Thursday, September 1, 2016

Amepere's Law and Magnetic Field around Conductor

We can also find out the magnetic field induction at any point due to a charge using the Ampere’s law.According to this rule the line integral of magnetic induction around a closed curve is permittivity of free space times the current in that closed loop.

Problem and solution

Let us consider a current carrying conductor in circular shape and we are interested in the magnetic field at the center of the coil. We can use the formula that we have derived to do that and we shall assume that the distance of the particle on the perpendicular axis is zero. It is because we are measuring it at the center of the coil. The problem is solved as shown below.

When we measure the line integral, we get the length of the wire around which we are measuring the magnetic field. We also need to measure the magnetic field only due to currents inside the closed loop. We need not worry about the currents outside as they do not produce any impact. We are measuring only due to the portion of currents that are in the closed loop.

The currents with in the loop which are coming into the loop are treated as positive and currents leaving the closed circuit shall be treated as negative.

Basing on this Ampere’s law, we can find the magnetic field around a closed straight current carrying conductor of infinite length as shown below.

Let us assume a conductor carrying a current “I” as shown in the figure. We would like to measure the magnetic field around it at a distance “r” from it.  We can consider the line integral around it as the circular path of the given radius and when we line integrate it; we get the length of that closed path. It is nothing but the circumference of the circle.

It is the dot product of the magnetic field and the component of the length due to which we need to measure the field as per the Amper’s law. Any way the field and the portion of the length are in the same direction and the angle is treated as zero.

In the place of that line integral of the component of the length, we need to write the circumference as shown and we can find the magnetic field as shown below.

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