Moving Charges and Magnetism

2.1 Core concepts

For most of human history electricity and magnetism were thought to be unrelated. The decisive experimental link came in 1820 when Hans Christian Oersted noticed that a compass needle placed near a wire was deflected the moment a current flowed through the wire (NCERT §4.1, p. 108). The deflection traced out a tangential pattern: the needle aligned itself along circles concentric with the wire, with the sense given by the right-hand rule. The conclusion was inescapable — moving charges (currents) produce magnetic fields in the space around them. This single observation triggered the development, over the next half-century, of the entire classical electromagnetic theory of Ampere, Faraday and Maxwell.

The complete force on a point charge q moving with velocity v through a region carrying simultaneously an electric field E and a magnetic field B is the Lorentz force:

F = q[E + v × B] (NCERT Eq. 4.3, §4.2.2, p. 109).

Three features of the magnetic part F_mag = q(v × B) deserve particular attention. (i) It is perpendicular to both v and B; it can change the direction of motion but never the magnitude of the velocity, so the magnetic force does no work. (ii) It vanishes whenever v is parallel (or anti-parallel) to B — only the component of v perpendicular to B contributes. (iii) It is zero for a stationary charge (v = 0). The SI unit of B is the tesla (T); 1 T = 1 N·s/(C·m). A handy non-SI unit is the gauss (1 G = 10⁻⁴ T); Earth's magnetic field at the surface is about 3 × 10⁻⁵ T = 0.3 G.

The same formula applied to a macroscopic current gives the force on a current-carrying conductor. A straight wire of length l carrying current I in an external uniform field B experiences the Laplace force

F = I l × B (NCERT Eq. 4.4, §4.2.3, p. 110),

where the direction of l is taken along the current. This is just the Lorentz force on the drifting charge carriers, summed over the wire's cross-section and length.

Motion of a charged particle in a magnetic field (NCERT §4.3, p. 112). When v is perpendicular to a uniform B, the magnetic force qvB acts always at right angles to v, providing exactly the centripetal force for circular motion of radius r:

qvB = mv²/r ⇒ r = mv/(qB).

The angular speed is ω = qB/m, independent of v, and the period is T = 2π/ω = 2πm/(qB). The cyclotron frequency ν_c = ω/(2π) = qB/(2πm) is the rate at which the particle circles the field line; remarkably, it depends only on the charge-to-mass ratio and B, not on the particle's speed or orbital radius. This is precisely the feature that makes the cyclotron particle accelerator work: a fixed-frequency oscillating electric field stays in resonance with the orbiting particle even as it gains energy.

If v has a component v_∥ along B, the parallel component is unaffected by the magnetic force, so the particle drifts steadily along B while spiralling around it — a helical trajectory. The pitch of the helix (distance advanced along B per revolution) is

p = v_∥ T = 2π m v_∥/(qB) (NCERT Eq. 4.6b, p. 113).

This is the basic mechanism behind aurora, magnetic mirror traps and the curving particle tracks seen in cloud chambers.

Biot–Savart law (NCERT §4.4, p. 113–114). The fundamental law relating a current element to the field it produces was inferred by Jean-Baptiste Biot and Félix Savart in 1820. For a current element I dl at position vector r from the field point P,

dB = (μ₀/4π) (I dl × r̂)/r² (NCERT Eq. 4.7a),

with magnitude |dB| = (μ₀/4π)(I dl sin θ)/r² where θ is the angle between dl and r̂. The proportionality constant is

μ₀/(4π) = 10⁻⁷ T m A⁻¹, so μ₀ = 4π × 10⁻⁷ T m A⁻¹,

the permeability of free space. Several immediate observations: the field is perpendicular to the plane containing dl and r̂; it vanishes along the direction of dl itself (sin 0 = 0); and superposition lets us build up the field of any current by integrating over all the elements.

Field on the axis of a circular loop (NCERT §4.5, p. 116). For a single circular loop of radius R carrying current I, the field at axial distance x from the loop's centre is

B(x) = μ₀ I R²/[2(x² + R²)^{3/2}] (NCERT Eq. 4.11),

directed along the axis. At the centre (x = 0):

B₀ = μ₀ I /(2R).

For N tightly-wound turns the field at the centre is B = μ₀ N I /(2R). Far from the loop (x ≫ R) the field falls off as 1/x³ — the loop behaves like a magnetic dipole of moment m = IA = πR²I.

Ampere's circuital law (NCERT §4.6, p. 118). Although the Biot–Savart law solves every problem in principle, the integration is usually messy. For configurations with enough symmetry, Ampere's law is much easier. It states:

∮ B·dl = μ₀ I_enc

around any closed loop, where I_enc is the total steady current passing through any surface bounded by the loop. Three classic applications:

