- Example: Changing Magnetic Flux
- Example: Generator
- Example: Motional emf
- Example: Terminal Velocity
- Simulation: Faraday’s Law
9.1 Magnetic Flux Fraday’s Law and Lenz Law
Earlier we have seen that if we place a current carrying loop inside of an external magnetic field, then the magnetic field generates a net torque on that current loop and under the influence of that torque, the loop rotates. So during the process, electrical potential energy is converted into the energy of motion or the kinetic energy. We call this type of system, which is arranged such that the loop rotates in one direction only, called as “electric motor”, a device which converts the electrical potential energy into kinetic energy.
Well, by looking at this system one can always ask the symmetric question such that, if you take the same system but in this case, instead of letting the current flow through the conducting loop, if we just crank the loop at a certain angle of velocity, do we end up with current flowing through this loop? The answer to that question is yes. As a matter of fact, this is a completely symmetrical system relative to the previous case. As you recall in the previous case, to be able to have the rotation of the loop in one direction only, one had to change the flow direction of current in every half cycle.
Now, in the second case, if we crank the loop, in other words, if we start with the kinetic energy, and crank it in one direction only, we will indeed induce current along this loop and that current will change direction of flow in every half turn as the loop rotates in one direction only. Therefore in this system, we will indeed end up with current. It will change direction in every half cycle and in the process, therefore, starting with kinetic energy, energy of the rotating loop will be converted into energy of moving charges, which is electrical potential energy. We call these systems, these devices, which makes this conversion, as “generators”.
Now we will understand why this happens in a moment, but before we go into that in detail, let’s consider a couple of interesting observations that one can make. Let’s assume that we have a coil which is connected to an instrument called a galvanometer. A galvanometer is nothing but a very sensitive ammeter, which detects low level of electric currents.
Now, obviously, in a circuit, something like this, there is no power supply. There’s no potential difference between any two points of this loop. We are not going to be able to detect any current. In other words, no deflection in the galvanometer’s dial in this position. Well, if we hold a bar magnet across from this loop, of course we know that the bar magnet will generate its own unique magnetic field lines and they’re going to be emerging from the north pole and entering into the south pole. So at a certain distance away from the loop, some of these lines will pass through the area surrounded by this loop.
Again, we will not observe any deflection in the dial of the galvanometer and if we move the magnet just next to the coil, in this case, all the field lines generated by this magnet is going to be passing through the area surrounded by this coil, but again, as long as we hold the magnet over in this position, we will not see any deflection in the galvanometer’s dial. So when the magnet is away, a small number of field lines are passing through the area surrounded by the coil and when the magnet is just next to the coil, then the maximum number of field lines are going to be passing through the area surrounded by this coil. In both cases, when the magnet is at rest, we will not observe any deflection in the pointer of the galvanometer.
As we do this observation, if we look carefully, we will see that when the magnet is moving towards the coil, then we will observe a deflection in the galvanometer’s dial. We will also see that whenever it is moving fast or faster to the coil, that deflection will get larger. Also, whenever we move the coil away from the coil, again, we will observe a deflection in the dial of the galvanometer, but at this time it is going to be in opposite direction.
Again, for slow velocity or low velocity as the coil moves away or moves closer, the deflection in the galvanometer’s dial will be small, but whenever it is moving faster, we will see that it will move for a larger deflection. The direction of these deflections are going to be in opposite directions depending on whether the magnet is moving towards the coil or away from the coil.
Of course the deflection of the galvanometer’s dial will tell us that during those instances, we end up with certain amount of current flowing through this coil. Also, since the deflections are in opposite directions relative to the motion of the magnet towards or away from the coil, will indicate that the direction of current flow are going to be in opposite directions relative to whether the magnet is moving towards the coil or moving away from the coil.
Indeed we end up with a certain induced current during the motion of the magnet relative to the coil. That current doesn’t have anything to do with the number of field lines passing through the area surrounded by this coil, but it has everything to do with how those field lines passing through the area surrounded by this coil is changing, how fast they are increasing or how fast they are decreasing. That brings us the concept of magnetic flux.
