Magnetic Dipole Moment: Definition, Theory, Derivation

Magnetic Dipole Moment is a quantity that is used to represent the orientation and strength of magnetic dipoles. It can also be defined as the torque that an object experiences when placed in a magnetic field. The electrons moving around the atomic nuclei also creates a magnetic dipole moment. The sum of the effects of all the electrons may cancel out the overall effect in the atom. In that case, it will not be a magnetic dipole. If these effects are not cancelled out then the atom has a permanent magnetic dipole moment, like in iron atoms. When millions of atoms are locked in the same spontaneous alignment then it forms a ferromagnetic domain. Magnetic needles and compasses are also a kind of magnetic dipoles.

Key Terms: Magnetic Moment, Electromagnets, Ferromagnetism, Magnetic Field, Magnetic Dipole Moment, Magnetic Dipole, Monopole, Quadrupole

Read More: Magnetic Properties of Materials

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Magnetic Moment

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It is the magnitude that determines the strength and the orientation of a magnet other than the object which has created the magnetic field. Examples of such objects are a loop of electric current (electromagnets), elementary particles such as electrons in an atom, permanent magnets, and many astronomical objects.

The word “magnetic moment” denotes the magnetic moment of a system, the component of a magnetic moment can be determined by the same magnetic dipole. For example, the magnetic north and south pole are divided by a small minor distance. Higher-order expressions might be needed with the dipole moment for prolonged objects. The magnetic dipole moment of the object is also defined as the torque experienced by that object in the magnetic field. Therefore the magnetic moment can be considered as a vector.

Magnetic Moment

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What is Magnetic Dipole Moment?

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A magnetic dipole moment can be defined as a vector aligning torque on an object in an external magnetic field. The relationship can be written as 

\(\tau = m \times B\)

Where,

• τ refers to the torque acting on the dipole
• m is the magnetic moment
• B is the external magnetic field

There is another substitute definition considering the thermodynamic calculation for magnetic dipole moment. The magnetic dipole moment of a structure is defined as the negative of the gradient of its energy, U_int, with respect to the outer magnetic field. We get

\(\mathbf{m}=-\hat{\mathbf{x}} \frac{\partial U_{\mathrm{int}}}{\partial B_{x}}-\hat{\mathbf{y}} \frac{\partial U_{\mathrm{int}}}{\partial B_{y}}-\hat{\mathbf{z}} \frac{\partial U_{\mathrm{int}}}{\partial B_{z}}\)

Magnetic Dipole Moment of Closed Circuit

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Magnetic Dipole

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It is the magnitude that determines the strength and magnetic orientation of a magnet or any magnetic object. Generally, a magnetic moment refers to a magnetic dipole moment, the object which has a dipole moment can be signified as a magnetic dipole.

A magnetic dipole is magnetic north and south pole which are separated by some distance.

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Theory Underlying Magnetic Dipole

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The field of any magnet can be represented by a series of terms of which every other term is more complicated than the previous one. The first term is a monopole (means isolated south or north pole) and dipole (means two opposite poles) and quadrupole (means four poles combined to form two dipoles). The degree of magnetic field for every term reduces progressively sooner with the distance than in the previous term. Therefore for a big enough distance, the first non-zero term will determine the field.

Magnetic Potential

It is a scalar quantity which is the characteristic of a point magnetic field, the negative gradient of which gives the value of magnetic field strength at that point.

In the magnetic pole model theory, the field related is a demagnetizing field which is represented by H and it is related by magnetic potential as

\(H(r) = - \nabla \psi\)

In the Amperian loop theory, the magnetic field is of magnetic induction represented by B such as

\(B(r) = \nabla \times A\)

These potentials can be measured for any kind of loop or magnetic charge distribution but the field should be restricted in a small region to give

\(\begin{aligned} &A(r, t)=\frac{\mu_{0}}{4 \pi} \int j\left(r^{\prime}\right)\left|r-r^{\prime}\right| d V^{\prime} \\ &\psi(r, t)=\frac{1}{4 \pi} \int \rho\left(r^{\prime}\right)\left|r-r^{\prime}\right| d V^{\prime} \end{aligned}\)

Here, P = magnetic pole density similar to J (current density)

For longer distances the first non zero terms is considered and the formula becomes

\(A(r) = \frac{\mu_0}{4\pi} \frac{m \times r}{|r|^3}\)

Where m is:

\(m = \displaystyle \frac{1}{2} \iiint Vr \times jdV\)

In this, r refers to the position vector, j is the electric current density and the integral is a volume integral. × is the vector cross product.

