Magnetic force, torque and magnetic materials

Why it matters

Magnetic forces and torques drive every moving-coil (PMMC) meter, galvanometer, loudspeaker and motor, and the force between conductors decides how busbars must be braced against short-circuit currents. Magnetic materials decide how much flux a core carries for a given coil current, and how much energy is lost each cycle in transformers, CTs and inductive sensors.

Key ideas

Force on a moving charge. A charge Q moving with velocity u in a field B feels F = Q u × B. The force is perpendicular to both u and B, so it changes the direction of motion but does no work and cannot change the particle's speed or kinetic energy. For a negative charge the force is reversed.

Force on a current element. A current is moving charge, so a wire element feels dF = I dl × B. For a straight wire of length L in a uniform field, F = I L × B, magnitude BIL sin θ. Fleming's left-hand rule gives the same direction.

Force between parallel conductors. Each wire sits in the other's field μ₀I/(2πd). Currents in the same direction attract; opposite directions repel. Force per metre = μ₀I1I2/(2πd).

Torque on a loop. In a uniform field the net force on a closed loop is zero, but there is a torque. Define the magnetic dipole moment m = N I S a_n (A·m²), with a_n normal to the loop by the right-hand rule. Then T = m × B, magnitude NIAB sin θ, where θ is the angle between the loop's normal and B. The torque is maximum when the plane of the loop is parallel to B (θ = 90°) and zero when the plane is perpendicular to B (θ = 0). In a PMMC meter a radial field keeps θ at 90°, so the deflecting torque NBAI is proportional to current — giving a linear scale.

Magnetisation. Bound atomic currents (electron orbits and spins) act as tiny dipoles. Their dipole moment per unit volume is the magnetisation M (A/m). B = μ₀(H + M); for a linear material M = χm H, so B = μ₀(1 + χm)H = μ₀μr H.

Classes of magnetic material:

• Diamagnetic (copper, silver, water, bismuth): χm small and negative (about −10⁻⁵), μr slightly below 1; weakly repelled. Superconductors are perfect diamagnets (Meissner effect).
• Paramagnetic (aluminium, platinum, oxygen): χm small and positive (10⁻⁵ to 10⁻³); weakly attracted; no retained magnetisation.
• Ferromagnetic (iron, nickel, cobalt and their alloys, ferrites): μr from hundreds to over 10⁵, non-linear and history-dependent. Only ferromagnets have domains. Above the Curie temperature they become paramagnetic.

Hysteresis. The B–H loop of a ferromagnet shows saturation, remanence Br (B left at H = 0) and coercivity Hc (H needed to bring B back to zero). The loop area is the energy lost per cycle per unit volume. Soft materials (silicon steel, ferrite, permalloy) have narrow loops — used in transformer, CT and inductor cores. Hard materials (alnico, NdFeB) have wide loops — permanent magnets for PMMC meters and motors. Eddy-current loss is reduced by laminating cores or using high-resistivity ferrites.

Magnetic boundary conditions. Normal B is continuous (B1n = B2n). Tangential H jumps by the surface current density: H1t − H2t = K; with no surface current, H1t = H2t.

Formulas

• F = Q (E + u × B) — Lorentz force on a charge (N); u in m/s, B in T.
• F = I L × B, |F| = BIL sin θ — straight conductor in a uniform field.
• F/L = μ₀ I1 I2 / (2πd) — between long parallel wires (N/m); d = spacing (m).
• m = N I A a_n — magnetic dipole moment (A·m²).
• T = m × B, |T| = N I A B sin θ — torque on a coil (N·m); θ between the coil's normal and B.
• B = μ₀(H + M) = μ₀μr H, M = χm H, μr = 1 + χm.
• B1n = B2n; H1t − H2t = K (K in A/m); tan θ1 / tan θ2 = μr1 / μr2 with no surface current.
• W_h = η B_max^1.6 f V — Steinmetz hysteresis loss (W); η is an empirical material constant — take it from a data book.

Worked examples

Example 1 (standard, torque). A 10-turn square coil of side 0.1 m carries 5 A in a uniform 0.2 T field; its normal makes 30° with B. Find the torque, and the maximum torque.

1. A = 0.1 × 0.1 = 0.01 m²; m = NIA = 10 × 5 × 0.01 = 0.5 A·m².
2. T = m B sin θ = 0.5 × 0.2 × sin 30° = 0.05 N·m.
3. Maximum (plane of coil parallel to B, θ = 90°): 0.5 × 0.2 = 0.1 N·m.

Example 2 (force between conductors). Two long parallel busbars 0.1 m apart each carry 100 A in the same direction. Find the force per metre.

1. F/L = μ₀I1I2 / (2πd) = 2 × 10⁻⁷ × 100 × 100 / 0.1 = 0.02 N/m, attractive.
2. Under a 10 kA fault current in both, the force rises by (100)² to 200 N/m — the reason busbar supports are rated for short-circuit forces.

Example 3 (GATE level, magnetic boundary). The plane z = 0 separates region 1 (z < 0, μr1 = 4) from region 2 (z > 0, μr2 = 1). There is no surface current. In region 1, H1 = 2a_x + 3a_z A/m. Find H2, B2 and M1.

1. Tangential H continuous: H2x = 2 A/m.
2. Normal B continuous: μr1 H1z = μr2 H2z → H2z = 4 × 3 / 1 = 12 A/m.
3. H2 = 2a_x + 12a_z A/m; B2 = μ₀H2 = 2.51a_x + 15.08a_z μT.
4. Region 1: χm = μr1 − 1 = 3, so M1 = 3H1 = 6a_x + 9a_z A/m.
5. Check: B1z = μ₀ × 4 × 3 = 15.08 μT = B2z ✓.

Common mistakes

• Measuring θ in the torque formula from the plane of the coil instead of its normal (this swaps sin and cos).
• Forgetting the sign of the charge: an electron's force is opposite to that given by u × B.
• Thinking a magnetic field can speed up a charge — it cannot; only E does work.
• Treating μr of iron as a constant: it depends on H and the magnetic history.
• Saying paramagnetic or diamagnetic materials have domains — only ferromagnets do.
• Confusing "same-direction currents attract" with like charges repelling.

For GATE IN

Expect: force on a charge or conductor (often with vector cross products), force between parallel wires, torque on a coil and the PMMC torque equation T = NBAI, magnetisation and susceptibility relations, magnetic boundary conditions at an iron–air interface, and identification of material classes from μr or χm. Practise cross products quickly and keep track of which angle a question gives.

Quick check

1. What work does a static magnetic field do on a moving charge?
2. Two parallel wires carry currents in opposite directions — attract or repel?
3. When is the torque on a coil in a uniform field zero?
4. What is the sign of χm for copper?
5. Which component of B is continuous across any interface?

Answers: 1. None. 2. Repel. 3. When the coil's plane is perpendicular to B (normal parallel to B). 4. Negative (diamagnetic). 5. The normal component.