(i) Derive an expression for the magnetic field at a point on the axis of a current carrying circular loop.

(ii) A coil of 100 turns (tightly bound) and radius 10 cm. carries a current of 1A. What is the magnitude of the magnetic field at the center of the coil?

OR

State Ampere’s circuital law. Consider a long straight wire of a circular cross section (radius a) carrying steady current I. The current I is uniformly distributed across this cross section. Using Ampere’s circuital law, find the magnetic field in the region r < a and r > a.

By, Biot-Savart Law, we know that,

The axis, is perpendicular to the plane, so, θ=90˚

r means the distance of the point from current location, i.e. (By, Pythagoras)

has a constant value.

dl is the infinitesimal part of the circular conductor. So, .

Now, by this equation we get the value for dB and by integrating dB we can get B.

But here, a simple visualization tells us that, all the cos components of B, cancel outs as we sum up running through the circular conductor. So, it is only the sin components that will count.

So, By Pythagoras, we know that,

(ii) We know that, magnetic field at center of circular coil is,

Here, n=100, r=10cm, I=1A and

Putting the values and calculating we get,

.

OR

Ampere’s Circuital Theorem: If current passes through a closed path, then Integral of Magnetic Field density along that path is equal to the product of current flowing within with the permeability of that medium.

For r<a,

As we know, I is distributed over , so, in , the amount of current is,

or

For r>a,

or

So, the graph is like,