Example- Magnetic field of a toroid
As we have seen earlier, from the solenoid example, by sending the same amount of current through a helical system, we can generate a very strong magnetic field, which is directly proportional to the number of turns that we have. We can generate another magnetic field geometry. In this case, if we take the solenoid and connect its both ends together, therefore generating a system something like this. If we look at this configuration from the horizontal cross sectional point of view, in other words if we just slice it down, we’re going to end up with two branches. An inner branch, something like this, and the outer branch which is going to be something like this.
As the current flows through one branch into the plane, then it is going to be coming out of plane through the other branch. In other words, let’s say if the current is coming out of plane through the inner branch, then it is going to be flowing into the plane through the outer branch like this. If we look at the magnetic field geometry generated by such a current flow, considering each one of these turns one by one, for the inner branch, current is coming out of plane and if we hold our right hand thumb in the direction of flow of current which is coming out of plane, then curling the right hand fingers about the thumb, we will see that for this current, the magnetic field lines are going to be in the form of concentric circles and circling in counterclockwise direction. Similarly the next one and then the next one, next one, and so on and so forth. They will be all circling in counterclockwise direction and so on and so forth.
If we consider the outer branch, in this case holding the right hand thumb pointing into the plane, and curling the right hand fingers about the thumb, we will see that the associated field lines are going to be circling in clockwise direction. We can easily see that these turns are going to be very near to one another and as a result of this, the magnetic field generated by one of these turns will overlap with the next one. In other words, they will add to one another, therefore it will generate a magnetic field line going along this circular direction in counterclockwise direction. They’re known as toroidal field lines.
A similar type of phenomenon will take place for the outer ones. The magnetic field lines will overlap, therefore they will generate a net magnetic field line something like this again circling in counterclockwise direction. If we look at over here, we see that the current is coming out through the inner branch and going into the plane over here and flowing through the back of the plane and coming out and then going into the plane again and again coming out of plane here, going into the plane there, and so on and so forth. This is how the current is flowing through this system.
Well we call these current systems as toroids. Now we can easily see that if the current is flowing in, let’s say clockwise direction through the turns of this toroid, then it generates magnetic field lines along this region, what we call in toroidal directions, and the direction of the magnetic field lines for such a current which is going in clockwise direction is counterclockwise direction. In other words, if the current is flowing in clockwise direction through this toroid, the magnetic field lines that it will generate through the system will be in counterclockwise direction. Of course the magnetic field will be tangent to the field line passing through the point of interest.
Now we will try to determine the magnetic field of a toroid. Okay. First, let’s give some dimensions to this toroid. Let’s say the inner radius is a and the outer radius is b. In other words, it is such that this radius is a and the outer radius is b. Let’s say that we’re trying to figure out the magnetic field at a specific point inside of this toroid. Let’s say somewhere over here at point p.
If we consider the magnetic field line passing through point p will be a magnetic field line in the form of a circular field line and it is going to be in toroidal direction. It will be in counterclockwise direction. Magnetic field vector at point p will be tangent to this field line, therefore it is going to be something like this. We would like to calculate the magnitude of this field line.
In order to do this, we’re going to apply Ampere’s law which is integral of b dot d l along a closed contour c is equal to Mu zero times i enclosed. We will choose a closed loop which will satisfy the conditions to apply Ampere’s law. In order to that, we will choose a loop in the form of a circle, which is coinciding with the field line passing through the point of interest.
Now if we choose a closed hypothetical loop that coincides with the field line passing through the point of interest, then the magnitude of the magnetic field at every point along this loop will have the same magnitude. In other words, if we consider for example the magnetic field at this point, it will be tangent to that field line and here it will be tangent to the field line like this. All these magnetic field vectors will have the same magnitude because they are tangent to the same field line. Therefore the first condition is satisfied in order to apply Ampere’s law. In other words, the d magnitude along loop c, and this is the loop c, is constant.
Furthermore, if we look at the angle between b and incremental displacement vector along this loop, we can easily see that that angle will always be zero degrees because d l is an incremental displacement vector along this loop c. Wherever we go the angle between b and d l will be zero degrees. Therefore we can say that the angle Theta is going to be zero degrees all of the time along c, therefore the second condition is also satisfied.
