# Magnetic Dipole Moment

A magnetic moment is a quantity that represents the magnetic strength and orientation of a magnet or any other object that produces a magnetic field. More precisely, a magnetic moment refers to a magnetic dipole moment, the component of the 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 metreβkilogramβ secondβampere is ampere-square metre. The unit in centimetreβgramβsecond unit system, is the erg (unit of energy) per gauss (unit of magnetic flux density). One thousand ergs per gauss equal to one ampere-square metre.

Following is the table explaining other dipole related concepts

## 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:

$$\begin{array}{l} B = \frac {ΞΌ_0 i R^2}{2(R^2~+~l^2 )^{\frac32}}\end{array}$$

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

$$\begin{array}{l} B = \frac{ΞΌ_0 i R^2}{2l^3 \left( \left(\frac{R}{l} \right)^2~+~1\right)^\frac 32 } \approx \frac {ΞΌ_0 i R^2}{2l^3}Β \equiv \frac {ΞΌ_0}{4Ο} \frac{2i (ΟR^2)}{l^3}\end{array}$$

Now, the area of the loop, A is

$$\begin{array}{l}A = ΟR^2 \end{array}$$

Thus, the magnetic field can be written as

$$\begin{array}{l} B = \frac {ΞΌ_0}{4Ο} \frac Β {2i A}{l^3} = \frac {ΞΌ_0}{4Ο } \frac {2ΞΌ}{l^3} \end{array}$$

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

$$\begin{array}{l} \overrightarrow{B} = \fracΒ {ΞΌ_0}{4Ο} \frac {2 \overrightarrow{ΞΌ}}{l^3}\end{array}$$

Notice the astounding similarity to the Β electric dipole field:

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

Unlike electric fields, magnetic fields do not have βcharge’ β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 a 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 electronβs existence.

We can generalize the magnetic moment for βNβ turns of the wire loop as

ΞΌ = NiA

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

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 canβt seem to be able to obtain just a north pole. Can you find the answer in this article?

### What Is the Magnetic Dipole Moment of a Revolving Electron?

The magnetic dipole moment of a revolving electron is given as:

The current of an electron revolving around a heavy nucleus is given as:

$$\begin{array}{l}I = \frac{e}{T}=\frac{e}{\frac{2\pi R}{V}}=\frac{eV}{2\pi R}\end{array}$$

The magnetic moment associated to the current of an electron revolving is given as:

$$\begin{array}{l}\mu _{l}=I_{A}=\frac{eV}{2\pi R}\times \pi R^{2}=\frac{eVR}{2}\end{array}$$

Substituting the angular momentum of the revolving electron we get,

l = mVR

$$\begin{array}{l}VR=\frac{l}{m}\end{array}$$

Therefore,Β

$$\begin{array}{l}\mu _{l}=\frac{el}{2m}\end{array}$$
$$\begin{array}{l}\frac{\mu_{l}}{l}=\frac{e}{2m}\end{array}$$

The above equation is known as Gyromagnetic ratio.

### How Does an Atom Behave as a Magnetic Dipole?

The electrons in an atom revolve around the nucleus in a closed orbit. The orbit around the nucleus is equivalent to a current loop as the electrons are charged particles. The electrons revolve in anticlockwise while the current revolves in the clockwise direction. This movement of electrons creates a south pole and north pole resulting in atom’s behavior as a magnetic dipole.

### What Is Magnetic Dipole Moment of a Current Loop?

The magnetic dipole moment of a current loop carrying current I with area A is given as the magnitude of m:

$$\begin{array}{l}\left | m \right | = IA\end{array}$$

The direction of the magnetic dipole moment is perpendicular to the plane of the current loop.

## Class 12 Physics magnetism and matter chapter analysis

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