What this chapter is about
This chapter explores the deep connection between electricity and magnetism. You already know that stationary charges produce electric fields. Here you learn that charges in motion—electric currents—produce magnetic fields and also experience forces when placed in external magnetic fields. This unification of electricity and magnetism was one of the great achievements of nineteenth-century physics.
The chapter builds on your understanding of electric current from earlier classes and extends it to explain how magnetic fields arise from currents in wires, loops and solenoids. You will learn the Biot-Savart law and Ampere's circuital law, which are the magnetic analogues of Coulomb's law and Gauss's law. You will also study how a current-carrying conductor or a moving charged particle experiences a force in a magnetic field, leading to devices like the galvanometer and the cyclotron.
After studying this chapter, you should be able to calculate the magnetic field due to common current configurations, find the force on a current-carrying wire or a moving charge, understand the torque on a current loop, and explain how instruments that measure current work.
Key ideas
- A moving charge creates a magnetic field around it; an electric current (many moving charges) produces a magnetic field whose pattern depends on the shape of the conductor.
- The Biot-Savart law gives the magnetic field dB at a point due to a small current element I dl: dB = (μ₀/4π) × (I dl sin θ)/r², directed perpendicular to the plane containing dl and the position vector r, following the right-hand rule.
- Ampere's circuital law states that the line integral of the magnetic field B around any closed loop equals μ₀ times the net current enclosed: ∮B · dl = μ₀ I_enclosed. It is most useful when symmetry simplifies the integral.
- A straight long wire carrying current I produces circular magnetic field lines; at perpendicular distance r, the magnitude is B = μ₀I/(2πr).
- A circular loop of radius R carrying current I produces, at its centre, a field B = μ₀I/(2R), directed along the axis of the loop (use right-hand curl rule).
- A solenoid of n turns per unit length carrying current I produces a nearly uniform field inside: B = μ₀nI, directed along its axis.
- The Lorentz force on a charge q moving with velocity v in a magnetic field B is F = q(v × B). This force is always perpendicular to v, so it changes direction but not speed.
- A current-carrying conductor of length L in a uniform field B experiences force F = I L B sin θ, where θ is the angle between the wire and the field.
- A current loop (magnetic dipole) in a uniform field experiences a torque τ = M × B, where M = NIA is the magnetic moment (N turns, area A).
Formulas and facts to remember
- Formula: dB = (μ₀/4π) (I dl sin θ)/r² · Meaning: Biot-Savart law: field due to a small current element
- Formula: B = μ₀I/(2πr) · Meaning: Field at distance r from a long straight wire
- Formula: B = μ₀I/(2R) · Meaning: Field at the centre of a circular loop of radius R
- Formula: B = μ₀nI · Meaning: Field inside a long solenoid (n = turns per metre)
- Formula: ∮B · dl = μ₀ I_enc · Meaning: Ampere's circuital law
- Formula: F = qvB sin θ · Meaning: Magnitude of Lorentz force on a moving charge
- Formula: F = BIL sin θ · Meaning: Force on a straight current-carrying wire in a field
- Formula: τ = NIAB sin φ · Meaning: Torque on a coil; φ is angle between M and B
- Formula: r = mv/(qB) · Meaning: Radius of circular path of a charge in a uniform field
- Formula: T = 2πm/(qB) · Meaning: Time period of circular motion; independent of speed
μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space.
Worked examples
Example 1: Magnetic field due to a long straight wire
A long straight wire carries a steady current of 5 A. Find the magnetic field at a point 10 cm away from the wire.
Solution
Use the formula for the field around a long straight conductor:
B = μ₀I/(2πr)
Given: I = 5 A, r = 10 cm = 0.10 m, μ₀ = 4π × 10⁻⁷ T m A⁻¹.
B = (4π × 10⁻⁷ × 5)/(2π × 0.10)
B = (2 × 10⁻⁶ × 5)/(0.10)
B = 10⁻⁵ T = 10 μT
The field is 10 μT, directed in circles around the wire (use right-hand grip rule to find direction).
Example 2: Motion of a proton in a magnetic field
A proton (mass 1.67 × 10⁻²⁷ kg, charge 1.6 × 10⁻¹⁹ C) enters a region of uniform magnetic field 0.20 T perpendicular to its velocity of 3.0 × 10⁶ m s⁻¹. Find the radius of its circular path and the time to complete one circle.
Solution
The magnetic force provides centripetal acceleration. Equating:
qvB = mv²/r → r = mv/(qB)
r = (1.67 × 10⁻²⁷ × 3.0 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.20)
r = (5.01 × 10⁻²¹)/(3.2 × 10⁻²⁰)
r ≈ 0.156 m ≈ 15.6 cm
Time period T = 2πm/(qB)
T = (2π × 1.67 × 10⁻²⁷)/(1.6 × 10⁻¹⁹ × 0.20)
T = (1.05 × 10⁻²⁶)/(3.2 × 10⁻²⁰)
T ≈ 3.28 × 10⁻⁷ s ≈ 0.33 μs
Example 3: Torque on a rectangular coil
A rectangular coil has 50 turns, each of area 4.0 × 10⁻³ m². It carries 2.0 A and is placed in a uniform field of 0.30 T with its plane making 60° with the field. Find the torque.
Solution
The angle between the magnetic moment M and B is φ. If the plane makes 60° with B, the normal to the plane makes 30° with B, so φ = 30°.
Torque τ = NIAB sin φ
τ = 50 × 2.0 × (4.0 × 10⁻³) × 0.30 × sin 30°
τ = 50 × 2.0 × 4.0 × 10⁻³ × 0.30 × 0.5
τ = 0.06 N m
The coil experiences a torque of 0.06 N m, tending to align M with B.
Common mistakes
- Confusing the angle in torque formula: the angle φ is between the area vector (normal) and B, not between the plane and B → always draw the normal first.
- Forgetting that magnetic force on a moving charge is perpendicular to velocity, so it does no work → kinetic energy stays constant; only direction changes.
- Using Ampere's law without checking symmetry; it simplifies calculations only when B is constant along the chosen loop → for irregular shapes, use Biot-Savart law.
- Mixing up the direction rules: for field from a current use right-hand grip (thumb along current), for force on a moving positive charge use right-hand palm rule (fingers along v, curl towards B, thumb gives F).
- Treating solenoid formula B = μ₀nI as valid everywhere; it applies only well inside a long solenoid → near the ends, the field is roughly half.
Quick revision
- Moving charges produce magnetic fields; stationary charges do not.
- Biot-Savart law is the fundamental relation; Ampere's law is its integral form useful with symmetry.
- Magnetic force on a charge is F = q(v × B); it changes direction, not speed.
- A current loop behaves like a magnetic dipole with moment M = NIA.
- Inside a solenoid, B = μ₀nI (uniform); outside, B ≈ 0.
- Cyclotron uses constant time period T = 2πm/(qB) to accelerate ions repeatedly.