0.10 Ampere's law  (Page 3/6)

Direction of surface

Since surface area vector is always normal, we may use the concept of normal unit vector n and denote surface vector as :

$$\mathbf{S}=\underset{}{\overset{‸}{n}}S\text{and}đ\mathbf{S}=\underset{}{\overset{‸}{n}}đS$$

Now, there can be infinite numbers of surfaces which can be drawn for a given closed boundary line. The choice of surface is easier to make if the imaginary closed line (loop) is in one plane. The surface in the same plane is generally chosen in that case. However, if the loop is not in one plane, then there is no simple choice. It does not matter then. The law is valid for all surfaces which are bounded by the loop.

Ampere loop and surfaces

If the consideration of magnetic field and current is done in a medium, then we need to substitute “ ${\mu }_0$ ” by “ ${\mu }_0{\mu }_r$ ”or “μ” representing permeability of the medium.

Ampere loop and enclosed current

One important consequence of freedom to draw imaginary loop is that it is our choice to keep a current inside or outside the loop. This appears to be a perplexing situation as we know that magnetic field at a point results due to magnetic fields due to each current. For illustration, let us consider five current carrying long conductors as shown in the figure, two of which are into the plane (shown by cross signs) and three are out of the plane of drawing (shown by filled circles). Now, we can draw valid Ampere loop in different ways to determine magnetic field at a point P in the plane of drawing as shown in the figure here.

Magnetic field at a point

Here, the currents are flowing perpendicular to the plane of drawing. The magnetic fields due to currents in long wires are in the plane of drawing as we can check by applying Right hand thumb rule. Since point P is not equidistant from the length elements of the loops drawn, the actual integration would be very rigorous and difficult. We shall, therefore, make only qualitative assertions here which are consistent with Ampere’s law. Further, we also make the simplifying assumptions that current in each wire is “I” and that we carry out integration in anticlockwise direction in each case. Let the magnetic field at point P is B as shown.

For the loop 1, there are two currents out of the page and one into the page. Thus, the net current is “I” flowing out of the page. For the loop 2, there are two currents out of the page and two into the page. Thus, the net current is zero. For the loop 3, there is no current at all. Thus, the net current is again zero. Now, how is it possible that integration of magnetic fields in three cases yields an unique value of magnetic field at P? The point to understand here is that when we integrate along a path, the sum of vector dot product “ B .d l ” for the complete closed path, due to currents lying outside the loop, cancels out. However, it does not cancel out for the currents inside. This is the reason Ampere's law considers only currents enclosed within the imaginary boundary.