3.4 Magnetic field due to a circular coil

Consider a circular coil of radius a carrying a steady current I. Let it be placed with its center at origin and plane perpendicular to the X axis.

Now imagine a small element AB of length dl. The magnetic field at a point P on the X axis at a distance x from the origin is given by

where r is the distance between the element and the point P.

If vector r makes an angle alpha with the X axis, then dB will be inclined to the +Y axis. Hence dB can be resolved into X and Y components given by

Now if we consider an identical element CD diametrically opposite to AB, the magnetic field due to CD will be  in a direction inclined to the Y axis.

The above expressions makes it clear that the Y component of magnetic field due to diametrically opposite identical elements cancels each other. Hence the net magnetic field at P is simply the vector sum of x components of all the elements of the ring.

Thus the magnetic field due to a coil at a point on the x axis depends on the current I, the radius of the coil and the distance of the point from the center.

If the point P selected at the center of the coil then x=0

The direction of magnetic field due to a circular coil is obtained by the right hand thumb rule which is stated as follows:

Curl the palm of right hand around the circular coil with the fingers pointing in the direction of current. Then the extended thumb gives the direction of magnetic field.

Thus an anti-clock wise current gives a magnetic field out of the coil and a clock wise current gives a magnetic field into the coil.

Current loop as a magnetic dipole

In the preceeding section we have seen that anti clock wise current gives outgoing flux, indicating south polarity. In short a current loop gives north polarity on one side  and south polarity on the other side resulting in a magnetic dipole.

The magnetic dipole moment of a current loop is defined as the product of electric current and area.Its SI unit is Am2.

Now magnetic field at a point on the axis of a circular coil given by the equation 10 can be rewritten as 

The above equation is similar to that of the electric field on the axial line of an electric dipole. This we can conclude that a current loop is equivalent to a magnetic dipole of moment equal to m=IA. This can be generalized to any geometric shape of the current loop.

Magnetic Field due to a long
straight current carrying conductor

Ampere's Circuital Law

3.3 Magnetic field due to a long straight current carrying conductor

Consider a long conductor carrying a steady current I. Let us now find the magnetic field on a point P at a distance 'a' away from the conductor. The long conductor can be imagined to be composed of very large number of small current elements. Now the magnetic field at the point is the vector sum of magnetic field due to all such current elements.

Let the perpendicular from P meet the conductor at O. Consider a small current element of length dl at a distance l below O. Let the current element make at an angle theta with the vector connecting dl and P.

Now the magnetic field at P due to the element is given by Biot-Savart law as

dB = (muzeroxIdl sin theta)/4xpiexrxr into the plane of the paper.
Now from the diagram we get

theta equal to 90-theta.
l equal to a tan theta
cos phi equal to a/r

using the above equations the variables in the Biot-Savarts law can be rewritten as

sin theta equal to sin(90- phi)=cos phi

The net magnetic field at P can be now obtained by integrating (4) between proper limits. The expression reveals that magnetic field due to a long straight conductor depends on current and the distance of the point from the conductor.

Now the direction of magnetic field can be obtained by a simple law known as right hand grip rule which is stated as follows:

Grasp the conductor in right hand with the extended thumb pointing in the direction of the current.Then the palm fingures around in the direction of the magnetic field.

Biot-Savart's Law

Magnetic Field due to 
a circular coil

3.2 Biot-Savart's Law

Soon after the qualitative study of magnetic effects of electric current, two French physicists Jean Baptiste Biot and Felix Savart performed qualitative study on the magnetic effect of electric current. They experimentally found that the magnetic effect at a point due to an electric current depends on the current strength, distance of the point from the current carrying conductor and geometry of the current carrying conductor. They stated their observations in the form of a law known as Biot-Savart's Law.

According to the Biot-Savart's law the magnitude of magnetic field dB at a point due to a current element ldl is directly proportional to

1.  the magnitude of current (I)
2. the length of current element (dl)
3. sine of the angle between the current element and the vector connecting the current element and the point  and
4. Inversely proportional to the square of the distance between them.

dB directly proportional to (Idl sin theta)/ r*r

or

The direction of magnetic field is perpendicular to the plane containing dl and r and is given by the right hand screw rule.

In the above expression muzero/4pie is the constant of proportionality and muzero is called the permeability of vaccum. Its value is 4piex 10 raised to -7 TmA raised to -1.

To represent a magnetic field the following convention is followed.

A magnetic field acting perpendicularly in to the plane of the paper is represented by the symbol o and a magnetic field acting perpendiularly out of the same of the paper is represented by the symbol dot.

The magnetic field at any point on the right of the conductor is into the plane of the paper and that on the left is out of the paper.

Biot-Savart law combined with the superposition principle is used to determine the magnetic field due to any current carrying conductor. In the following sections we shall obtain expressions for magnetic field due to a straight long conductor and also due to a circular coil carrying current using Biot-Savart law.

Magnetic Field due to a long
straight current carrying conductor

Introduction