Example-Infinite straight current carrying wire
Applications of Ampere’s law. As a first example, let’s consider the same example that we did by applying the Biot savart law, which was the case of infinitely long, straight, current carrying conductor or a wire. As you recall, we have considered a very long current carrying wire carrying the current in outward direction and it is going to plus infinity in the upper boundary and minus infinity in the lower boundary. We were interested with the magnetic field that this current generates some r distance away from the wire.
When we look at this current carrying wire from the top view, that the current is coming out of plane, and holding right hand thumb in the direction of flow of current and curling the right hand fingers about the thumb or about the wire, we have seen that this generates concentric circular magnetic field lines and they are going to be rotating in counterclockwise direction from the top view.
So for our point of interest which is r distance away from the wire, there will be a magnetic field line in the form of a circle passing through this point with radius of big R. Something like this.
Let me try to make this a good circle. All right. Okay. Here’s our wire at the center carrying the current in out of plane direction and the radius of this circle is r. That is the magnetic field line which is circling in counterclockwise direction by using right hand rule and passing through the point of interest. So at the point of interest, just right over here, the magnetic field will be tangent to this field line therefore it is going to be pointing in this direction. From this point of view, the magnetic field is going to be tangent to the field line and if I try to draw that in three dimensional sense, something like this, therefore the field line is circling in counterclockwise direction, going into the plane right here and then moving on behind the plane in counterclockwise direction and coming out at this point.
Therefore the magnetic field right at this point will be tangent to this field line and pointing into the plane. Here we’re going to try to apply Ampere’s law which is b dot d l integrated over a closed contour is equal to Mu zero times i enclosed. Therefore we will first choose a hypothetical closed loop such that it will satisfy the conditions in order to apply Ampere’s law. For this case, I am going to choose a closed loop and I will show that loop in the form of dashed lines, such that it will coincide with the magnetic field line passing through the point of interest. In other words, I choose this closed loop c in this form and let’s check whether it will satisfy the conditions to apply Gauss’s law or not.
Well d l is an incremental displacement vector along this loop by definition and we see that at the point of location, b and d l are going to be in the same direction, so the angle between them will be zero. If we go to any other location, like here for example, the magnetic field will be tangent and there also we will see that the magnetic field and incremental displacement vector will be pointing in the same direction and angle between them will be zero. Again, if we choose any other location b is tangent to the field line, and d l is an incremental displacement vector along that loop. Again at that point, they are going to be parallel to one another, or they will be in the same direction I should say. Therefore the angle between b and d l will always be zero degrees as we go along this loop.
Furthermore, since as long as we are on this loop, we are going to be along this specific magnetic field line the same distance away at every point from the source. As a result of that, the magnitude of the magnetic field vector will always be the same as we go along this loop. So both conditions, condition one, which was b magnitude is constant along this loop, is satisfied as well as the condition two, the angle Theta between b and d l is constant along the loop. These all along loop c are all satisfied which we just write down over here. These two conditions are satisfied by this loop c, therefore we can apply Ampere’s law to calculate the magnetic field generated by this current carrying wire.
Now the left hand side, b dot d l, is b magnitude times d l magnitude times cosine of the angle between them. That angle is zero everywhere. Cosine of zero is just 1 and this whole quantity should be equal to Mu zero times i enclosed. Since the magnitude of the magnetic field is constant along this loop c, we can take it outside of the integral and that will give us b times integral of d l over this closed loop c is equal to Mu zero times i enclosed.
Okay. Integral of d l integrated over this loop c means that we’re adding all these incremental displacement vectors, d l’s along this loop to one another. If we do that, in other words we’re adding the magnitude of all these displacement vectors to one another along this loop. If we do that of course we’re gonna end up with the length of that loop and in this case it is the circumference of the circle, which is going to be equal to 2 Pi times r. On the left hand side we will have b times 2 Pi r and on the right hand side, i enclosed means the net current passing through the area surrounded by this loop c and that is this shaded surface.
When we look at that surface we see that there is only one current passing through that surface and that is the current carried by our wire. Therefore i enclosed is equal to i. So the right hand side of the equation will be equal to Mu zero times i. If we solve for b, we will have b is equal to Mu zero i over 2 Pi r. That will give us the magnitude of the magnetic field at that point which is big R distance away from the wire. For that specific point, again we see that it is pointing into the plane, so we can write this down in vector form also by saying that into the plane. If you compare this result with the result that we obtained earlier by applying Biot savart law for the same physical system, you will see that the answers are exactly the same.
As you recall, in that problem, we had to take rather lengthy integral to be able to achieve this solution, but in this case, by applying Biot savart law, it is basically three or four steps. That is the beauty of Ampere’s law. Whenever it is applicable, then it is very simple to apply.