• Long straight wire: a coaxial circular Amperian loop of radius r gives B (2πr) = μ₀ I ⇒ B = μ₀ I/(2πr) (NCERT Eq. 4.14, p. 119). The field forms tangential circles, as Oersted first observed.
• Thick wire with uniform current density: for r > a (outside), the result is the same as a thin wire (∝ 1/r); for r < a (inside), the enclosed current scales as r²/a², giving B = μ₀ I r / (2π a²) — a linear growth from zero at the axis to B_max at the surface (NCERT Example 4.7, p. 120).
• Long solenoid: a rectangular Amperian loop with one side parallel to the axis inside the solenoid and the opposite side far outside (where B ≈ 0) gives B = μ₀ n I inside, where n is the number of turns per unit length (NCERT Eq. 4.16, §4.7, p. 122). The field is uniform throughout the bulk of a long solenoid and is the workhorse field for laboratory experiments. A toroid is a closed solenoid bent into a ring; by the same Amperian argument B = μ₀ n I inside the core and zero outside (NCERT p. 122). Force between two parallel currents (NCERT §4.8, p. 123–124). Two long parallel wires a distance d apart carrying currents I_a and I_b exert magnetic forces on each other. The first wire creates a field B_a = μ₀ I_a/(2πd) at the position of the second; the second wire then experiences a force per unit length f = μ₀ I_a I_b / (2π d) (NCERT Eq. 4.19, p. 123). Parallel currents attract (the opposite of the rule for like electric charges!); antiparallel currents repel. This very effect defines the SI unit of current: the ampere is the steady current that, flowing in each of two long parallel wires 1 m apart in vacuum, produces a force of 2 × 10⁻⁷ N per metre of length on each. Torque on a current loop (NCERT §4.9.1, p. 124–126). Place a rectangular loop of N turns carrying current I in a uniform magnetic field B. Its area vector A makes angle θ with B. The forces on opposite arms cancel, so the net force is zero, but a couple acts producing a torque τ = N I A B sin θ, or in vector form τ = m × B, with magnetic moment m = N I A (NCERT Eq. 4.23). The moment m is taken along the area-vector normal (right-hand rule). The torque tries to align m with B; it is zero in stable equilibrium (m ∥ B), zero in unstable equilibrium (m ∥ −B), and maximum when m ⊥ B. Magnetic dipole equivalence (NCERT §4.9.2, p. 128–129). A planar current loop behaves exactly like a magnetic dipole of moment m = NIA. At large axial distance x ≫ R, B_axial = (μ₀/4π)(2m/x³); on the equatorial line, B_equ = (μ₀/4π)(m/x³). These have exactly the same form as the electric dipole field with the replacement (1/4πε₀) → (μ₀/4π) and p → m — the deep electrostatic-magnetic analogy. Moving-coil galvanometer (MCG) (NCERT §4.10, p. 130–131). A rectangular coil of N turns of area A is suspended in a radial magnetic field (a soft-iron core curves the field so that B is always tangential to the coil's motion). The magnetic torque NIAB is balanced by the elastic torque kφ of a spring, giving deflection φ proportional to current: φ = (NAB/k) I. The current sensitivity is dφ/dI = NAB/k; the voltage sensitivity is dφ/dV = NAB/(kR), where R is the coil's resistance. A galvanometer is converted to an ammeter by connecting a small shunt resistance r_s in parallel: most of the circuit current bypasses the meter through the shunt, and the parallel combination R_G r_s/(R_G + r_s) ≈ r_s is much smaller than R_G, so the ammeter has very low resistance and barely disturbs the circuit. It is converted to a voltmeter by connecting a large resistance R in series, so the meter draws negligible current; the voltmeter has very high resistance and is connected across the element whose voltage is being measured.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 4.1 (p. 108): Magnetic field lines around a long straight current as concentric circles; iron-filing pattern that Oersted observed.
• Fig. 4.2 (p. 109): Right-hand rule for direction of v × B for positive and negative charges.
• Fig. 4.5 / 4.6 (p. 112): Circular motion of a charge perpendicular to B; helical motion when v has a component along B.
• Fig. 4.7 (p. 113): Geometry of the Biot–Savart law showing dl, r and the dB direction perpendicular to their plane.
• Fig. 4.9 / 4.10 (p. 115–116): Axial field of a circular current loop; the closed-loop magnetic field-line pattern that makes a current loop look like a tiny bar magnet.
• Fig. 4.13 / 4.14 (p. 119–120): Amperian loops for a thick wire (r < a and r > a), and B vs r plot showing linear growth inside, 1/r decay outside.
• Fig. 4.15 / 4.16 (p. 121): Solenoid — stretched-out side view showing inter-turn cancellation; rectangular Amperian loop abcd used to derive B = μ₀ n I.
• Fig. 4.17 (p. 122): Two parallel wires with their mutual force, the basis of the ampere definition.
• Fig. 4.18 / 4.19 (p. 125): Rectangular loop in B with torque from a couple on opposite arms.
• Fig. 4.20 (p. 130): MCG with radial field, soft-iron core and restoring spring; Figs. 4.21–4.22 — shunt for ammeter, series R for voltmeter.

2.4 Common confusions / NTA trap points

• Cyclotron frequency ν_c = qB/(2πm) is independent of speed v and radius r — distractors often insert v or r. Note ω = qB/m, while ν_c = ω/(2π).
• Field at the centre of a circular loop is B = μ₀ I/(2R) (not μ₀ n I, which is the solenoid result, and not μ₀ I/(2π R), which is the straight-wire result). The factor of π distinguishes them.
• B = μ₀ I/(2π r) is for a long straight wire; the same formula does NOT apply at the centre of a loop.
• For a thick wire, inside (r < a) B grows linearly with r; outside (r > a) B falls as 1/r. Students often invert this.
• Parallel currents attract, antiparallel currents repel — the opposite of the rule for like/unlike electric charges.
• The magnetic force does no work because it is always perpendicular to v; kinetic energy and speed are unchanged, only direction of momentum changes.
• For a loop in a uniform field, net force is zero but torque is non-zero (unless m ∥ B or m ∥ −B). Stable equilibrium is m ∥ B; unstable is m ∥ −B.
• Doubling N in a galvanometer doubles current sensitivity but leaves voltage sensitivity unchanged (because the coil resistance also doubles).
• Ammeter: low shunt in parallel; voltmeter: high resistance in series. Reversing these gives wrong meters that disturb the circuit.
• Biot–Savart involves a cross product; the field is perpendicular to both dl and r̂, never along dl itself.
• The Lorentz force vector form has E + v × B (with a plus), not E − v × B.
• In a uniform field a current loop experiences zero net force but a torque; in a non-uniform field both force and torque can be non-zero.

2.5 Key formulas table