Q.A bar magnet of length '$\ell$' and magnetic moment '$P_m$' is bent in the form of an arc as shown in figure. The new magnetic dipole moment will be :

Magnetic Dipole Moment – From Intuition to Precision

Think of a bar magnet. It has a north pole and a south pole. If you place it in a magnetic field, it tries to turn — the north pole is pulled one way, the south pole the opposite way. That turning effect (torque) is the most basic sign that something is a magnetic dipole.

A current loop behaves exactly the same way. A circular wire carrying current, when placed in a magnetic field, also feels a torque and tries to align itself. That is the deep insight: a tiny current loop and a bar magnet are the same kind of object — a magnetic dipole.

The Intuitive Picture

Imagine a small, flat loop of wire carrying a steady current $I$. The loop has an area $A$. The direction of the loop is defined by its area vector $\mathbf{A}$ — perpendicular to the plane of the loop, following the right-hand rule (curl your fingers along the current, thumb points along $\mathbf{A}$).

Now place this loop in a uniform magnetic field $\mathbf{B}$. What happens?

• If the loop is perpendicular to $\mathbf{B}$, nothing turns — it's already aligned.
• If the loop is parallel to $\mathbf{B}$, it feels maximum torque, trying to flip it perpendicular.
• If the loop is at some angle, the torque is somewhere in between.

That torque depends on three things: the current $I$, the area $A$, and the angle between the loop and the field. The combination $I A$ is the magnetic dipole moment of the loop.

For a bar magnet, the same idea applies: $\mathbf{m}$ points from the south pole to the north pole (yes, that's the convention — the moment points northward), and its magnitude tells you how strong the dipole is.

The Precise Statement

A magnetic dipole moment $\mathbf{m}$ is a vector that characterises the strength and orientation of a magnetic dipole. For a current loop:

where $I$ is the current and $\mathbf{A}$ is the area vector (magnitude = area, direction = perpendicular to the loop by right-hand rule). For a bar magnet, $\mathbf{m}$ points from south to north, and its magnitude is roughly $m = p \, l$, where $p$ is the pole strength and $l$ is the separation between poles.

What Happens in a Magnetic Field?

Two key results follow directly from the definition.

Torque: The field tries to align the dipole with itself. The torque is:

The magnitude is $\tau = m B \sin\theta$, where $\theta$ is the angle between $\mathbf{m}$ and $\mathbf{B}$. Maximum torque when they are perpendicular ($\theta = 90^\circ$), zero when aligned ($\theta = 0^\circ$).

Potential Energy: A dipole in a field has energy that depends on its orientation:

The lowest energy ($U = -mB$) is when $\mathbf{m}$ is parallel to $\mathbf{B}$ — the stable equilibrium. The highest energy ($U = +mB$) is when they are antiparallel — the unstable equilibrium. …