Bar Magnet as an Equivalent Solenoid

Panel (a): Axial field of a finite solenoid

The left panel shows a finite solenoid — a cylindrical coil of wire — drawn in three dimensions. A dashed horizontal line runs through the centre of the solenoid, labelled O at the centre, and extends to a point P on the right. The solenoid has total length $2l$, with half-length $l$ marked from O to the right end. Its radius is $a$, shown as a vertical line from the axis to the top edge of the solenoid near a thin current-element ring.

A thin shaded ring of width $dx$ is highlighted at a distance $x$ from O along the axis. This ring represents a single current loop of the solenoid. The distance from O to P is labelled $r$. At P, a symbol $\otimes$ indicates the magnetic field points into the page (the field direction along the axis of a solenoid).

The purpose of this panel is to set up the calculation of the axial magnetic field of a finite solenoid. By treating the solenoid as a stack of $N$ circular current loops, each of magnetic moment $d m = (N \, dx / 2l) \cdot I \cdot (\pi a^2)$, and integrating their contributions along the axis, one obtains the total field at P. At large distances ($r \gg l, a$), the result simplifies to the far-axial field of a bar magnet:

where:

• $\mu_0$ is the permeability of free space,
• $m$ is the magnetic moment of the solenoid (or bar magnet),
• $r$ is the distance from the centre O to the point P.

This formula is equation (5.1) in the textbook. It shows that the solenoid behaves like a magnetic dipole at large distances, exactly as a bar magnet does.

Panel (b): Magnetic needle in a uniform field

The right panel shows a uniform magnetic field $\mathbf{B}$ represented by horizontal dashed lines with rightward arrowheads. A slim magnetic needle (a small bar magnet) is placed in this field, tilted at an angle $\theta$ relative to the field direction. The north pole (N) is at the upper tip, the south pole (S) at the lower tip. A dashed reference line is drawn along the field direction, and the angle $\theta$ is marked between the needle and this line.

This panel illustrates the torque experienced by a magnetic dipole (the needle) in a uniform external field. The torque tends to align the needle with the field. The magnitude of the torque is:

where:

• $m$ is the magnetic moment of the needle,
• $B$ is the magnitude of the uniform field,
• $\theta$ is the angle between the needle’s axis and the field.

By measuring the deflection of the needle, one can determine either $B$ (if $m$ is known) or $m$ (if $B$ is known). This is the experimental arrangement described in the caption.

Physical idea …