# Magnetic Dipole Moment

A magnetic moment is a quantity that represents the magnetic strength and orientation of a magnet or other object that produces a magnetic field. More precisely, a magnetic moment refers to a magnetic dipole moment, the component of magnetic moment that can be represented by a magnetic dipole. A magnetic dipole is a magnetic north pole and a magnetic south pole separated by a small distance.

Magnetic dipole moments have dimensions of current times area or energy divided by magnetic flux density. The unit for dipole moment in metre–kilogram– second–ampere is ampere-square metre. The unit in centimetre–gram–second electromagnetic system, is the erg (unit of energy) per gauss (unit of magnetic flux density). One thousand ergs per gauss equal one ampere-square metre.

## Derivation of Magnetic Dipole Moment Formula

Magnetic Dipole moment- The magnetic field, B due to a current loop carrying a current i of radius, R at a distance l along its axis is given by:

B = $\frac {μ_0 i R^2}{2(R^2~+~l^2 )^{\frac32}}$

Now if we consider a point very far from the current loop such that l>>R, then we can approximate the field as:

B = $\frac{μ_0 i R^2}{2l^3 \left( \left(\frac{R}{l} \right)^2~+~1\right)^\frac 32 }$$\frac {μ_0 i R^2}{2l^3}$  ≡ $\frac {μ_0}{4π} \frac{2i (πR^2)}{l^3}$

Now, the area of the loop, A is

A = $πR^2$

Thus, the magnetic field can be written as

B = $\frac {μ_0}{4π} \frac {2i A}{l^3}$  = $\frac {μ_0}{4π } \frac {2μ}{l^3}$

We can write this new quantity μ as a vector that points along the magnetic field, so that

$\overrightarrow{B}$ = $\frac {μ_0}{4π} \frac {2 \overrightarrow{μ}}{l^3}$

Notice the astounding similarity to the  electric dipole field:

$\overrightarrow{E}$ = $\frac {1}{4 \pi \epsilon_0} \frac {2 \overrightarrow {p}}{r^3}$

Thus we call this quantity $\overrightarrow{μ}$  the magnetic dipole moment. Unlike electric fields magnetic fields do not have ‘charge ‘counterparts. In other words there are no sources or sinks of magnetic fields, there can only be a dipole. Anything that can produce a magnetic field comes with both a source and a sink i.e. there is both a north pole and south pole. In many ways, the magnetic dipole is the fundamental unit that can produce a magnetic field.

Most elementary particles behave intrinsically as magnetic dipoles. For example, the electron itself behaves as a magnetic dipole and has a Spin Magnetic Dipole moment. This magnetic moment is intrinsic as the electron has neither an area A (it is a point object) nor does it spin around itself but is fundamental to the nature of the electron’s existence.

We can generalize the magnetic moment for ‘N’ turns of the wire loop as

μ = NiA

The magnetic field lines of a current loop look similar to that of an idealized electric dipole:

Magnetic field lines

If you have ever broken a magnet into two parts, you would have found that each piece forms a new magnet. The new pieces also contain a north and a south pole. You just can’t seem to be able to obtain just a North Pole. Can you find the answer in this article?

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#### Practise This Question

A coil of 25 turns of dimensions 20cm × 5cm carries 4A current and lies in the x- y plane. Its placed in a B - field of 3^i+4^j Find the torque.