Source: Chapters 11 and 12
Tags: magnetic field, magnetic force, Lorentz force, right-hand rule, current loop, Biot-Savart law, Ampere's law, solenoid, toroid, Hall effect, paramagnetism, ferromagnetism, diamagnetism
Difficulty: Intermediate
Prerequisites: Chapters 5-10 notes (electric field, current, circuits). Cross products and right-hand rule from vector mathematics.
Electricity and magnetism are two faces of the same phenomenon, and this pair of chapters is where the magnetic side comes into focus. Chapter 11 covers what magnetic fields do to moving charges and current-carrying wires. Chapter 12 covers where magnetic fields come from: moving charges and currents create them. Together, these chapters set the stage for electromagnetic induction (Ch. 13), which links changing magnetic fields back to electric effects and ultimately leads to Maxwell's equations.
A magnetic field exerts a force on moving charges that is perpendicular to both the velocity and the field (use the right-hand rule). Current-carrying wires create magnetic fields, described by the Biot-Savart law for specific geometries and by Ampere's law for symmetric ones. Solenoids produce uniform internal fields, and materials respond to magnetic fields differently depending on whether they are paramagnetic, ferromagnetic, or diamagnetic.
Magnetic field (B)
The field that exerts a force on moving charges. Measured in tesla (T). Field lines run from north to south outside a magnet and form continuous loops.
Magnetic force on a charge
F = qv x B. The force is perpendicular to both the velocity and the field. Its magnitude is F = qvB sin θ. Think of it as: the field pushes the charge sideways, never speeding it up or slowing it down, only changing its direction.
Right-hand rule
Point your fingers in the direction of v, curl them toward B; your thumb points in the direction of F (for a positive charge). For a negative charge, reverse the direction.
Lorentz force
The total electromagnetic force on a charge: F = qE + qv x B.
Radius of curvature
A charged particle moving perpendicular to a uniform magnetic field follows a circular path. The radius is r = mv/(qB).
Period of circular motion
T = 2πm/(qB). Independent of velocity, which is why cyclotrons work.
Pitch of a helix
When a charged particle has a velocity component along B, it spirals. The pitch is p = v_∥ T = v cos θ × T.
Magnetic force on a current-carrying wire
F = IL x B, where L is a vector along the wire in the direction of current. Magnitude: F = BIL sin θ.
Magnetic dipole moment (m)
For a current loop: m = NIA, where N is the number of turns, I is the current, and A is the area. Direction follows the right-hand rule around the current loop.
Torque on a current loop
τ = m x B. Magnitude: τ = mB sin θ. The loop tends to align its dipole moment with the field.
Potential energy of a magnetic dipole
U = -m · B = -mB cos θ.
Hall effect
When a current-carrying conductor is placed in a magnetic field, a voltage (Hall voltage) develops across it, perpendicular to both the current and the field: V_H = BIw/(nqA), where w is the width. In simple terms, the magnetic force pushes charges to one side of the conductor, creating a measurable voltage.
Biot-Savart law
The magnetic field contribution from a small current element: dB = (μ₀/4π) I dl x r̂ / r². This is the magnetic analogue of Coulomb's law.
Permeability of free space (μ₀)
μ₀ = 4π x 10⁻⁷ T·m/A. The magnetic constant that plays the same role as ε₀ does for electric fields.
Ampere's law
∮ B · dl = μ₀ I_enclosed. The line integral of B around a closed path equals μ₀ times the current threading through that path. Useful when there is sufficient symmetry.
Solenoid
A long wire wound into a helix. Inside a long solenoid, the field is uniform: B = μ₀nI, where n is the number of turns per unit length.
Toroid
A donut-shaped coil. The field inside is B = μ₀NI/(2πr), where N is the total number of turns and r is the distance from the centre of the toroid.
Paramagnetism
A material in which atomic dipoles weakly align with an applied field, slightly strengthening it. The effect disappears when the field is removed.
Ferromagnetism
A material in which magnetic dipoles align strongly and lock together, producing a permanent magnetisation even after the field is removed. Iron, nickel, and cobalt are ferromagnetic.
Diamagnetism
A material with no net atomic magnetic moment. An applied field induces a weak opposing field. The effect is very small and always present, but usually masked by para- or ferromagnetism.
Magnetic susceptibility (χ)
A dimensionless number describing how a material responds to an applied field. Positive for para- and ferromagnetic materials, negative for diamagnetic.
Magnetic permeability (μ)
μ = (1 + χ)μ₀. Replaces μ₀ in formulas when the field is inside a material. B = μnI for a solenoid filled with that material.
F = qv x B. Magnitude: F = qvB sin θ.
The force is always perpendicular to velocity, so it does no work and cannot change the particle's speed, only its direction.
Direction determined by the right-hand rule (reverse for negative charges).
A charge moving perpendicular to a uniform B follows a circle of radius r = mv/(qB).
Period: T = 2πm/(qB). This does not depend on speed.
If the velocity has a component parallel to B, the particle traces a helix. The parallel component is unaffected by the force. The pitch is p = v cos θ × T.
F = IL x B. Magnitude: F = BIL sin θ.
For a straight wire in a uniform field, the force is uniform along the wire.
Dipole moment: m = NIA.
Torque: τ = m x B = NIAB sin θ. Maximum when the loop plane is parallel to B.
Potential energy: U = -m · B. Minimum when m is aligned with B, maximum when anti-aligned.
