9.2 Induced Electric Fields
Earlier we have seen that if a magnetic flux passing through an area surrounded by a conducting loop is changing then we end up with an induced electromotive force enhanced induced current along that loop directly from Faraday’s law. If we consider magnetic field, let us say, pointing into the plane, something like this, then let’s assume that it is increasing into the plane. And if we place a closed conducting loop inside of this region, let’s consider a circular loop, then the flux through the area surrounded by this loop is going to be changing. As a result of this we will end up with an induced electromotive force from Faraday’s law, which will in turn cause an induced current flowing through this loop.
And from Lenz’s law, once we determine the direction of the current, which says that it has to be flowing in such a direction that it will oppose its course, and its course is the increasing magnetic field into the plane, therefore the only way that it can oppose that by generating a current whose magnetic field will be in opposite direction to this external magnetic field, and in order to have the current to flow in this direction along this loop by using right-hand rule, we want the magnetic field coming out of the plane through the area surrounded by this loop, therefore the current has to be flowing in counterclockwise direction throughout this loop.
Well, when we look at this process, this is in a way equivalent process to say that that the same magnetic field that we have over here pointing into the plane, and if we just consider one of the charges that’s moving along this path, for a charge, for a point charge to move along this direction, along this circular trajectory, we have to have an electric field in the medium, in a circular form, like this. In other words, that electric field is going to exert the Coulomb force on that charge so that the charge is going to move in the direction of that force. In order to go along a circular orbit like this, that force has to be always tangent to a circle, something like this. To do that the corresponding electric field line has to be in circular form.
So this is going to be, in other words, equivalent to the case that we’re going to have an induced electric field line, which will be in the form of a circle. Something like this. And if we place a charge, for simplicity let’s consider a positive charge here, and that charge will be under the influence of Coulomb force, which will be tangent always to this field line, and that field line is the electric field, and at every point along this trajectory under the influence of this Coulomb force, the charge is going to move along this circular electric field line.
Therefore, we can link this case to the next case by saying that while we start with a change in magnetic field which is increasing into the plane, we end up with electric field in the form of concentric electric field lines. In other words, change in magnetic field is generating electric field, whose electric field lines are in the form of concentric circles in this medium. Of course if B is not changing, if it is constant, then the magnetic flux through this circular wire, or through the area surrounded by the circular wire will be equal to a constant, and since it is not changing, from Faraday’s law, derivative of a constant will give us 0, we’re not going to end up with any induced electromotive force and hence the current also. But if it is changing, then that change will result with the induction of an electric field in the medium, like this.
So if we summarize up to this point, we can say that for this case, i induced will show up in counterclockwise direction. That is from Lenz’s law and Faraday’s law. And i induced is of course directly associated with the induced electric field, and we can see, by linking this first case to this case, this is in a way equivalent to this case, because if we just consider one charge’s motion and it will look like this. And in this case a changing magnetic field produced electric field.
If we look at this part, let’s say our wire has the radius of r, therefore we’re looking at the field line, induced field line, with the same radius r over here. From Faraday’s law, induced EMF is the change in flux with respect to time, and if we look at this process over here of course a work will be done by the Coulomb force in moving the charge along this slope. Work done during that process, if we denote that by W, will be equal to force dotted with the displacement, integrated overall the path, let’s call it F dot dl, integrated overall path.
Here, dl is an incremental displacement vector in the direction, or along the direction of path, and where we will go we will see that the angle between F and dl is going to be equal to 0, so the work done is going to be equal to therefore integral of the magnitude of F times dl times cosine of the angle between them, which is 0 degree. Cosine of 0 is just 1, and the magnitude of F is constant over here, and that’s the Coulomb force which is equal to q times the induced electric field E.
So moving on, work done will be equal to qE times dl, and these quantities are constant we can take it outside of the integral. Therefore the work done becomes qE times integral of dl along this path. Of course the path is a closed path, and as we go all along this path, and adding all these dl‘s to one another along that path, we will end up with the length of that path and that is the circumference of that circle, which is 2πr. So the work done will be equal to qE times 2πr.
On the other hand, as you recall from the definition of potential, which we defined the potential as work done per unit charge, one can also express this work done as charge times the potential, or the voltage, or the potential difference. Well, that potential, or the voltage, is the induced voltage ε. So we have two expressions for the work. One of them is induced EMF times the charge, and the other one is q times E times 2πr. Since the left-hand sides of these equations are equal, we can equate the right hand sides. In doing so we will have q times ε is equal to qE times 2πr. Here the charges will cancel on both sides, and induced EMF will be equal to E times 2πr.
Well, in other words, the product of the electric field magnitude, and the length of the loop, we can generalize this expression by saying that induced EMF is equal to integral over a closed path of E dot dl. The reason for that is that once the electric field is generated as a result of change in magnetic field in the region, those field lines are going to be always closed field lines. Therefore the path integral will be taken over a closed path, or closed loop.
Well, induced EMF was also equal to, from Faraday’s law, dΦB over dt, in other words rate of change of magnetic flux. We can combine these two expressions to express the Faraday’s law in its most general form by saying that integral of E dot dl over a closed contour, or closed loop, will be equal to minus, and again, we can generalize this N number of terms, by multiplying with the number of terms N over here, times the rate of change of flux. So this is the most general form of Faraday’s law of induction.
Now, it is simply stating that if the magnetic flux is changing through the area surrounded by a conducting loop, then we’re going to end up with an induced electromotive force. Now, if we take the integral of E dot dl along a closed conducting loop, that will also give us the induced electromotive force along that loop. This quantity will be equal to 0 if the flux through the area surrounded by that loop is constant. It will be different than 0 if it is changing, if the flux is changing with time.
Well, this is going to bring us to an interesting point, which will make us, to give a new look to the electric potential. Let’s say a new look at electric potential concept. And that is electric potential has meaning only for electric fields set up by static charges. It does not have any meaning for electric fields set up by induction. Let’s make a note of this by saying that electric potential has meaning only for electric fields set up by static charges. It has no meaning for electric fields set up by induction.
Now, for static charges, we had positive charges collected at some point, and then negative charges collected at some point, like this. Once we have a charge separation in this way, or polarization, then we immediately end up with a net electric field pointing from positive charge to negative charge, and as you recall, since the potential associated with the positive charge is greater than the potential associated with the negative charge, or negative point charge, then we end up with a certain potential difference of V volts between these two points, and that was equal to Vf minus Vi, and which was equal to work done in moving the charge from initial to final point. Let’s say this is initial, this is final, on per unit charge, and that was also equal to minus, negative of E dot dl integrated from initial to final point.
So in the case of induction, we end up with closed electric field lines. In other words, for the static charges, the electric field lines are always open field lines, originate from positive enter into the negative terminal. But through induction, as we have seen moments ago, we end up with closed field lines like this. And the electric field is tangent to these field lines at every point.
Well, in this case if we look at the potential difference between two points, that will be equal to minus integral of E dot dl over a closed loop, because the field line is a closed loop. It means that we’re going to start a certain point, initial point, and then we end up with the same point. The final point will be the same point that we started with. Therefore, the potential between initial and final point will be the same and this is going to give us always 0. That’s why the electric potential, or the potential difference doesn’t have any meaning for the electric field set up through induction.
This quantity from Faraday’s law, is such that induced EMF is equal to integral E dot dl over a closed loop and that is also equal to minus change in magnetic flux with respect to time. So this becomes equal to 0 whenever the magnetic flux is constant. If it is changing, then we will end up with a non-zero result, and that is going to be equal to the induced electromotive force. So in this sense, we have to always recall that potential difference does not have any meaning for the electric fields which are set up by induction.