The Bar Magnet

A Dipole and Its Field Lines

When iron filings are sprinkled on a glass sheet placed over a short bar magnet, they arrange themselves in a pattern that reveals two poles, mimicking the positive and negative charge of an electric dipole. One end is designated the north pole and the other the south pole; a freely suspended magnet aligns these poles approximately along the geographic north and south. The same pattern of filings appears around a current-carrying solenoid, the first hint that the two objects are physically equivalent.

The filings trace out magnetic field lines, a visual realisation of the field B. Their properties are precise and examinable. The lines form continuous closed loops, unlike electric-dipole lines that begin on a positive charge and end on a negative one. The tangent at any point gives the direction of B there, and the local crowding of lines indicates the field's strength. Field lines never intersect, since two tangents at a crossing would make the field direction ambiguous.

Crucially, not all field lines emanate from the north pole or converge on the south. Inside the magnet the lines run from south to north, completing each loop, so the net flux through a closed surface drawn around either pole is zero. This continuity is the experimental basis for Gauss's law of magnetism, treated in the sibling note.

A second, important caution accompanies the term "field line." In some older texts these are called magnetic lines of force, a name NCERT deliberately avoids. Unlike the electrostatic case, the tangent to a magnetic field line does not give the direction of the force on a moving charge: the magnetic force is $q\mathbf{v}\times\mathbf{B}$, always perpendicular to B. The lines map the field's direction and, by their density, its strength, but they are not force trajectories. This distinction is a favourite source of conceptual NEET items, where a figure of "lines of force on a charge" is offered as a distractor.

Bar Magnet as an Equivalent Solenoid

A current loop acts as a magnetic dipole, and Ampere's hypothesis holds that all magnetic phenomena can be explained in terms of circulating currents. The resemblance of the field lines of a bar magnet and a solenoid suggests that the magnet is, in effect, a stack of such circulating currents. Cutting a bar magnet in half is like cutting a solenoid: one obtains two smaller magnets, each with weaker but complete dipole properties, the field lines remaining continuous.

The analogy is made firm by computing the axial field of a finite solenoid at a far point. At large distance the solenoid's axial field has the same form as the experimentally measured far axial field of a bar magnet. The conclusion NEET tests is compact: the magnetic moment of a bar magnet equals the magnetic moment of the equivalent solenoid that produces the same field.

This equivalence is more than a picture; it is the route by which solenoid numericals are converted into dipole numericals. A closely wound solenoid of $N$ turns, cross-sectional area $A$, carrying current $I$, has magnetic moment $m = NIA$. Once that single number is known, the solenoid may be treated exactly as a bar magnet: it experiences the same torque $mB\sin\theta$ in an external field and stores the same energy $-\mathbf{m}\cdot\mathbf{B}$. The reverse reading is equally useful for examination logic: a permanent bar magnet's moment can be quoted in $\text{A m}^2$ precisely because it is the moment of the circulating currents it stands in for.

Dipole in a Uniform Field: Torque

Place a small needle of known magnetic moment m in a uniform field B. The two poles feel equal and opposite forces, so the net force is zero, but the forces form a couple. The resulting torque is

$$ \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}, \qquad \tau = mB\sin\theta $$

where $\theta$ is the angle between m and B. This restoring torque vanishes when the needle aligns with the field ($\theta = 0$) and is maximum when it lies perpendicular to the field ($\theta = 90^\circ$). The torque is what drives the oscillations of a compass needle and underlies the oscillation-period numericals NEET favours.

Magnetic Potential Energy

Rotating the dipole against the restoring torque stores potential energy. Integrating $\tau = mB\sin\theta$ over the angle gives the magnetic potential energy

$$ U_m = \int \tau\, d\theta = -\,mB\cos\theta = -\,\mathbf{m}\cdot\mathbf{B} $$

Taking the constant of integration as zero fixes the zero of energy at $\theta = 90^\circ$, where the needle is perpendicular to the field. The energy is then minimum, $-mB$, at $\theta = 0^\circ$ (the most stable position, moment parallel to field) and maximum, $+mB$, at $\theta = 180^\circ$ (the most unstable position, moment antiparallel). The choice of the $90^\circ$ reference is a convention; as in electrostatics the zero of potential energy may be fixed wherever convenient, so only differences in $U$ carry physical meaning. The work an external torque must do to rotate the dipole from $\theta_1$ to $\theta_2$ is $W = U(\theta_2) - U(\theta_1) = mB(\cos\theta_1 - \cos\theta_2)$, which is the quantity NEET asks for in "work to align" problems.

Axial and Equatorial Fields

For a short bar magnet of size $l$ and moment $m$, the field at a distance $r \gg l$ from its midpoint takes two standard forms. On the axis (the end-on position), the field points along the moment:

$$ \mathbf{B}_A = \frac{\mu_0}{4\pi}\,\frac{2\mathbf{m}}{r^3} $$

On the equatorial line (the normal bisector, broadside-on), the field points opposite to the moment:

$$ \mathbf{B}_E = -\,\frac{\mu_0}{4\pi}\,\frac{\mathbf{m}}{r^3} $$

At equal distance the axial field is exactly twice the equatorial field in magnitude, a ratio NEET probes repeatedly.

The Electrostatic Analog

The equations for torque, energy, and field on a magnetic dipole map term-for-term onto those of an electric dipole. The magnetic field at large distance is obtained from the electric-dipole result by the substitutions $\mathbf{E} \to \mathbf{B}$, $\mathbf{p} \to \mathbf{m}$, and $1/\varepsilon_0 \to \mu_0$. Memorising the analogy converts every electrostatics dipole formula into its magnetic counterpart.