Then if we write down the left hand side in explicit form, we will have b magnitude d l magnitude times cosine of the angle between them which is zero degrees integrated along this loop c will be equal to Mu zero times i enclosed. Well cosine of zero is just 1 and b is constant along this loop, so we can take it outside of the integral. Therefore we end up with b times integral over loop c of d l is equal to Mu zero times i enclosed.
Integral of d l over loop c means all the incremental displacement vector magnitude d l’s, these distances, are added to one another along this loop c. If we do that, we’re going to end up with the length of that loop. In this case, it is the circumference of this loop c. Let’s say that our point of interest is r distance away relative to the center of the toroid, so the radius of this loop is little r. Then from here, this integral is going to give us the circumference of that circle which is 2 Pi r. So the left hand side will be b times 2 Pi r and that will be equal to Mu zero times i enclosed.
Well the left hand side of the Ampere’s law is done. Now we will look at the right hand side. i enclosed is the net current passing through the region or surface surrounded by Amperian loop or the closed loop c. That region is this shaded region. So we will look at the net current passing through this surface. When we look at that region, we see that it encloses all the turns of this toroid. In other words, all the turns of this toroid are coming out of this surface. We know that each turn is carrying current i, therefore the total current passing through that surface will be equal to total number of turns of the toroid times current i.
If we say that n represents the total number of turns of toroid, then i enclosed is going to be equal to, since all these turns are passing through the surface of interest, and each one of them carries current i, therefore i enclosed will be equal to n times i. From here, b becomes equal to Mu zero, for i enclosed we will have n times i divided by 2 Pi r. That is the magnitude of the magnetic field that a toroid will generate.
If we look at over here, we see that the magnetic field will be proportional with 1 over r and r is the distance from the center of the toroid. It means that since b is proportional to 1 over r, the distance, then as the distance increases, the strength of magnetic field decreases. If we look at our diagram over here, it means that we’re gonna end up with a stronger magnetic field nearer to the inner wall relative to the magnetic field nearer to the outer wall. That can easily be understood directly from the geometry.
Once we put the solenoid in this form by connecting both its ends, you can visualize this as a slinky connected from both ends, something like this. The number of turns in the inner region will be nearer to one another in comparing to the region of the outer part. In other words, we’re gonna end up with better overlapping of fields generated due to each turn over here along this inner wall in comparing to the outer wall. That will result with a stronger magnetic field nearer to the inner wall of the toroid in comparing to the regions which are nearer to the outer wall.
When we go from inner wall to outer wall, we will see that the strength of the magnetic field generated by the current flowing through the toroid will decrease. If we look at the magnetic field outside of the toroid, which we can easily do that by placing our, let’s say, loop passing through a point which is located outside of the toroid, let’s say at this p prime. For such a region, once we place our Amperian loop, such that passing through that point and without even considering the left hand side of Ampere’s law, if we look at the right hand side, to be able to get the net current passing through the area surrounded now by this loop c 2, we can easily see that n i of current is coming out of the surface or the area surrounded by this loop, which is basically this surface here, and also the n times i of current is going into the plane.
In that case that net current flowing through this surface will be equal to n i mines n i and then they will cancel and we will end up with zero. If we look at here, let’s say magnetic field b outside of the toroid is going to be such that from Ampere’s law, b dot d l, in this case we will use loop c 2, which is a circular loop passing through the point located outside of the toroid, will be equal to Mu zero times i enclosed and in this case, i enclosed will be equal to, if n i is coming out of plane, n i is going into the plane and that will give us the zero. As a result of this, b outside of a toroid will always be equal to zero.
By the same token, we can look at the magnetic field in this region, in other words over here. So we choose our Amperian loop passing through a point located in this region, and then when we look at the i enclosed through that region, we will see that that will be equal to zero because none of the current flowing through the toroid will go through that region. Therefore here also magnetic field is zero. Therefore here again, the strength of the magnetic field is directly proportional with the number of turns. The greater the number of turns will result with the greater the magnetic field generated from the same current i, but as a major difference from the solenoid magnitude was is that that the magnetic field is not constant inside of the toroid. It will change from point to point whereas compared to the solenoid, the magnetic field strength was the same everywhere inside of the solenoid.