3.6 Applications of Ampere's Circuital Law

In section 3.3 we have determined the magnetic field intensity due to current carrying straight conductor of infinite length by applying Biot-Savart's law. However in symmetric cases it is more convenient to use amperes law for determining the magnetic field. Here we shall try Ampere's law to obtain magnetic fields.

For a straight long conductor

Consider a straight long conductor carrying steady current I. In order to find the magnetic field at a point P at a distance r from the conductor, considering a circular path containing the point with the conductor as axis. 

As the magnetic field at all points on the circular path is tangential and of the same magnitude we get

Line integral of Bxdl equal to B x line integral of dl=Bx2xpixr

The current enclosed by the circular path is sigma i=I

The result is same as that of equation.

For a solenoid

A current carrying uniform wire wound in the form of helix is called solenoid. In fact it is a combination of number of circular coils spread over a length. We have seen that a circular coil behaves as a magnetic dipole. Now let us investigate the magnetic field due to a solenoid. A solenoid of definite length behave exactly in the same way as that of a bar magnet. However if we consider an infinitely long solenoid the magnetic field inside the solenoid is uniform and outside the solenoid the solenoid the magnetic field vanishes. Let us now determine the magnetic field due to an infinitely long solenoid.

Consider an infinitely long solenoid of n turns/ unit length and imagine a rectangular path abcd of lengthl and breadth b

The line integral of magnetic field along the closed path abcd is given by equation.

Since the magnetic field along cd is zero and along the path bc and da the magnetic field is perpendicular to the path all the terms in the expression except the first reduces to zero. For the path ab of length l the magnetic field is uniform.

The net current enclosed by the path abcd is sigma i=nlI

where nl is the number of turns passing inside the path abcd.

If the core of the solenoid is filled with a medium of relative permeability mu, then the above equation is modified as 

B=muoxmurxnxI

The above expression makes it clear that the magnetic field inside a long solenoid is uniform and depend on the number of turns per unit length, current and medium of the core.

Toroid

Practically a solenoid of infinite length is not possible. But if the ends of a solenoid is curved to form a ring such that the free ends touches each other, then every points inside it is identical and may be considered as an end less solenoid. Such an arrangement is called a Toroid. Let us now determine the magnetic field inside a toroid.

Consider a toroid of inner radius r1 and outer radius r2 with N turns carrying a current I. Imagine a circular path of radius a=(r1+r2)/2.

As the magnetic field inside a toroid is uniform line integral of Bxdl=Bx2xpixa.

The net current enclosed by the path is sigma i= NI.

Putting N/2xpixa =n the number of turns per unit length, the above expression becomes

B=muzeronI

If the core of the toroid is filled with a medium of relative permeability mur, then the above equation is modified as 

B=muzeroxmurxnxI

Thus the magnetic field inside a toroid is uniform and depends on the medium, number of turns per unit length and electric current intensity.

Ampere's Circuital Law

Lorentz Force