Bar Magnet

A bar magnet is the simplest macroscopic object that produces a magnetic field. It behaves as a magnetic dipole — two equal and opposite poles separated by a fixed distance. Its study introduces the language of dipole moment, field lines and the deep statement that magnetic monopoles do not exist.

Concept

A bar magnet has two ends called the North (N) and South (S) poles. Like poles repel and unlike poles attract. The pole strengths $\pm q_m$ are separated by a length $2l$ (the magnetic length), giving a magnetic dipole moment

$\vec m = q_m\,(2\vec l),$

directed from S to N inside the magnet. The SI unit of $|\vec m|$ is $A\,m^2$, and the unit of pole strength is $A\,m$.

Equivalently, every magnetic dipole is a tiny current loop. A planar loop of area $A$ carrying current $I$ has moment

$\vec m = I\vec A,$

with $\vec A$ oriented by the right-hand rule. For an $N$-turn coil, $\vec m = NI\vec A$.

Properties of magnetic field lines

1. They are continuous closed loops — leaving N outside, re-entering at S, and returning inside the magnet from S to N.
2. They never cross.
3. The tangent gives the direction of $\vec B$.
4. Line density at a point is proportional to $|\vec B|$ there.
5. The closure of lines is the geometric content of $\nabla\cdot \vec B = 0$.

No monopoles. Cutting a bar magnet in half always yields two smaller magnets, each with both N and S. Isolated magnetic charges have never been observed.

Derivation

Consider a coil of radius $R$, $N$ turns, carrying current $I$. The current creates a magnetic moment

$m = NIA = NI\,\pi R^2.$

Atoms in a ferromagnet have electronic currents (orbital + spin) acting like microscopic loops; when these align across the body, the loop moments add to give a macroscopic $\vec m$. Hence the equivalence between a bar magnet and a current-carrying solenoid:

$m_{\text{magnet}} = m_{\text{solenoid}} = NIA.$

For a uniformly magnetised cylinder of magnetisation $M$ (moment per volume) and volume $V$,

$m = M V.$

Worked Example

A circular coil of $N = 200$ turns, radius $r = 5.0$ cm, carries $I = 0.40$ A. Its magnetic moment is

$m = NIA = 200 \times 0.40 \times \pi (0.05)^2 = 0.628\,A\,m^2.$

A bar magnet of length $10$ cm and pole strength $q_m = 6.28\,A\,m$ has the same moment:

$m = q_m (2l) = 6.28 \times 0.10 = 0.628\,A\,m^2.$

Common Confusions

• Pole strength is a convenient fiction. The fundamental object is the dipole moment of a current loop.
• $\vec m$ points from S to N inside the magnet; many students draw it outside, from N to S.
• Field lines are continuous everywhere, including inside the magnet — sketches that stop the lines at the poles are wrong.
• Magnetic and geographic poles of Earth are not the same, and the magnetic south is near the geographic north.

Key Takeaways

• A bar magnet is a magnetic dipole with $\vec m$ from S to N.
• $\vec m = q_m (2\vec l)$ for a pole model or $\vec m = NI\vec A$ for a current loop — they are equivalent.
• Magnetic field lines form closed loops; $\nabla\cdot\vec B = 0$.
• Magnetic monopoles do not exist macroscopically.