Bar magnets, magnetic poles and field geometry

A bar magnet produces a magnetic field whose external lines are conventionally drawn leaving its north pole and entering its south pole, continuing within the magnet so that the complete lines form loops. Unlike positive and negative electric charges, isolated magnetic north or south poles have not been observed in standard classical experiments. Cutting an ordinary bar magnet produces smaller magnets, each with its own north and south regions, rather than an isolated monopole.

The field direction at a point is the direction a small north-seeking compass pole would be pushed or, more accurately, the equilibrium alignment of a test compass with the local field. The field is strongest near typical pole regions in a simple bar-magnet drawing but the precise spatial field depends on magnet geometry and magnetization. A common examination error is to draw magnetic field lines beginning or ending at an isolated magnetic charge; the magnet’s closed-line character is the essential difference from electrostatic sketches.

Magnetic dipole moment and torque in a field

A sufficiently small bar magnet or current loop can be modeled as a magnetic dipole. Its magnetic dipole moment m has units A·m² and points from magnetic south toward magnetic north for the usual bar-magnet convention. In a uniform external magnetic field B, the torque on an ideal dipole is τ=m×B, with magnitude mB sinθ. The torque tends to align the dipole moment with the applied field.

The potential energy is U=−m·B, which is smallest when m is parallel to B. A perfectly uniform B produces no net force on an ideal rigid dipole, although the torques may cause rotation. A nonuniform field can produce both net force and torque. The expression must not be confused with electric dipole moment p, measured in coulomb-metres. A bar magnet is a model of distributed currents and magnetic moments; the magnetic-pole picture is a convenient analogy, not an assertion of independently existing positive and negative magnetic charges.

Axial and equatorial fields of a short magnetic dipole

For a compact magnetic dipole of moment magnitude m observed at distances much greater than its size, the axial field magnitude in vacuum is B_axial=(μ₀/4π)(2m/r³), directed along the dipole axis in the appropriate sense. On the equatorial plane the field magnitude is B_equatorial=(μ₀/4π)(m/r³), directed opposite to the dipole moment vector for the usual dipole orientation. Both are far-field approximations.

At equal large distances from the same ideal short dipole, the axial field magnitude is twice the equatorial field magnitude. This ratio is often tested conceptually. The 1/r³ dependence means the dipole field decays more rapidly than the 1/r field of an infinite current-carrying wire or the 1/r² field of a magnetic field component associated with specific source geometries. Never use the dipole far-field expression close to a bar magnet where its finite size is comparable to observation distance.

Gauss’s law for magnetism

Gauss’s law for magnetism states ∮B·dA=0 over any closed surface. It expresses the absence of net magnetic flux emanating from a closed surface and is equivalent in ordinary classical field theory to the divergence equation ∇·B=0. Field lines that enter a closed surface also leave it, accounting for zero net outward flux. It does not say the magnetic field itself is zero everywhere on that surface.

An imaginary sphere placed around a bar magnet has magnetic field entering through some parts and exiting through others. Their oriented fluxes cancel, although the field at individual points can be significant. This is analogous to Gauss’s law for electric fields in mathematical form but has a crucial different right-hand side: electric flux equals enclosed electric charge divided by ε₀, whereas closed magnetic flux is zero without any magnetic-monopole source. Use oriented area normals when interpreting the sign of flux.

Magnetisation and magnetic field intensity

A magnetic material contains atomic-scale magnetic moments that can become aligned in response to an applied field. Magnetisation M is defined as magnetic dipole moment per unit volume, with SI units A/m. Magnetic field intensity H is an auxiliary field related to free current and magnetisation; in SI, B=μ₀(H+M) within matter. The relation separates the externally driven field and the response of the material, though their distributions can be complex in real geometries.

For an isotropic linear material in an appropriate weak-field region, M=χ_m H and hence B=μ₀(1+χ_m)H=μ₀μ_rH, with μ_r=1+χ_m. These simple proportionalities cannot be blindly extended to ferromagnets close to saturation or to materials with hysteresis. A material’s magnetisation depends on prior history, temperature, shape and external field. Field intensity H and magnetic induction B have different SI units: A/m for H and tesla for B.

Magnetic susceptibility, permeability and classification

Magnetic susceptibility χ_m measures the response M/H in a simple linear isotropic regime and is dimensionless in SI. Relative permeability μ_r is also dimensionless. Diamagnetic materials have χ_m slightly negative, paramagnetic materials have χ_m positive and typically small, while ferromagnetic materials may have very large, nonlinear, history-dependent magnetic responses. The classification reflects microscopic magnetic moments and their interactions.

A useful basic distinction is that diamagnetism typically produces a moment opposing the applied field, whereas paramagnetism tends to align existing moments with it. Ferromagnetism involves cooperative alignment and domain formation. In exam problems claiming μ_r=1+χ_m, verify the assumed linear material model; it is not a general constant relationship over all fields for a ferromagnet. Also avoid claiming every paramagnetic solid strongly attracts an ordinary hand magnet; many paramagnetic responses are far too small to notice without sensitive instruments.

Diamagnetism and its microscopic explanation

Diamagnetism occurs because an applied magnetic field modifies electronic orbital motion and creates an induced magnetic moment that tends to oppose the applied field. Ordinary diamagnets have weak negative susceptibility and are slightly repelled by stronger-field regions. Examples commonly discussed include bismuth, copper and water, although every material has some microscopic diamagnetic contribution to its overall response.

Diamagnetic response is generally much weaker than the ferromagnetic attraction exhibited by iron under familiar conditions. Describing diamagnetic substances as ‘unmagnetic’ is imprecise because their induced response is small but real. Superconductors exhibit a distinctive Meissner effect, expelling magnetic field from their bulk under ideal conditions; this is stronger and fundamentally different from ordinary weak diamagnetism. In practical magnetic levitation, field gradients and stability matter as much as a material’s sign of susceptibility.

