Magnetism and Moving Charges (AAMC 4C)
Electromagnetism studies the magnetic effects of moving charges and the forces magnetic fields exert on currents and individual moving charges. The AAMC MCAT content outline highlights three areas: magnetic flux, magnetic flux density, and the motion of a charged particle in a magnetic field. These are high-yield Chem/Phys topics.
Magnetic Flux Density
The magnetic flux density (also called the magnetic field B) at a point is defined by the force experienced by a current-carrying conductor or a moving charge there. Its SI unit is the tesla (T): 1 T = 1 N A−1 m−1 = 1 kg s−2 A−1.
Force on a current-carrying conductor
F = BIL sinθ
where I is the current, L is the length of the wire in the field, and θ is the angle between B and the wire. The force is maximum when the wire is perpendicular to B (θ = 90°) and zero when parallel.
| Source | Formula for B | Notes |
|---|---|---|
| Long straight wire | B = μ0I / (2πr) | r = perpendicular distance; circles around wire (right-hand rule) |
| Centre of circular loop | B = μ0I / (2r) | r = loop radius; perpendicular to plane |
| N-loop coil | B = μ0NI / (2r) | N times the single-loop value |
| Long solenoid (inside) | B = μ0nI | n = turns per metre; uniform along axis |
The constant μ0 = 4π × 10−7 T m A−1 is the permeability of free space.
Magnetic Flux
The magnetic flux through a surface is the total number of field lines crossing it:
Φ = B · A = BA cosθ
where θ is the angle between B and the area vector (normal to the surface). SI unit: weber (Wb); 1 Wb = 1 T m2.
- If the surface is perpendicular to B (θ = 0), Φ is maximum: Φ = BA.
- If parallel to B (θ = 90°), Φ = 0.
- For a coil of N turns, flux linkage = NΦ.
Motion of Charged Particle in Magnetic Field
A charge q moving with velocity v in a magnetic field B feels the Lorentz force:
F = q v × B |F| = qvB sinθ
The force is always perpendicular to v, so it changes direction but not speed. Magnetic forces do no work on a free charge.
Three special cases
- v parallel to B: F = 0. The charge moves in a straight line; B has no effect.
- v perpendicular to B: The force provides centripetal acceleration. The charge moves in a circle.
- v at an angle to B: The component along B is unchanged; the perpendicular component drives circular motion. Result: a helical path.
Circular motion: key formulas
For v perpendicular to B, equating qvB to mv2/r:
r = mv/(qB)
The angular frequency and period are
ω = qB/m, T = 2πm/(qB) = 2π/ω
Crucially, the period is independent of v. This is the principle of the cyclotron: faster particles trace bigger circles but take exactly the same time to complete a half-loop.
Crossed E and B fields select a single speed: only particles with v = E/B pass undeflected (electric and magnetic forces cancel). Selected ions then enter a region with B alone and curve with radius r = mv/(qB), allowing m/q to be measured precisely.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter. Read the explanation even when you get the answer right, it's where the deeper concept lives.
Q1. A charged particle moving in a uniform magnetic field experiences a force that is:
F = q v × B. The cross-product is perpendicular to both v and B, so the force never does work and the speed stays constant. The force is maximum when v is perpendicular to B and zero when parallel.
Q2. The radius of the circular path of a charged particle in a magnetic field is given by:
qvB = mv2/r ⇒ r = mv/(qB). The period T = 2πm/(qB) is independent of v, same time for any speed, just different circles.
Q3. SI unit of magnetic flux is:
Flux Φ is measured in webers (Wb = T m2). Flux density B is measured in teslas. Henry is the unit of inductance; gauss is a CGS unit (104 G = 1 T).
Q4. A solenoid has 500 turns per metre carrying 2 A. The field inside is approximately:
B = μ0nI = 4π × 10−7 × 500 × 2 = 4π × 10−4 ≈ 1.26 × 10−3 T. The field is uniform along the length and zero outside (long-solenoid approximation).
Q5. An electron and a proton enter the same magnetic field perpendicularly with the same kinetic energy. Which has the smaller circular radius?
Same KE: mv2/2 same, so mv = √(2mE). Therefore r = mv/(qB) = √(2mE)/(qB) ∝ √m. Electron has smaller mass, hence smaller radius. (Charges have equal magnitude.)
Quick Recap
- Magnetic flux: Φ = BA cosθ (Wb).
- Flux density (field): B in tesla; F = BIL sinθ on a wire.
- Solenoid: B = μ0nI; long straight wire: B = μ0I/(2πr).
- Lorentz force: F = qv × B; magnitude qvB sinθ; does no work.
- v ⊥ B ⇒ circle of radius r = mv/(qB); period T = 2πm/(qB), independent of v.
- v at angle to B ⇒ helical motion.
- Velocity selector: undeflected if v = E/B.