Magnetic Fields and Forces: every key term you need (+ practice quiz)

More AP Physics C: E&M guides

• Conductors, Capacitors, and Dielectrics
• Electric Circuits
• Electromagnetic Induction and Maxwell's Equations

Magnetic Field

Vector field B (tesla, T) produced by moving charges and currents; exerts forces only on moving charges and current-carrying wires.

Magnetic Force on a Charge

F = qv × B: magnitude qvB sinθ, direction from the right-hand rule, always perpendicular to both v and B.

No Work by Magnetic Force

Because F ⊥ v always, the magnetic force does zero work; it changes a charge's direction but never its speed or kinetic energy.

Right-Hand Rule (Force)

Point fingers along v, curl toward B; the thumb gives F for a positive charge. Reverse the result for negative charges.

Circular Motion in B

A charge moving perpendicular to a uniform B travels a circle of radius r = mv/(qB), with the magnetic force supplying the centripetal force.

Cyclotron Frequency

f = qB/(2πm): the orbit frequency in a uniform field is independent of the particle's speed and radius — the principle behind cyclotrons.

Helical Motion

Velocity components parallel to B are unaffected, so a charge with both parallel and perpendicular components spirals along the field lines.

Velocity Selector

Crossed E and B fields pass only particles with v = E/B undeflected, since qE balances qvB for that single speed.

Mass Spectrometer

After a velocity selector, particles curve with r = mv/(qB); measuring r separates ions by mass-to-charge ratio.

Hall Effect

Current in a magnetic field pushes carriers to one side of a conductor, creating a transverse voltage whose sign reveals the carrier charge.

Force on a Current Wire

F = IL × B: a straight segment of length L carrying current I in field B feels force ILB sinθ, perpendicular to both.

Force on a Curved Wire

In a uniform field, the force on any wire equals that on the straight chord joining its endpoints; a closed loop feels zero net force.

Torque on a Current Loop

τ = μ × B with magnetic moment μ = NIA; a uniform field exerts no net force but twists the loop to align μ with B — the motor principle.

Magnetic Moment

μ = NIA, perpendicular to the loop's plane by the right-hand rule (curl fingers with current, thumb gives μ); energy U = −μ·B.

Biot–Savart Law

dB = (μ₀/4π)(I dl × r̂)/r²: each current element contributes a field falling off as 1/r², perpendicular to both dl and r̂.

Permeability of Free Space

μ₀ = 4π×10⁻⁷ T·m/A, the magnetic constant appearing in Biot–Savart and Ampère's laws.

Field of a Long Straight Wire

B = μ₀I/(2πr): circles the wire per the right-hand rule (thumb along I, fingers curl with B), falling off as 1/r.

Field at Center of a Loop

A circular loop of radius R carrying I produces B = μ₀I/(2R) at its center, along the loop axis.

Field on a Loop's Axis

B = μ₀IR²/[2(x² + R²)^(3/2)] on the axis of a current loop — a standard Biot–Savart integral; reduces to μ₀I/(2R) at x = 0.

Force Between Parallel Wires

Per unit length F/L = μ₀I₁I₂/(2πd); parallel currents attract, antiparallel currents repel.

Definition of the Ampere

Historically defined via the parallel-wire force: two long wires 1 m apart each carrying 1 A exert 2×10⁻⁷ N per meter on each other.

Ampère's Law

∮B·dl = μ₀I_enc: the line integral of B around any closed loop equals μ₀ times the current threading the loop.

Amperian Loop

Closed path chosen to exploit symmetry (circles around wires, rectangles for solenoids) so B is constant and parallel or perpendicular to dl.

Field Inside a Thick Wire

For uniform current density, Ampère's law gives B = μ₀Ir/(2πR²) inside radius R — rising linearly, then falling as 1/r outside.

Solenoid Field

Inside a long solenoid, B = μ₀nI (n = turns per length), uniform and axial; the ideal exterior field is negligible.

Toroid Field

Inside a toroid with N total turns, B = μ₀NI/(2πr), circulating around the doughnut and confined to its interior.

Gauss's Law for Magnetism

∮B·dA = 0: magnetic flux through any closed surface is zero — there are no magnetic monopoles, so field lines always close on themselves.

Magnetic Field Lines

Form closed loops (no start or end points); outside a bar magnet they run N to S, inside they run S to N.

Ferromagnetism

Materials like iron whose atomic moments align in domains, hugely amplifying applied fields; the basis of permanent magnets and electromagnet cores.

Superposition of B Fields

Fields from multiple currents add as vectors; canceling or reinforcing points between parallel wires are found by summing μ₀I/(2πr) terms with directions.

DC Motor Principle

A current loop in a magnetic field experiences torque NIAB sinθ; a commutator reverses the current every half turn to keep the torque direction constant.

Charge in Combined Fields

The full Lorentz force is F = qE + qv × B; electric force can do work and change speed while the magnetic part only steers.