Magnetic Dipole Moment - Definition, Formula, FAQs

A magnetic dipole is similar to a very small magnet with two poles; the North and South poles that form a magnetic field. It helps us to get knowledge about magnets, current loops, and moving charges for magnetic force and effects. Magnetic dipole strength is characterized by a magnetic dipole moment. A number of magnetic phenomena can be described by studying the concept of magnetic dipoles. Let's understand this concept in detail.

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This Story also Contains

1. What is Magnetic Dipole?
2. What is Magnetic Dipole Moment?
3. Derivation of Magnetic Dipole Moment

What is Magnetic Dipole?

A system that generates a magnetic field and has two poles (North and South) is referred as a Magnetic dipole. It is a fundamental concept in Physics that describes the behavior of magnetic materials and the field generated by moving charges or current loops.

What is Magnetic Dipole Moment?

A magnetic dipole is defined as a pair of equal and opposite magnetic charges or, it can be understood, as a bar magnet that produces a magnetic field. The strength of a magnetic dipole is measured by its magnetic dipole moment. It is represented by $\mu$.

The SI unit of magnetic dipole moment is ampere-square meter $\left(A \cdot \mathrm{~m}^2\right)$.

Formula to Calculate Magnetic Dipole Moment

$$
\mu=I \cdot A
$$

where:

$I$ is the current in the loop (in amperes)

$A$ is the area of the loop (in square meters)

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Derivation of Magnetic Dipole Moment

Torque on a Current carrying Loop: A current-carrying loop in a uniform magnetic field ( $B$ ) experiences torque:

$$
\tau=I \cdot A \cdot B \cdot \sin \theta
$$

where:
$I=$ current,
$A=$ area of the loop,
$\theta=$ angle between the magnetic field and the normal to the loop.

Vector Form of Torque: The torque can also be expressed as:

$$
\vec{\tau}=\vec{\mu} \times \vec{B}
$$
Comparing the two, the magnetic dipole moment is:

$$
\mu=I \cdot A
$$
Special Cases:
Circular Loop: $\mu=I \cdot \pi r^2$,
Multiple Loops: $\mu=N \cdot I \cdot A$, where $N=$ number of turns.

Thus, the magnetic dipole moment quantifies the loop’s ability to produce a magnetic field.

Magnetic Field Due to Dipole

The magnetic field produced by a dipole decreases with distance and has a specific pattern. Mathematically, it is represented as:

$$
B=\frac{\mu_0}{4 \pi} \cdot \frac{2 \mu \cos \theta}{r^3}
$$

in the axial direction, and:

$$
B=\frac{\mu_0}{4 \pi} \cdot \frac{\mu \sin \theta}{r^3}
$$

in the equatorial direction.
Here:
$\mu_0$ is the permeability of free space
$\mu$ is the magnetic dipole moment
$\theta$ is the angle between the dipole axis and the point of observation
$r$ is the distance from the dipole

Magnetic Dipole in a Uniform Magnetic Field

Magnetic field when placed in a uniform magnetic field can be expressed as:

1. Torque ( $\tau$ ): A torque aligns the dipole with the magnetic field direction, given by:

$$
\tau=\mu \times B
$$

or $\tau=\mu B \sin \theta$, where $\theta$ is the angle between $\mu$ and $B$.
2. Potential Energy $(U)$ : The potential energy of the dipole in the field is:

$$
U=-\mu \cdot B
$$

or $U=-\mu B \cos \theta$.