Earlier, we have defined the concept of flux through different cases. First we considered the motor flux and then we introduced the electric field flux as we study the Gauss’s law. Here, using similar type of approach, we will define magnetic flux. The magnetic flux is define as, as you remember, to be able to define a flux we need an area, a surface, and the vector field dotted with that surface will give us the flux of those vectors through that surface.
The magnetic flux is defined as the integral of B dot dA. As you recall, we have defined electric field flux as the integral of E dot dA integrated over a closed surface s. Here the magnetic flux is defined as the integral of B dot dA over an open surface. That difference is directly coming from the properties of magnetic field lines.
In the case of electric field lines, we could always end up with an open electric field lines, because we could always isolate a positive charge or a negative charge. As you recall, the electric field lines originate from the positive charge and goes radially outward to infinity and for the negative charge, that was just the opposite. In order to define the total flux associated with those field lines, we have to choose a closed surface which will enclose all those field lines.
Whereas in the case of magnetic field lines, since there cannot be any magnet monopole, that the magnetic poles are always in the form of dipoles, then the magnetic field lines always close upon themselves. If we choose a closed surface to block, then the number of field lines entering into that closed surface will always be equal to the number of field lines leaving that surface. Therefore the net flux will always give us 0 for a closed surface.
As a result of that, to be able to define magnetic field flux, we always choose an open surface, in other words, a surface which does not enclose a volume. So here, in this example, the main quantity associated with the induced current along this loop is not the magnitude of the magnetic field or not the area surrounded by this coil, but how fast or how slow that the magnetic flux through the area surrounded by the coil is changing.
This phenomenon was discovered by Michael Faraday at 1831 and it is formulated as the induced electromotive force, which causes the induced current to appear along this closed conducting loop, once the magnetic flux is changing through the area surrounded by the loop is equal to minus rate of change of magnetic flux. So here ε represents the induced electromotive force and ΦB is, again, the magnetic flux and the negative sign in this relationship shows up due to another law, which is known as “Lenz’s law”. This whole relationship is again known as “Faraday’s law of induction”.
All right. Well this is the case if we have only one turn for our coil. If we have n number of turn, in that case, Faraday’s law takes the form of –N times dΦB over dt, because in this case, flux through each loop is going to link to the next loop and therefore the induced electromotive force is going to be equal to N times the rate of change of flux. In other words, flux through the area surrounded by each one of these turns.
Before we proceed with the Lenz law, let’s also go ahead and introduce the unit of magnetic flux. Since ΦB is the product of magnetic field and area in a SI unit system, therefore it will have the units of tesla times meter squared. We have a special name for this product. It is called “weber”. We are going to approbate this unit with “Wb”. Therefore we can express the Faraday’s law as induced electromotive force in a circuit as equal to the negative rate of change of magnetic flux.
If we look at the units of the right-hand side of Faraday’s law, we will see indeed it will result in the dimensions of potential difference. Since ε is equal to the number of turns times the rate of change of flux, that will be equal to webers per second, and in explicit form that will be tesla meters squared per second, which is going to be equal to the unit of tesla and the explicit form of tesla, we can express that by looking at the magnetic force.
Magnetic force was equal to qv cross B and in SI units system therefore, we have newtons on the left-hand side, coulombs meters per second and the unit of magnetic field, if all of these quantities are in SI units system, then the unit of magnetic field was in tesla. So we can express tesla in explicit form as newtons times seconds divided by coulomb per meter and then we have times meters squared per second. Here this meter and that meters squared will cancel, seconds will cancel, and moving on, ε is going to be equal to newtons per coulomb.
Newtons is the force which is mass times acceleration and that will be equal to then kilograms meters per second squared and divided by coulombs left in the denominator, we have another meters over here. So kilogram meters squared per seconds squared or newton meter is nothing but the unit of energy, work, so it’s going to be equal to joule. We will have joules per coulomb. Joule per coulomb is, by definition, electrical potential energy per unit charge, by definition is nothing but the electric potential. Therefore, this will end up in units of volts in SI unit system. Indeed this relationship will have the dimensions of electric potential.
