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### Course: Physics library>Unit 13

Lesson 4: Magnetic flux and Faraday's law

# Emf induced in rod traveling through magnetic field

An emf induced by motion relative to a magnetic field is called a motional emf. This is represented by the equation emf = LvB, where L is length of the object moving at speed v relative to the strength of the magnetic field B.

## Want to join the conversation?

• If the area we were concerned with was the magenta area, then why wouldn't we want the magnetic field induced by the current to go down INSIDE the magenta area? Sal shows the current coming toward us because it would produce a field going down on the area to the left of the rod. This means the field is going UP inside the magenta area which adds to the existing flux. Isn't the magenta area the area of interest??
(28 votes)
• I think you are mistaken. The magenta area is only valuable for finding out the change in flux. When it comes to Lenz's law, we are concerned with the area to the left of the rod inside the loop. This is because the area on the right of the rod is not a loop, and thus, current cannot flow through it. Since the change in flux is increasing in the upwards direction, Lenz's law tells us that the induced current's induced magnetic field must resist that, and point downwards on the inside of the loop. Using the right hand rule, we can see that we end up with an induced current in the clockwise direction.
(36 votes)
• After listening to this video, I got a little bit confused about the difference between the contents in this video and the Lorentz Force. Are these two processes exactly inverse? Please, who can help me! :)
(10 votes)
• The Lorentz Force is the force that makes the eletrons move, creating current on the wire. The force realizes work in the charges in the lenght L of the wire on the magnetic field, then if we divide the work by Q to get the work done per charge, we get J/C, which is the same as Volt. The Lorentz Force and the content in this video are two different ways to do the same thing. :)
(7 votes)
• How does the coil moving slightly to the right increase magnetic Flux? It's just moving from one area to another, how is that 'increasing' area?
(10 votes)
• Hello K.M.,

Here "area" is the area of the loop. The important thing to notice is that the number of flux lines contained within the loop is changing. In fact "change" is the critical thing to notice for all of these electromagnetic devices. Without change there will be no induced current...

Regards,

APD
(4 votes)
• If two electric fields,for example,if 2 transformers are kept close,will there be a disturbance in the working of the transformers?
(5 votes)
• Yes, you can induce "noise" into a transformer that way. The permeability of iron is much higher than the permeability of air, though, so the effect will be quite small.
(4 votes)
• Shouldn't it be counter clock-wise? The purple arrows he drew seem to be going counter clock-wise.
(5 votes)
• Sal assumes a current direction when he first draws the purple arrow, but explains it violates Lenz's law. Later in the video, he switches the directions and explains how the clockwise direction is consistent with Lenz's law. According to Lenz's law, the induced current is always such that the induced magnetic field opposes the CHANGE in the existing magnetic field. In this case, the magnitude of the magnetic field is constant, but the flux increases because the area is increasing phi=BA
(1 vote)
• To be honest i have no question regarding your explanation. But i was wandering Sir, What if i want to derive an expression of motional induced E.M.F in a conducting rod rotated in magnetic field? I found it on internet an expression as E.M.F = 1/2*B*L^2W Where W (omega) = dA/dt where A = angle subtended in time t. That went pretty up above my head.
(3 votes)
• Hello Roushan,

It may help to simplify the model and talk about a machine that moves in a straight line:

Assumption #1: Let the velocity (v) be perpendicular to the magnetic field (B) where B is a vector

Assumption #2: Let the conductor be parallel with the vector v x B where B is a vector

Out model simplifies to e = Bvl

e is EMF
B is magnitude of the B field
v is velocity
l is length of the wire

Now to your question, ω can be substituted for the velocity term. Please give it a try. Place a point on the tip of the second hand of a clock. The angular frequency aka ω describes the velocity of this point.

Regards,

APD
(1 vote)
• So how do we determine the direction of the MAGNETIC FORCE acting on the bar? Is it opposite or same direction with the velocity of the bar?
(1 vote)
• There's more than one way to do it, but the easy way is to use Lenz' law, which tells us that the induced EMF will oppose the change that is creating it. That means it has to, in this case, try to make the bar stop moving.

This makes sense because energy has to be conserved. If charge is moving, somehow energy is getting put into that system. Where does it come from? Whoever is pushing the rod. For that person to be doing work, they have to be exerting force in the direction that the rod is moving. But why should force be needed to keep the rod moving at constant velocity? It must be that there's an opposing force that the person doing the pushing has to fight against. If that person stops pushing, the opposing force will cause the rod to come to rest.

Think about what would happen if somehow the force was in the same direction as the velocity. You would give the rod a little push, and then that force would push it some more in the same direction, and then the rod would go faster so that force would get bigger, so the rod would go faster, etc etc etc. A tiny push would send the rod zooming off. Where would that energy come from to do that?
(4 votes)
• When Sal is talking about the change in flux over time and he says that this is equal to delta B times the area over a change in time, he then simplifies this to B times delta A over a change in time. However, since the magnetic field is not changing, wouldn't delta B equal zero?
(2 votes)
• it's not (ΔB) * A, it's Δ(B*A). The change in the product of field and area, the change in flux. Since B is constant with respect to time, you can pull it out and get B*Δ(A). Now repeat the same process, you get B*ΔA = B*Δ(l*x). Pull out l because l is constant, and you get B*l*Δx, which is simply v*Δt so you get B*l*v*Δt, but the Δt cancels with the Δt in the denominator, and you are left with Blv
(1 vote)
• Why is the magnetic field strength (B) constant? If the rod induces its own magnetic field in the opposite direction of the initial magnetic field, then wouldn't the resultant magnetic field be less? The formula for emf = BvL where B is the initial magnetic field strength? Is the induced magnetic field strength considered in the calculation?
(2 votes)
• the EXTERNAL magnetic field is constant. The moving bar will create its own field, yes, which will be superimposed on the external field B. And the induced field is considered in flux calculations, yes. But solving that would probably involve a lot of complex stuff and probably a differential equation or two, so we can leave it aside for now.
(1 vote)
• wait ,how the current can be MAKING..current can create magnetic field but can magnetic field create current.i am confusing in here.i mean when sal tells us here is a magnetic field means a currnt has made this magnetic field
(2 votes)
• yes. Current creates magnetic fields and changing magnetic fields can create currents too. And these currents create their own magnetic fields, and so on
(1 vote)