The first non zero term of scalar potential is given as

\(\psi (r) = \frac{m.r}{4\pi |r|^3}\)

M can also be represented in the form of magnetization field as

\(m = \displaystyle \iiint MdVm\)

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Derivation of Magnetic Dipole Formulae

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The magnetic field B, at a distance l from its axis and field of radius R due to a current-carrying loop having current i is given by

\(B = \displaystyle \frac{\mu_0 iR^2}{2(R^2 + l^2)^{\frac{3}{2}}}\)

For simplification let us assume that the distance of the point is very far from the current-carrying loop.

\(B=\frac{\mu_{0} i R^{2}}{2 l^{3}\left(\left(\frac{R}{i}\right)^{2}+1\right)^{\frac{3}{2}}} \approx \frac{\mu_{0} i R^{2}}{2 l^{3}}=\frac{\mu_{0}}{4 \pi} \frac{2 i\left(\pi R^{2}\right)}{l^{3}}\)

The area of the loop is given as

A = R²

The magnetic field can be written as

\(B=\frac{\mu_{0}}{4 \pi} \frac{2 i A}{l^{3}}=\frac{\mu_{0}}{4 \pi} \frac{2 \mu}{l^{3}}\)

which is similar to the Electric dipole field

\(\bar{E}=\frac{1}{4 \pi \epsilon_{0}} \frac{2 \bar{p}}{r^{3}}\)

Therefore μ→is called a magnetic dipole moment. Different Electric fields do not have charge counterparts as magnetic fields. Hence there is no source or sink for the magnetic fields, only a dipole exists in this case. Most elementary particles behave as magnetic dipoles. The electron acts as a magnetic dipole and has a spin magnetic moment. It is inherent directly from the electron, it does not have any area and it doesn’t even spin around itself. The magnetic property is fundamental to the electron.

The formula for a magnetic moment for N turns of an electric loop carrying current i is

μ = NiA

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Internal Magnetic Field of a Dipole

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The two types of the dipole, one is the current-carrying coil and the other is magnetic poles, both give the same value for the magnetic field at a far distance. Far distance means farther from the source. But inside the source, they give different values. 

If a magnetic dipole is made from a current-carrying coil by making it smaller and smaller. But keeping the value of product * current even, the value of the limiting field is given by

\(B(x)=\frac{\mu_{0}}{4 \pi}\left[\frac{3 n(n \cdot m)-m}{|x|^{3}}+\frac{8 \pi}{3} m \partial(x)\right]\)

If a magnetic dipole is made by taking the north pole and south pole, bringing them as close as possible then the limiting field is given by

\(H(x)=\frac{1}{4 \pi}\left[\frac{3 n(n . m)-m}{|x|^{3}}+\frac{4 \pi}{3} m \partial(x)\right]\)

Both these fields are related by B = μ₀(H+M) where M(x) = mδ(x) is magnetization.

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Things to Remember

• A magnetic dipole moment is a quantity that is used to represent the orientation and strength of magnetic dipoles.
• When millions of atoms are locked in the same spontaneous alignment then it forms a ferromagnetic domain.
• The word “magnetic moment” denotes the magnetic moment of a system, the component of a magnetic moment can be determined by the same magnetic dipole.
• A magnetic dipole moment can be defined as a vector aligning torque on an object in an external magnetic field.
• Different Electric fields do not have charge counterparts as magnetic fields.
• There are two types of dipole, one is the current-carrying coil and the other is magnetic poles.
• A magnetic dipole is magnetic north and south pole which are separated by some distance.

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Previous Year Questions

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