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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