Current Loop as Magnetic Dipole

Consider a flat, planar loop of wire carrying a steady current $I$. If the loop encloses an area $A$, its magnetic dipole moment is defined as $\vec{\mu} = I A \hat{n}$, where $\hat{n}$ is the unit normal to the plane of the loop. The direction of $\hat{n}$ is given by the right-hand rule: curl the fingers of your right hand in the direction of current flow and your thumb points in the direction of $\vec{\mu}$. The SI unit of magnetic dipole moment is the ampere-square-metre ($\text{A}\,\cdot\,\text{m}^2$), sometimes called the joule per tesla ($\text{J/T}$).

If the wire is wound into $N$ turns (a coil), each turn contributes the same dipole moment, so the total magnetic dipole moment is $\vec{\mu} = N I A \hat{n}$. This is why electromagnets with many turns are so much stronger than single loops. A solenoid of $N$ turns, cross-sectional area $A$, and current $I$ has exactly this dipole moment, and at large distances it produces a magnetic field indistinguishable from a bar magnet with the same $\vec{\mu}$.

When a magnetic dipole $\vec{\mu}$ is placed in a uniform external magnetic field $\vec{B}$, it experiences a torque that tries to align it with the field: $$\vec{\tau} = \vec{\mu} \times \vec{B}$$ The magnitude is $\tau = \mu B \sin\theta$, where $\theta$ is the angle between $\vec{\mu}$ and $\vec{B}$. This torque is zero when the dipole is aligned with $\vec{B}$ (stable equilibrium at $\theta = 0$) and maximum when the dipole is perpendicular to $\vec{B}$. This is exactly how a compass needle behaves: the needle is itself a magnetic dipole that the Earth's field exerts a torque upon.

The potential energy of a magnetic dipole in an external field is obtained by integrating the torque: $$U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\theta$$ The negative sign means energy is lowest (most stable) when $\vec{\mu}$ is parallel to $\vec{B}$ ($\theta = 0$, $U = -\mu B$) and highest when anti-parallel ($\theta = 180°$, $U = +\mu B$). This energy expression is crucial in nuclear magnetic resonance (NMR) and MRI, where the energy difference between aligned and anti-aligned proton dipoles is detected as a radio-frequency signal.

At distances $r$ much larger than the size of the loop, the magnetic field produced by the dipole has components (in spherical polar coordinates with the dipole at the origin pointing along $\hat{z}$): $$B_r = \frac{\mu_0}{4\pi} \frac{2\mu \cos\theta}{r^3}, \quad B_\theta = \frac{\mu_0}{4\pi} \frac{\mu \sin\theta}{r^3}$$ Notice the $1/r^3$ dependence — dipole fields fall off much faster than the $1/r^2$ of a monopole. This is why the field of a small bar magnet weakens rapidly with distance. These equations are the magnetic analogue of the electric dipole field, with $\mu_0/4\pi$ replacing $1/(4\pi\epsilon_0)$ and $\mu$ replacing the electric dipole moment $p$.

Electrons orbiting atomic nuclei constitute tiny current loops, and their orbital motion gives each atom a magnetic dipole moment. Additionally, electrons possess an intrinsic magnetic moment due to quantum spin: $\mu_s = g_s \frac{e}{2m_e} \hbar$, where $g_s \approx 2$ is the electron g-factor and $\hbar$ is the reduced Planck constant. In ferromagnetic materials like iron, quantum-mechanical exchange interactions cause neighbouring atomic dipoles to align, forming magnetic domains. The macroscopic magnetism of a magnet is therefore the cooperative effect of $\sim 10^{23}$ atomic current loops all acting as magnetic dipoles.

• \vec{\mu} — Magnetic dipole moment (A·m² (or J/T)): Vector quantity characterising the strength and orientation of the loop's magnetic effect.
• N — Number of turns (dimensionless): The number of complete loops of wire wound in the coil.
• I — Current (A (amperes)): The steady electric current flowing through the wire.
• A — Area enclosed by loop (m²): The area of the flat surface bounded by the current loop.
• \hat{n} — Unit normal vector (dimensionless): Direction perpendicular to the plane of the loop, found by the right-hand rule.

Problem. A circular coil has 50 turns, a radius of 4.0 cm, and carries a current of 2.5 A. (a) Calculate the magnetic dipole moment of the coil. (b) The coil is placed in a uniform magnetic field of magnitude 0.30 T with its axis making a 60° angle with the field. Find the torque on the coil and its potential energy in this orientation.

Answer. Magnetic dipole moment 0.628 A·m²; torque 0.163 N·m; potential energy −0.094 J.

• Magnetic Force on a Current-Carrying Wire — The torque on a current loop in a field is derived from the forces on its straight segments; understanding $\vec{F} = I\vec{L} \times \vec{B}$ is a prerequisite.
• Biot–Savart Law — The far-field dipole equations are derived by applying the Biot–Savart Law to a small current loop and taking the limit $r \gg R$.
• Magnetic Domains and Ferromagnetism — Ferromagnetism arises when atomic magnetic dipoles (current loops) align cooperatively; the dipole moment concept directly underpins this topic.
• Electric Dipole Moment — The magnetic dipole is the direct magnetic analogue of the electric dipole; their field structures and energy expressions are mathematically identical.
• Magnetic Field of a Solenoid — A solenoid is a stack of current loops; its total dipole moment $\mu = NIA$ and its external field matches the dipole field at large distances.