## Video transcript

- [Voiceover] I have an interesting set up over here. I have a magnetic field that is constant, and it's going straight out of the surface of this loop. The magnitude of the magnetic field at any point of the surface is going to be 'B'. What's interesting here is this loop that we have, This right part of the loop is movable. You can imagine it's a cylinder that can roll to the right, and the magnitude of it's velocity, we're going to say is lowercase 'v'. And this cylinder, lets say it has length 'L' Given that, you can see that we're going to have a change in magnetic flux. Why are we gonna have a change in magnetic flux, or a change in magnetic flux through the surface? Well if this thing is moving to the right, when it's speed is 'v' it can be any units, meters per second, or whatever it is. Even though the magnetic field itself is constant, you're actually going to be changing the area, the area that is, I guess you could, contained by this loop, and so that is going to give us a change in flux And if you have a change in flux that is going to induce an electromotive force, or it's going to induce a voltage in this loop which will cause a current to flow. Let's think about what that electromotive force that's going to be induced is going to be. I'm just gonna rewrite Faraday's law right over here. Faradays law: negative 'N' times our change in flux, and we're talking about our change in flux over change in, let me write that a little bit neater, our change in flux over change in time. The 'N' is the number of loops we're talking about. So in this case, 'N' is just going to be one and the negative, I've already complained a little bit about in previous videos. This is really just reminding us the math, when you're not using proper vector mathematics, this just reminds us that this 'Emf' is going to cause a current to go through this loop, and the magnetic field that is induced by that current will go against the direction of our change in flux. That's just something to remind us. What we really care about is our change in flux over change in time. What is that going to be? Let me just write it here, our change in flux over change in time. Our change in flux in this case is going to be our change in our, the magnitude of our magnetic field that is going perpendicular to the surface times the area of our surface over our change in time. What is this going to be equal to? 'B' is constant, it's not going to change over the time that we care about. So the change in 'B' times 'A' is really going to be 'B' times our change in area. 'B' times our change in area over how much time goes by. So what is our change in area going to be? Let's say we let delta 't' , let's say delta 't' happens, what is going to be our change in area? Let's let delta 't' go on. If delta 't' goes on, this cylinder, or if we wait for 't' units of time, or if 't' units of time go by, let me do this in another color, I'm gonna use this color right over here. How far will we go after delta 't'? We know the magnitude of our velocity if you multiply the magnitude of your velocity times your change in time, that will give you your distance. Your change in area is going to be the amount of distance it's travelled times the length of the rod, which is just that. Our change in area, which is that area right over there, our change in area over that time is going to be the distance this rod goes. Notice this rod is going in a direction that is perpendicular to the direction of the magnetic field that's an important thing to realize. We have our change in area is going to be this dimension which is how far the rod travels times the length of the rod, that's how much area we gain, that's how much our area increases, times the length of the rod, that's our change in area. We can substitute that back over here. This is going to be equal to, we got our 'B', we have our change in time, and our change in area we just said is going to be, Let me just write it this way, I could write is as the length of our rod times the magnitude of our velocity, or our speed times our change in time. Well, change in time divided by change in time, those cancel out. Our change in flux over that time or our average rate of change in flux we see simplifies to the length of our rod times the magnitude of our velocity, or our speed, times the magnitude of the magnetic field that is going perpendicular to the surface. This is something that you will see many times in your physics class, this whole notion of if your rod going in a perpendicular direction to a magnetic field, it induces an electromotive force of 'LvB' This is where it's coming from, it's coming directly from Faraday's law. You can say, okay, if that's happening, what direction is the current going to actually flow in? If the magnetic field isn't changing, but since the area is increasing, the flux is increasing in the upward direction. You can say the flux is increasing in that direction. The current that gets induced, and the magnitude of the current is going to be, based on how much resistance we have, but the current is going to induce a magnetic field that goes against our change in flux. Let's see, if the current went that way, what will happen? I'm going to take my right hand out, put my thumb in that direction, and if I were to loop my fingers around, this would, if we went that way, this would describe, this would induce a magnetic field that goes like this. That actually would enhance the flux going through the, or change in flux, going the same direction of the change in flux, We've already talked about this, would violate the law of conservation of energy. The current's gonna go in the other direction. It's gonna go, in this case is going to go clockwise, because the current is going to induce a magnetic field. Once again, I could take my right hand out, and try to draw my right hand, and we can see now, the direction my fingers go in are that way. Notice we're looking at the inside of the surface, the surface area was increasing the flux in the upward direction, getting more of the magnetic field in the upward direction, being contained in the area. The magnetic field induced by the current that is induced, or that current that's being caused by the electromotive force and the magnitude of the force is going to be dependent on our resistance, that's going to go in the other direction, it's going to go down that way. Or it's going to induce a magnetic field that is gonna go downwards. So the current needs to be going clockwise.