Paramagnetism and Curie’s law

Paramagnetic substances contain atomic or molecular magnetic moments that tend to align with an external magnetic field. Thermal agitation opposes alignment, so for many ideal dilute paramagnets in appropriate regimes susceptibility approximately obeys Curie’s law χ_m=C/T, where C is a material constant and T must be in kelvin. Their susceptibility is positive and usually weak compared with strong ferromagnets.

The Curie relation is a model with limits, not a universal law of all real paramagnets. Some materials exhibit temperature-independent paramagnetism or more complex interactions. Examples often cited include aluminium and oxygen gas. A paramagnetic substance placed in a nonuniform field can be pulled toward a stronger-field region because its induced alignment is favourable, but the magnitude depends on its susceptibility and geometry. Do not confuse Curie’s law for paramagnets with Curie temperature, which characterises the loss of ferromagnetic ordering in many materials.

Ferromagnetism, domains and spontaneous order

Ferromagnetic materials such as iron, cobalt and nickel can exhibit strong collective alignment of magnetic moments. A sample may contain domains, each magnetized strongly in a different direction, so its total net moment can be small before application of a field. An external field changes domain orientations and boundaries, producing large magnetization and often a nonlinear response. This cooperative ordering is a microscopic material property, not merely stronger ordinary paramagnetism.

Domain behaviour depends on crystalline structure, defects and sample shape. Heating a ferromagnet above its Curie temperature ordinarily destroys its long-range ferromagnetic ordering, and the material becomes paramagnetic in the relevant temperature range rather than losing all possible magnetic response. The material may recover ferromagnetism after cooling, subject to changes in microstructure and applied field. Magnets used in engineering are selected for desired remanence, coercivity and maximum operating temperature rather than susceptibility alone.

Hysteresis, coercivity and retentivity

When a ferromagnet is magnetised and then the applied H field is reduced, B or M may not retrace the same path followed during magnetisation. This history dependence is hysteresis. Remanence describes the residual magnetic induction or magnetization after the applied field is removed. Coercivity describes the magnitude of reverse field needed to bring magnetization or induction to a chosen zero, depending on the particular convention used.

The area enclosed by a B–H hysteresis loop represents energy dissipated as heat per unit volume in one complete magnetization cycle under the usual magnetoquasistatic interpretation. A narrow loop with low coercivity is desirable for many transformer cores that reverse magnetisation repeatedly, while a wide loop with larger coercivity helps permanent magnets resist demagnetisation. The ideal distinction between soft and hard magnetic materials concerns magnetic behaviour, not how easy the sample is to scratch mechanically. Saturation means magnetization approaches a maximum and does not increase linearly without bound.

Earth’s magnetic field, declination and dip

Earth has a large-scale magnetic field that can be approximately described as a tilted dipole near its surface, although the real geomagnetic field is more complex. A compass responds to the local horizontal component and points approximately toward magnetic north, which may differ from geographic north. Magnetic declination is the angle between the geographic north direction and the magnetic meridian. Magnetic inclination or dip is the angle between the local field and horizontal plane.

At some locations the magnetic field has a strong downward or upward vertical component, whereas near a magnetic equator the inclination is approximately zero. Magnetic poles move over time, so fixed compass values should not be asserted without geographical and date context. A dip needle measures inclination; a compass suspended to rotate horizontally primarily indicates the horizontal magnetic direction. Questions that involve ‘magnetic north’ and ‘geographic north’ require clear distinction, especially in navigation and earth-science applications.

Magnetic shielding, soft iron and practical applications

High-permeability materials can redirect magnetic flux, reducing field strength in protected regions for certain low-frequency or static configurations. Such shielding is not the same as electrostatic shielding: magnetic fields cannot simply be blocked by an ideal thin conductor in the DC limit. Soft magnetic materials are useful in transformer cores, motor stators and electromagnets because magnetization can be changed with comparatively small hysteresis loss.

Hard magnetic materials with high coercivity are useful in permanent magnets and data-storage systems where magnetic states must persist. Modern ferrites and engineered alloys are chosen according to frequency, saturation, loss and temperature requirements. Designing a magnetic circuit requires geometry and demagnetising fields as well as material permeability. Competitive questions often oversimplify ‘soft iron’ as having no remanence; a more accurate statement is that appropriate soft magnet materials generally have low coercivity and relatively narrow loops compared with permanent-magnet materials.

Exam reasoning with magnetic materials and fields

To solve a magnetic material question, decide whether it asks about microscopic response, field-line topology, a dipole torque, a B–H curve or a compass orientation. First check whether the problem describes an ideal linear isotropic material. If yes, χ_m and μ_r relations may be used; if not, a nonlinear hysteresis curve or qualitative material behaviour is needed. Then distinguish positive and negative susceptibility and describe the likely material class.

For a dipole field problem, specify whether the observation point is on the axis or equatorial plane and sufficiently distant. For hysteresis, identify whether a question asks about coercivity, remanence, energy loss or saturation. For terrestrial magnetism, identify whether the angle is declination in the horizontal plane or inclination relative to horizontal. A correct conceptual classification usually solves these questions faster than memorising a large set of numerical values without context.

Formula sheet with model assumptions

The simple susceptibility and permeability relationships apply only to the indicated linear isotropic model. Dipole field formulas are far-field approximations.

Solved numerical applications

UPSC examination PYQ concept practice

The following are paraphrased NDA General Ability Test-style topics and closely related magnetic-material concepts. They are not verbatim original UPSC papers or authenticated exact question identifiers; check official papers for direct attribution.

20 UPSC-style original MCQs with explanations