Okay. Earlier, we said that in Faraday’s law we have a negative sign on the right-hand side, and that negative sign appears due to another law which is known as Lenz’s law. Lenz’s law simply states that an induced electromotive force or current in a closed conducting loop will appear in such a direction of flow that it will oppose its cause. In other words, it will oppose the change that produces it.
Okay. Let’s look at this law through an example. Let’s assume that we have an external magnetic field pointing into the plane and we place a conduction loop, a circle or wire, let’s say, inside of this region. For the first case, let us assume that this external magnetic field is uniform. In other words, it is not changing. Therefore, the magnetic field passing through the area surrounded by this loop, which is this area, is not going to be changing. It also means that magnetic flux through this area is constant. If the flux through this area is constant, then since induced EMF is equal to –dΦB over dt for a one turn coil like this, and the derivative of a constant is 0, so there will not be any induced EMF along this conducing loop and it means that i induced also is going to be equal to 0.
Now let us take this same system. Magnetic field is into the plane, but in this case, B is not uniform, but it is increasing into the plane. Magnetic field is increasing. In this case, when we place our conducting loop inside of this region, since the magnetic field is increasing into the plane, the flux through the area surrounded by this conducting loop will increase. It means that ΦB is increasing. It means that it is going to be changing with time. As a result of this, we are going to end up with some induced electromotive force along this loop, so ε is going to be different than 0 and naturally this induced electromotive force will cause an induced current. That too will be different than 0.
Let’s look at the direction of flow of this induced current. Well, change in flux, from Faraday’s law, is going to generate the induced electromotive force. An induced electromotive force will generate the induced current. Lenz’s law says that this current will show up along this conducting loop such that it will try to oppose its cause. The cause of this induced current is the increase in magnetic flux. The magnetic flux is increasing due to the increase in magnetic fields.
Therefore, the induced current is going to flow through this loop such that it will try to oppose the increase in this external magnetic field. The only way that it can do that, by generating a magnetic field, that it will be in opposite direction to the direction of the external magnetic field. Therefore, the magnetic field of the induced current should be pointing out of plane.
Well, by using the right-hand rule, if the magnetic field through the area surrounded by this conducting loop is coming out of plane and we know that the magnetic field lines are always in the form of concentric circles going around the wire, so if the magnetic field line is coming out of here for this induced current, it will be going into the plane outside of the loop. To be able to have field lines in that direction, we simply hold our right hand fingers in the direction of magnetic field lines which are rotating in clockwise direction and the thumb will give us the direction of the associated current flow.
In that case therefore, the current has to be flowing in counterclockwise direction for this case. Because if the current is moving in counter clockwise direction, using the thumb in the direction of flow of current, and rotating the right hand fingers about the thumb, we’ll see that the associated field lines are going to be coming out of plane through the area surrounded by this current and going into the plane outside of the loop. As a result of this, the magnetic field in the surface of interest will be opposing the external magnetic field, therefore trying to decrease its strength or avoid its increase, but it will never achieve high enough value to do that, so we end up with an induced electromotive force and the associated current which will be flowing through this conducting loop in counterclockwise direction.
For the last case, if we consider the magnetic field, again, into the plane, but in this case, B is decreasing, since magnetic field is changing, it will cause flux through the area surrounded by this conducting loop to change and change in a way that, due to the decreasing magnetic field, flux will decrease through the area surrounded by this loop. It means that we’re going to end up again with an induced electromotive force and also therefore induced current.
If we try to determine the direction of flow of current, again, from Lenz’s law, the current should flow in such a direction that it will oppose its cause. Its cause is the decrease in magnetic field. Therefore it will try to avoid that decrease. In order to do that, it should generate magnetic fields that will be in the same direction with the direction of this external magnetic field. By using, again, the right-hand rule, to be able to have field lines going into the plane through the area surrounded by this loop, current has to be flowing in clockwise direction. Therefore the induced current will appear in this case as it is flowing in clockwise direction.