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AP Physics C: E & M · Unit 5 of 6
Magnetic Fields
14–20% of the exam4 lessons · 53 min21 terms
What this unit covers
The topics below follow the published Physics C: E&M course framework for Unit 5. This unit is worth 14–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.
Biot–SavartAmpère’s lawForces on currentsSolenoids
Lessons in this unit
- Magnetic Force on Moving Charges13 min · 3 objectivesCompute the magnetic force with F = qv × B and the right-hand rule · Analyze circular motion of a charge in a uniform magnetic field · Explain why magnetic forces change direction but never speed
- The Biot–Savart Law14 min · 3 objectivesState the Biot–Savart law and identify the direction of dB from the cross product · Derive the on-axis field of a current ring by integration · Derive the field of a long straight wire and apply B = μ₀I/(2πd)
- Forces on Currents & Parallel Wires12 min · 3 objectivesDerive F = IL × B from the force on individual drifting charges · Find force directions on current-carrying wires with the right-hand rule · Compute the force per length between parallel wires and predict attraction or repulsion
- Ampère’s Law: Wires, Solenoids & Toroids14 min · 3 objectivesState Ampère’s law and explain the role of symmetry in applying it · Derive the field inside and outside a thick current-carrying wire · Derive B = μ₀nI for an ideal solenoid and B = μ₀NI/(2πr) for a toroid
Formulas in Unit 5
Magnetic force on a charge
→F = q→v × →B, F = qvB·sinθ
F is always perpendicular to both v and B. Maximum force at θ = 90°; zero force for motion along the field.
Circular motion in a magnetic field
r = mv/(qB), T = 2πm/(qB)
Radius grows with momentum mv; the period depends only on m, q, and B — not on how fast the particle moves.
Biot–Savart law
d→B = (μ₀/4π) · I d→l × r̂ / r², μ₀ = 4π × 10⁻⁷ T·m/A
Superpose by integrating over the whole current path. |dl × r̂| = dl·sinφ, where φ is the angle between the element and the line to the point.
Key Biot–Savart results
Straight wire: B = μ₀I/(2πd) Center of ring: B = μ₀I/(2R) On axis: B = μ₀IR²/[2(x² + R²)^(3/2)]
Wire field lines are circles around the wire: point the right thumb along I and the fingers curl in the direction of B.
Force on a current-carrying wire
→F = I→L × →B, F = BIL·sinθ
Right-hand rule: fingers along I, curl toward B, thumb gives F. A wire parallel to B feels nothing.
Force between parallel wires
F/L = μ₀I₁I₂/(2πd)
Currents in the same direction attract; opposite directions repel. Newton’s third law holds: each wire feels the same magnitude of force.
Ampère’s law
∮ →B·d→l = μ₀ I_enc, μ₀ = 4π × 10⁻⁷ T·m/A
Only currents that pierce the surface bounded by the loop count toward I_enc. Curl the right-hand fingers along the traversal direction; the thumb gives the positive current direction.
Solenoid and toroid fields
Solenoid: B = μ₀nI (inside, uniform; ≈0 outside) Toroid: B = μ₀NI/(2πr) (inside the windings)
n = N/ℓ is turns per meter for the solenoid; N is the total turn count for the toroid, whose field falls as 1/r across its interior and vanishes outside.
Every term in Unit 5
All 21 terms we publish for Magnetic Fields, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.
- Biot-Savart law
- dB = (μ₀/4π)(I dl × r̂)/r². The magnetic analogue of computing E by integrating over a charge distribution.
- Ampère's law
- ∮B·dl = μ₀I_enc. Useful only where symmetry makes B constant along a chosen closed path — the magnetic counterpart to Gauss's law.
- Magnetic force on a charge
- F = qv × B, magnitude qvB sin θ, perpendicular to both. Zero when velocity is parallel to the field.
- Why magnetic force does no work
- It is always perpendicular to velocity, so it changes direction but never speed or kinetic energy.
- Circular motion in a magnetic field
- r = mv/qB and the period T = 2πm/qB is independent of speed — the principle behind the cyclotron.
- Velocity selector
- Crossed electric and magnetic fields pass only particles with v = E/B, since the two forces cancel at that speed alone.
- Force on a current-carrying wire
- F = ∫I dl × B, which reduces to BIL sin θ for a straight wire in a uniform field.
- Torque on a current loop
- τ = μ × B, where the magnetic moment μ = NIA. The basis of the electric motor and the galvanometer.
- Field of a long straight wire
- B = μ₀I/2πr, circling the wire with direction given by the right-hand rule.
- Field at the center of a circular loop
- B = μ₀I/2R, perpendicular to the plane of the loop.
- Field inside a solenoid
- B = μ₀nI, uniform and independent of position, where n is turns per unit length.
- Force between parallel wires
- Parallel currents attract and antiparallel currents repel, with force per unit length μ₀I₁I₂/2πd.
- Choosing an Ampèrian loop
- Pick a path along which B is constant and either parallel or perpendicular to dl — a circle for a wire, a rectangle for a solenoid.
- Ampère's law vs Biot-Savart
- Ampère is fast where symmetry allows; Biot-Savart is general but requires a vector integral over the current distribution.
- Field of a toroid
- B = μ₀NI/2πr inside and zero outside, which is why toroidal inductors confine their field and do not interfere with neighbors.
- Field of a finite wire segment
- Requires Biot-Savart with angular limits; it reduces to μ₀I/2πr as the segment becomes infinitely long.
- Right-hand rules summary
- For the field of a current, thumb along I and fingers curl. For force, fingers along v or I, curl toward B, thumb gives F on positive charge.
- Mass spectrometer
- A velocity selector fixes v, then a magnetic field bends the beam to radius r = mv/qB, separating isotopes by mass.
- Cyclotron frequency
- f = qB/2πm, independent of speed and radius — which is what allows a fixed-frequency accelerating voltage to work.
- Hall effect
- Magnetic force pushes carriers to one side of a conductor until the resulting electric field balances it, producing a measurable transverse voltage.
- Magnetic dipole
- A current loop behaves like a bar magnet with moment μ = NIA, experiencing torque μ × B and potential energy −μ·B in a field.
What examiners penalize here
- Ampère’s-law FRQs award points for the argument, not just the answer: (1) name the symmetry and draw the Amperian loop, (2) justify that B is constant and parallel to dl on it, so ∮B·dl = B·(length), (3) compute I_enc — for distributed currents use the current density times the enclosed area, (4) solve. Skipping step 3’s enclosed-fraction logic is the most common lost point on thick-wire problems.
Practice Physics C: E&M
Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.
Questions about this unit
How much of the AP Physics C: E & M exam is Unit 5?
Unit 5, Magnetic Fields, is worth 14–20% of the Physics C: E&M multiple-choice section according to the published course framework. Across all 6 units that makes it one of the heaviest units on the exam, and worth front-loading.
What topics are covered in Physics C: E&M Unit 5?
Magnetic Fields covers Biot–Savart, Ampère’s law, Forces on currents and Solenoids. We publish 21 terms with definitions for this unit, all of them on this page.
How should I study Physics C: E&M Unit 5?
Read the 4 lessons below first — about 55 minutes — then drill the 21 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.
All 6 units of AP Physics C: E & M
Unit names, topics and exam weights follow the published College Board course framework for AP Physics C: E & M. AP® is a trademark registered by the College Board, which does not endorse this site.