This document gives a clear introduction to the concept of the magnetic dipole. This document is suitable for any graduate course in basic electrostatics. This material covers topics like magnetic pole strength, magnetic properties of a current loop, magnetic dipole, magnetic dipole moment, geometr...
Introduction to Magnetostatics. The Magnetic
Pole Strength and Magnetic Dipole
1 The Magnetic Dipole
A current loop carrying a current i, produces a magnetic field
µ0 2µ
µ
B loop = (1)
4π d3
A is the magnetic dipole moment of the loop and A is the vector area of
at an axial point. µ = iA
the loop.
The electric field produced by an electric dipole at an axial point is
1 2pp
E dipole = (2)
4πε0 d3
Note that the electric field due to an electric dipole and the magnetic field due to a current
loop have similar mathematical structures.
We know that the electric dipole comprises two equal and opposite charges separated by a
small distance. The similarity of the mathematical structure of the field due to a current loop
and the field due to an electric dipole allows us to define two magnetic charges (positive and
negative). Thus, the current loop acts as a magnetic dipole. The positive magnetic charge is
called the north pole and the negative magnetic charge is called the south pole. The magnetic
charge is denoted by the letter m. Just as a charge Q placed in an electric field E experiences
a force QEE , a magnetic charge m placed in a magnetic field of flux density B experiences a
force mBB. The force on the positive magnetic charge (north pole) is directed along the field
and that on the negative magnetic charge is opposite to the field.
A magnetic pole of magnetic charge m produces a magnetic field
µ0 m
|B
B| = (3)
4π r2
at a distance r from it. The field is radially outward if the magnetic charge is positive (north
pole), and radially inwards if it is south pole. Two equal and opposite magnetic charges sepa-
rated by a small distance is called a magnetic dipole.
The magnetic dipole moment is
µ = md (4)
1
, where d is the separation between the poles. It is directed from the south to the north pole.
Circular current loops behave as magnetic dipoles. A loop carrying a current i may be replaced
by a magnetic dipole of moment
µ = mdn̂n = iA
A (5)
where n̂n is a unit vector directed from the south pole to the north pole. A is the vector area of
the loop. It has a magnitude equal to the area of the loop and a direction from -m to m (south
to north pole). Similarly, a solenoid can be replaced by a magnetic dipole with pole strength
m and the distance of separation approximately equal to the length of the solenoid. In order
to find the north and south poles of a current loop or solenoid, the following trick is useful.
Looking into the loop, if the current is flowing in the anti-clockwise direction, then the pole
facing us is the north pole.
Note that the quantity m is commonly referred to as pole strength. Its unit is Ampere meter
(A m).
2 Permanent Magnets
There are materials which can retain magnetism. They are called permanent magnets. A
magnet in the shape of a bar or a rod is called a bar magnet. The poles of a bar magnet are
not exactly on the edges. It is located somewhat inward. The distance between poles of a
magnet is called the magnetic length. The distance between the ends of a magnet is called the
2
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