally curl in the direction of the induced current. This rule helps visualize how the induced current creates a magnetic field that opposes the original change in magnetic flux, as stated by Lenz's Law.

Can you provide an example of Lenz's Law in action?

Consider a bar magnet moving downward through a loop of wire. As the magnet approaches, the magnetic flux through the loop increases. According to Lenz's Law, the induced current will create a magnetic field opposing this increase. Using the right-hand rule, if the induced magnetic field points upward, the induced current will flow counterclockwise when viewed from above. This counterclockwise current generates a magnetic field that opposes the downward motion of the bar magnet, demonstrating Lenz's Law in action.

Why is the negative sign important in Faraday's Law?

The negative sign in Faraday's Law, $\epsilon =\; -n\frac{\mathrm{\Delta \Phi }}{\mathrm{\Delta t}}$, is crucial because it indicates the direction of the induced emf and current. This negative sign embodies Lenz's Law, which states that the induced current will always oppose the change in magnetic flux. Without this negative sign, the equation would only provide the magnitude of the induced emf, not its direction. The negative sign ensures that the induced current creates a magnetic field that counteracts the change in the original magnetic flux, maintaining the conservation of energy.