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 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 metreâ€“kilogramâ€“ secondâ€“ampere is ampere-square metre. The unit in centimetreâ€“gramâ€“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.

Following is the table explaining other **dipole related concepts**

Electric Dipole |

Torque on an Electric Dipole in an Uniform Electric Field |

The Electric Field of a Dipole |

## 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 = \( \frac {Î¼_0 i R^2}{2(R^2~+~l^2 )^{\frac32}}\)

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

B = \( \frac{Î¼_0 i R^2}{2l^3 \left( \left(\frac{R}{l} \right)^2~+~1\right)^\frac 32 }\) â‰ˆ \( \frac {Î¼_0 i R^2}{2l^3}\) Â â‰¡ \( \frac {Î¼_0}{4Ï€} \frac{2i (Ï€R^2)}{l^3}\)

Now, the area of the loop, A is

A = \( Ï€R^2 \)

Thus, the magnetic field can be written as

B = \( \frac {Î¼_0}{4Ï€} \frac Â {2i A}{l^3} \) Â = \( \frac {Î¼_0}{4Ï€ } \frac {2Î¼}{l^3} \)

We can write this new quantity Î¼ as a vector that points along the magnetic field, so that

**\( \overrightarrow{B}\) = \( \fracÂ {Î¼_0}{4Ï€} \frac {2 \overrightarrow{Î¼}}{l^3}\)**

Notice the astounding similarity to the Â electric dipole field:

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

Thus, we call this quantity \(\overrightarrow{Î¼}\) Â the magnetic dipole moment. Unlike electric fields, magnetic fields do not have â€˜charge â€˜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 electronâ€™s existence.

We can generalize the magnetic moment for â€˜Nâ€™ turns of the wire loop as

**Î¼ = 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 canâ€™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:

\(I = \frac{e}{T}=\frac{e}{\frac{2\pi R}{V}}=\frac{eV}{2\pi R}\)The magnetic moment associated to the current of an electron revolving is given as:

\(\mu _{l}=I_{A}=\frac{eV}{2\pi R}\times \pi R^{2}=\frac{eVR}{2}\)Substituting the angular momentum of the revolving electron we get,

l = mVR

\(VR=\frac{l}{m}\)Therefore,Â \(\mu _{l}=\frac{el}{2m}\) \(\frac{\mu_{l}}{l}=\frac{e}{2m}\)

**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 clockwise direction. This movement of electrons creates a south pole and north pole resulting in atom’s behaviour 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:

\(\left | m \right |\) = IA

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

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