Chapter 4: Moving Charges and Magnetism | Summary

The total force on a charge q moving with velocity v in the presence of both electric field E and magnetic field B is given by the Lorentz force: F = q[E(r) + v × B(r)]. The magnetic part of this force, q(v × B), is perpendicular to both velocity and field, does zero work on the particle, and vanishes when the charge is at rest or moving parallel to B. The direction of the magnetic force is given by the right-hand screw rule for the vector cross product.

For a current-carrying straight conductor of length l in an external magnetic field B, the force is F = Il × B, where the vector l has magnitude equal to the length and direction along the current. This expression is derived by summing the Lorentz forces on all charge carriers inside the conductor. The SI unit of magnetic field is the tesla (T), defined such that a force of 1 N acts on a charge of 1 C moving at 1 m/s perpendicular to B. A smaller unit, gauss (1 G = 10⁻⁴ T), is also used.

Since the magnetic force is always perpendicular to velocity, it does no work and does not change the speed of a charged particle—only its direction. When a particle moves perpendicular to a uniform magnetic field B, the magnetic force acts as centripetal force, producing uniform circular motion with radius r = mv/(qB). The cyclotron frequency ν = qB/(2πm) is independent of speed and energy, a fact exploited in the design of the cyclotron accelerator.

If the particle has a velocity component parallel to B in addition to the perpendicular component, the parallel component is unaffected by the field while the perpendicular component produces circular motion. The combined effect is helical motion, with the pitch (distance moved along the field per revolution) given by p = v‖T = 2πmv‖/(qB).

The Biot-Savart law gives the magnetic field dB produced by an infinitesimal current element I dl at a point P at displacement r: dB = (μ₀/4π)(I dl × r)/r³. The magnitude is |dB| = (μ₀/4π)(I dl sinθ)/r², where θ is the angle between dl and r. The constant μ₀ = 4π × 10⁻⁷ T m/A is the permeability of free space. The principle of superposition applies, so the total field is obtained by integrating over the entire conductor.

Like Coulomb's law, the Biot-Savart law is a long-range law (inverse square dependence). However, unlike Coulomb's law, the magnetic field is produced by a vector source (I dl), is perpendicular to the displacement vector r, and has an angle dependence (sinθ factor) absent in electrostatics. Using Biot-Savart law, the magnetic field at the centre of a circular loop of radius R carrying current I is B = μ₀I/(2R), while on the axis at distance x it is B = μ₀IR²/[2(x²+R²)^(3/2)].

Ampere's circuital law states that the line integral of the magnetic field B around any closed loop C equals μ₀ times the total current Ie enclosed: ∮B·dl = μ₀Ie. For a simplified amperian loop where B is tangential and constant along length L, the law reduces to BL = μ₀Ie. This law is to Biot-Savart law what Gauss's law is to Coulomb's law—both relate the field at the boundary to the source in the interior, and Ampere's law can be derived from Biot-Savart law.

For a long straight wire of radius a carrying current I uniformly distributed: outside the wire (r > a), B = μ₀I/(2πr); inside the wire (r < a), B = (μ₀I/2πa²)r. For a long solenoid with n turns per unit length carrying current I, the field inside is B = μ₀nI (uniform), and the field outside approaches zero. The solenoid is widely used to generate a uniform magnetic field.

Two long parallel conductors carrying currents Ia and Ib separated by distance d exert magnetic forces on each other. The force per unit length between them is fba = μ₀IaIb/(2πd). Parallel currents attract each other, and antiparallel currents repel—opposite to the rule for electric charges (like charges repel). This result is consistent with Newton's third law for steady currents.

The ampere is defined based on this force: it is the current in each of two very long, straight, parallel conductors of negligible cross-section placed 1 m apart in vacuum that produces a force of 2 × 10⁻⁷ N per metre of length. This definition was adopted in 1946. The SI unit of charge, the coulomb, is defined as the charge flowing through a cross-section when a steady current of 1 A flows for 1 second.

A rectangular current loop of area A carrying current I placed in a uniform magnetic field B experiences zero net force but a torque τ = IAB sinθ = m × B, where m = IA is the magnetic moment and θ is the angle between m and B. This is analogous to an electric dipole in a uniform electric field. For N turns, the magnetic moment is m = NIA. Torque vanishes when m is parallel (stable equilibrium) or antiparallel (unstable equilibrium) to B.

A circular current loop behaves as a magnetic dipole at large distances. On the axis at distance x >> R, the field is B ≈ μ₀(2m)/(4πx³), analogous to the axial field of an electric dipole. Unlike an electric dipole which is composed of two electric monopoles, the magnetic dipole (current loop) is the most elementary magnetic element—magnetic monopoles are not known to exist. Ampere proposed that all magnetism arises from circulating currents.

The moving coil galvanometer (MCG) consists of a coil with many turns suspended in a uniform radial magnetic field, with a soft iron core to increase field strength. When current I flows, a magnetic torque τ = NIAB is balanced by the restoring torque kφ of a spring, giving deflection φ = (NAB/k)I. Current sensitivity is φ/I = NAB/k, and voltage sensitivity is φ/V = NAB/(kR). Doubling N doubles current sensitivity but does not change voltage sensitivity (since R also doubles).

A galvanometer is converted to an ammeter by connecting a small shunt resistance rs in parallel (most current bypasses through rs), giving effective resistance ≈ rs. It is converted to a voltmeter by connecting a large resistance R in series, giving effective resistance ≈ R. An ideal ammeter has zero resistance; an ideal voltmeter has infinite resistance. These conversions allow measurement of both small and large currents and voltages.

Chapter 4 establishes the foundational principles linking moving charges, currents, and magnetic fields. The Lorentz force law unifies electric and magnetic interactions on a moving charge. The Biot-Savart law and Ampere's circuital law provide complementary methods for calculating magnetic fields from current distributions—the former for arbitrary configurations by integration, the latter for symmetric configurations by choosing appropriate amperian loops. The solenoid emerges as the practical device for generating uniform fields, while the cyclotron illustrates the energy-independence of cyclotron frequency. Torque on current loops leads to the concept of magnetic dipole moment, bridging macroscopic current loops and atomic magnetic behavior, and forms the basis of the moving coil galvanometer.

For board examinations, students must be proficient in applying the Biot-Savart law to compute fields of straight wires, circular loops, and solenoids; using Ampere's law for symmetric configurations; deriving expressions for radius, frequency, and pitch in circular and helical motion; calculating torque and magnetic moment of current loops; and converting a galvanometer to ammeter or voltmeter. The analogies between Biot-Savart and Coulomb's law, between Ampere's and Gauss's law, and between magnetic and electric dipoles are recurring themes in higher examinations and provide deep conceptual insight.