Hall voltage: V_H = BId/(nqA), where d is the width perpendicular to both I and B.
The Hall effect is used to measure magnetic fields, identify charge carrier type, and determine carrier density.
General form: dB = (μ₀/4π) I dl x r̂ / r².
Straight wire of length L at distance R: B = (μ₀I)/(2πR) for an infinitely long wire.
Force between two parallel wires carrying currents I₁ and I₂, separated by distance d: F/L = μ₀I₁I₂/(2πd). Parallel currents attract; anti-parallel currents repel.
Centre of a circular current loop of radius R: B = μ₀I/(2R).
∮ B · dl = μ₀ I_enclosed.
Choose an Amperian loop that exploits symmetry (just as you choose a Gaussian surface for Gauss's law).
Inside a solenoid: B = μ₀nI (uniform, along the axis).
Inside a toroid at radius r: B = μ₀NI/(2πr).
Outside both solenoid and toroid: B ≈ 0.
Paramagnetic: dipoles partially align with B, producing a small enhancement. Susceptibility χ is small and positive.
Ferromagnetic: dipoles lock together, producing strong permanent magnetisation. Susceptibility is large and positive.
Diamagnetic: no net dipole. Applied field induces a small opposing field. χ is small and negative.
Magnetic permeability: μ = (1 + χ)μ₀. For a solenoid with a magnetic core: B = μnI.
Quantity | Formula |
|---|---|
Force on charge | F = qvB sin θ |
Circular orbit radius | r = mv/(qB) |
Period | T = 2πm/(qB) |
Force on wire | F = BIL sin θ |
Dipole moment | m = NIA |
Torque on loop | τ = mB sin θ |
Biot-Savart (infinite wire) | B = μ₀I/(2πR) |
Force between parallel wires | F/L = μ₀I₁I₂/(2πd) |
Centre of circular loop | B = μ₀I/(2R) |
Ampere's law | ∮ B · dl = μ₀I_enc |
Solenoid | B = μ₀nI |
Toroid | B = μ₀NI/(2πr) |
Electric motors work by the torque on a current loop in a magnetic field. Mass spectrometers separate ions by their radius of curvature in a uniform field (r = mv/qB), which is how isotopes are identified. MRI machines use the alignment of nuclear magnetic dipoles in a strong field to produce detailed images of soft tissue.
Students often think the magnetic force can speed up a charged particle. It cannot. Because F is always perpendicular to v, it changes the direction of motion but never the speed or kinetic energy.
The right-hand rule is for positive charges. For electrons, you must reverse the direction of the force. Students frequently forget this reversal.
Inside a solenoid, the field is uniform and parallel to the axis, not radial. Students sometimes draw the field pointing outward from the axis.
Ampere's law gives zero for the enclosed current if the Amperian loop does not actually thread through any current, even if there is a strong field nearby. The field and the enclosed current are related only through the closed-loop integral.
⚠️ The cross product in F = qv x B means direction matters. Practise the right-hand rule until it is automatic.
⚠️ Circular motion problems (find r, T, or the charge-to-mass ratio) are common. Know r = mv/(qB) cold.
⚠️ For Biot-Savart and Ampere's law problems, identify the geometry first (straight wire, loop, solenoid, toroid) and apply the correct formula.
⚠️ Know the three types of magnetism and their distinguishing features (sign and size of χ, whether magnetisation is permanent).
True or false: A stationary charge in a magnetic field experiences a force.
False. The magnetic force requires the charge to be moving (F = qv x B; if v = 0, F = 0).
Fill in the blank: The magnetic field inside a solenoid is B = ________.
μ₀nI.
True or false: Parallel currents in the same direction attract each other.
True.
Fill in the blank: The radius of a charged particle's circular orbit in a magnetic field is r = ________.
mv/(qB).
Q: A proton (m = 1.67 x 10⁻²⁷ kg, q = 1.6 x 10⁻¹⁹ C) moves at 3 x 10⁶ m/s perpendicular to a 0.5 T magnetic field. What is the radius of its circular path?
A: r = mv/(qB) = (1.67 x 10⁻²⁷)(3 x 10⁶) / [(1.6 x 10⁻¹⁹)(0.5)] = 0.063 m ≈ 6.3 cm.
Q: A solenoid has 400 turns per metre and carries 2 A. What is the magnetic field inside?
A: B = μ₀nI = (4π x 10⁻⁷)(400)(2) = 1.005 x 10⁻³ T ≈ 1.0 mT.
Q: Two parallel wires carry 5 A in the same direction, separated by 0.1 m. What is the force per metre between them?
A: F/L = μ₀I₁I₂/(2πd) = (4π x 10⁻⁷)(5)(5)/(2π × 0.1) = 5 x 10⁻⁵ N/m, attractive.
The magnetic force on moving charges connects to the Hall effect (used in sensors) and to the motion of charged particles in electromagnetic waves (Ch. 16). Ampere's law is one of Maxwell's four equations. The torque on a current loop is the operating principle of motors and galvanometers. These concepts feed directly into electromagnetic induction (Ch. 13), where changing magnetic fields create electric fields.
magnetic field, tesla, magnetic force, Lorentz force, cross product, right-hand rule, circular motion, cyclotron, radius of curvature, current loop, dipole moment, torque, Hall effect, Hall voltage, Biot-Savart law, Ampere's law, solenoid, toroid, parallel wires, permeability, μ₀, paramagnetism, ferromagnetism, diamagnetism, magnetic susceptibility, PHYS 212