Moving Charges and Magnetism

In this chapter: Lorentz force, force on a current-carrying conductor, motion of a charge in a magnetic field, Biot-Savart law, field on the axis of a circular loop, Ampere's circuital law, field of a long wire and a solenoid, force between parallel currents, torque on a current loop, the loop as a magnetic dipole, the moving coil galvanometer and its conversion to an ammeter or voltmeter.

Magnetic force on a moving charge

Oersted noticed in 1820 that a compass needle is deflected by a current in a nearby wire. Moving charges (currents) create magnetic fields, and a magnetic field exerts a force only on moving charges. For a charge q moving with velocity v in fields E and B, the Lorentz force is

• No magnetic force on a charge at rest, or on one moving parallel or antiparallel to B.
• For a negative charge the force is opposite to that for a positive charge.
• The SI unit of B is the tesla: 1 T = 1 N s C⁻¹ m⁻¹. A smaller unit is the gauss, 1 G = 10⁻⁴ T. The earth's field is about 3.6 × 10⁻⁵ T.

Force on a current-carrying conductor

Adding the forces on all drifting electrons in a straight wire of length l carrying current I gives

Motion of a charge in a uniform magnetic field

The magnetic force is always perpendicular to the velocity, so it does no work and cannot change the speed or kinetic energy. It only bends the path.

The time period and frequency do not depend on the speed or the radius: a faster particle moves on a bigger circle but takes the same time for one revolution. If the velocity has a component along B, that component is unaffected and the path becomes a helix. The radius uses the perpendicular component (r = mv_⊥/qB) and the pitch is p = v_∥T = 2πmv_∥/qB.

Worked example: An electron moving at 3 × 10⁷ m s⁻¹ enters a field of 6 × 10⁻⁴ T at right angles. Find the radius of its path and its frequency of revolution.
Solution: r = mv/(eB) = (9 × 10⁻³¹ × 3 × 10⁷)/(1.6 × 10⁻¹⁹ × 6 × 10⁻⁴) = (2.7 × 10⁻²³)/(9.6 × 10⁻²³) ≈ 0.28 m. Frequency ν = eB/(2πm) = (9.6 × 10⁻²³)/(2π × 9 × 10⁻³¹) ≈ 1.7 × 10⁷ Hz, which does not depend on the speed.

Biot-Savart law

The field produced at a point by a small current element I dl is

Like Coulomb's law it is an inverse square law, but the field is perpendicular to both dl and r, and there is no field along the line of the element itself (θ = 0). The constants are linked by 1/√ε₀μ₀ = c.

Field on the axis of a circular loop

The direction follows the right-hand thumb rule for a loop: curl the fingers along the current and the thumb gives the field along the axis. Seen from one face, anticlockwise current makes that face behave like a north pole.

Ampere's circuital law

The line integral of B around any closed loop equals μ₀ times the net current passing through the surface bounded by that loop. Its role is the same as Gauss's law in electrostatics: it is always true, but it gives B easily only when there is enough symmetry.

• Long straight wire: take a circle of radius r around the wire, so B · 2πr = μ₀I and B = μ₀I/2πr. The field lines are circles centred on the wire. Grip the wire in your right hand with the thumb along the current; the fingers curl in the direction of B.
• Long solenoid with n turns per unit length: taking a rectangular loop half inside and half outside gives B = μ₀nI inside. The field inside is uniform and along the axis, and the field outside is nearly zero. At the ends of a long solenoid, B drops to about half this value.

Force between two parallel currents

The field of wire a at wire b is μ₀I_a/2πd, and wire b feels a force I_bLB in that field. Per unit length:

Parallel currents attract and antiparallel currents repel, which is the reverse of the rule for charges. For two wires 1 m apart each carrying 1 A, the force is 2 × 10⁻⁷ N per metre. This was the older definition of the ampere; since 2019 the ampere is defined by fixing the value of e, but the number is still correct.

Torque on a current loop and the magnetic dipole

A rectangular loop in a uniform field feels zero net force, but a torque acts on it. For a coil of N turns and area A carrying current I, define the magnetic moment m = NIA, with A along the normal given by the right-hand rule (unit A m²). Then

The torque is maximum when the plane of the coil is parallel to B and zero when the plane is perpendicular to B. The expression is exactly like p × E for an electric dipole. Far from a small loop, the field on its axis is B = μ₀2m/(4πx³), the same form as the axial field of an electric dipole, so a current loop behaves as a magnetic dipole.

An electron orbiting a nucleus is a tiny current loop. Its magnetic moment is related to its orbital angular momentum by μ_l = (e/2m_e) l; the smallest value, (μ_l)_min = eh/(4πm_e) = 9.27 × 10⁻²⁴ A m², is called the Bohr magneton.

Moving coil galvanometer

A coil pivoted in a radial magnetic field (made using concave pole pieces and a soft iron core) always has its plane parallel to B, so the torque is NIAB. A spring provides a restoring torque kφ, and at equilibrium

Doubling the number of turns doubles the current sensitivity, but it also roughly doubles the coil resistance, so the voltage sensitivity may not change.

• Ammeter: connect a small resistance, the shunt r_s, in parallel with the galvanometer: r_s = I_gG/(I − I_g). An ideal ammeter has zero resistance.
• Voltmeter: connect a large resistance R in series: R = V/I_g − G. An ideal voltmeter has infinite resistance.

Worked example: A galvanometer of resistance 12 Ω gives full-scale deflection for 3 mA. How would you convert it into an ammeter reading up to 3 A, and into a voltmeter reading up to 18 V?
Solution: Ammeter: shunt r_s = (3 × 10⁻³ × 12)/(3 − 0.003) ≈ 0.012 Ω in parallel. Voltmeter: series resistance R = 18/(3 × 10⁻³) − 12 = 6000 − 12 = 5988 Ω.

Common mistakes: (1) Forgetting to reverse the force direction for an electron. (2) Saying a magnetic field can change a particle's speed; it changes only the direction. (3) Taking θ in τ = NIAB sin θ as the angle between the plane of the coil and B; it is the angle between the normal and B. (4) Using the field at the centre of a loop, μ₀I/2R, for a point on the axis. (5) Connecting the shunt in series or the voltmeter resistance in parallel.

JEE and NEET focus

• Radius, period and pitch of a charged particle in a magnetic field; comparing proton, deuteron and alpha particle.
• Fields at the centre of loops and arcs, near straight wires and inside a solenoid, with correct directions.
• Force per unit length between parallel wires.
• Torque and magnetic moment of a coil; potential energy of a magnetic dipole, U = −m · B.
• Galvanometer sensitivity and conversion to ammeter and voltmeter.

Practice questions