amazon.co.ukA-Level Physics revision: magnetic flux density, F = BIL, force on moving charges, induced EMF, Faraday's and Lenz's laws, transformers, alternating currents and RMS values - AQA 7408 3.7.5, OCR A H556 6.3, Edexcel 9PH0 Topic 7.
Magnetic flux density B is defined by the force on a current-carrying wire: F = BIL sin θ, where θ is the angle between the wire and the field - B is the force per unit current per unit length on a wire perpendicular to the field, in tesla (AQA 7408 3.7.5.1; Edexcel 9PH0 Topic 7; OCR A H556 6.3). Full force when perpendicular (θ = 90), zero when parallel to the field (θ = 0). Direction from Fleming's left-hand rule: first finger = field (N to S), second finger = current, thumb = force (motion) - all three mutually perpendicular. This is the motor principle: the force on the coil sides of a motor makes it rotate (commutator reverses the current each half-turn to keep the rotation going).
Flux: Φ = BA cos θ - the perpendicular component of the field through an area, in webers (1 Wb = 1 T m^2). For a coil of N turns the flux linkage is NΦ. Tilting a coil relative to the field changes cos θ, hence the flux - the geometry that drives induction (section 3).
A wire of length 0.50 m carries 3.0 A at 30 degrees to a uniform field of 0.20 T. Find the force.
Line 1: F = BIL sin θ (M1: equation with the angle).
Line 2: F = 0.20 × 3.0 × 0.50 × sin 30 = 0.20 × 3.0 × 0.50 × 0.5 = 0.15 N (M1 A1), direction from Fleming's left-hand rule.
Line 3: Parallel to the field (θ = 0): F = 0 - the sine factor carries the geometry (A1: interpretation).
A charge Q moving at speed v perpendicular to a field experiences F = BQv (AQA 7408 3.7.5.1-3.7.5.2). Derivation from the wire force: a wire of length L with n carriers per volume (area A) carrying current I contains nAL carriers, each moving at drift speed v; the total charge moving is nALq, and the current I = nqAv. The force on the wire F = BIL = B(nqAv)L = (nALq)Bv - the force on the total moving charge. Per charge: F = BQv. The force is always perpendicular to the velocity, so it changes direction but not speed - it does no work. Result: circular motion in a uniform field, with the magnetic force as the centripetal force:
BQv = mv^2/r, so r = mv/(BQ) - radius grows with momentum and falls with charge and field strength. This is how the mass of ions is measured in mass spectrometers (r ∝ m at fixed v, Q, B) and how particles are steered in accelerators.
Cyclotron: two D-shaped electrodes (dees) with an alternating pd across the gap in a perpendicular field: the field bends each particle into a semicircle of radius r = mv/(BQ) inside a dee; the alternating pd accelerates it across the gap each crossing, increasing v and hence r - but the period is independent of speed (T = 2πm/(BQ)), so the alternating pd frequency stays fixed even as the orbits grow: the key design fact. Velocity selector: crossed E and B fields give electric force qE and magnetic force BQv in opposition; particles pass undeflected when qE = BQv, i.e. v = E/B - a speed filter.
An electron (m = 9.11 × 10^-31 kg, e = 1.60 × 10^-19 C) travels at 2.0 × 10^7 m/s at right angles to a field of 0.50 mT. Find the radius of its circular path.
Line 1: Magnetic force provides the centripetal force: BQv = mv^2/r (M1: force balance).
Line 2: r = mv/(BQ) (M1: rearrangement).
Line 3: r = (9.11 × 10^-31 × 2.0 × 10^7)/(0.50 × 10^-3 × 1.60 × 10^-19) = 1.82 × 10^-23/8.0 × 10^-23 = 0.23 m (A1) - 23 cm.
Faraday's law: the induced EMF is proportional to the rate of change of flux linkage: ε = -N dΦ/dt (AQA 7408 3.7.5.3; OCR A H556 6.3). Lenz's law: the direction of the induced current opposes the change in flux causing it (the minus sign) - push a magnet towards a coil and the coil's face becomes the same pole, repelling the magnet. Lenz's law is energy conservation: if the induced current aided the change, it would amplify motion and create free energy. Induction experiments: magnet moved in/out of a coil (EMF only while moving - faster motion gives larger EMF because dΦ/dt is larger), a wire moving through a field (ε = BLv, derivation: flux swept per second = BLv, so EMF = rate of flux cutting), a coil rotating in a field (the generator, below).
Derivation of ε = BLv (straight wire): a wire of length L moving at speed v perpendicular to a field B sweeps out an area Lv each second, cutting flux Φ = BLv per second. Flux linkage change per second is the induced EMF: ε = BLv.
Derivation of the rotating-coil EMF: a coil of N turns, area A, rotating at angular speed ω in field B: the angle between the field and the coil's normal is θ = ωt, so flux linkage NΦ = BAN cos(ωt). Faraday's law: ε = -d(NΦ)/dt = -d[BAN cos(ωt)]/dt = BANω sin(ωt) - a sinusoidal EMF with peak ε_max = BANω, maximum when the coil's plane is parallel to the field (flux zero but changing fastest), zero when the plane is perpendicular (flux maximum, rate of change zero). This is the AC generator: slip rings output the alternating EMF.
A 200-turn coil of area 0.05 m^2 rotates at 50 Hz in a 0.4 T field. Find the peak EMF.
Line 1: ω = 2πf = 2π × 50 = 314 rad/s (M1: angular speed from frequency).
Line 2: ε_max = BANω (M1: generator equation).
Line 3: ε_max = 0.4 × 200 × 0.05 × 314 = 1257 V ≈ 1.26 kV (A1).
Line 4: The EMF is zero when the coil plane is perpendicular to B (flux maximum, dΦ/dt = 0) - a standard interpretation point (A1).
A transformer couples two coils on a shared laminated iron core (AQA 7408 3.7.5.4): alternating current in the primary creates a changing flux; the core carries the changing flux through the secondary, where Faraday's law induces an EMF.
Transformer equation derivation: the same changing flux passes through both coils, so each turn links dΦ/dt. Primary: V_p = N_p dΦ/dt; secondary: V_s = N_s dΦ/dt (ideal, no losses). Dividing: V_s/V_p = N_s/N_p - the voltage ratio equals the turns ratio. Step-up (N_s > N_p) raises voltage and lowers current (power conservation for an ideal transformer: V_pI_p = V_sI_s); step-down does the reverse. Efficiency losses: eddy currents induced in the iron core (minimised by laminating - thin insulated layers break the eddy paths), resistance of the windings (thick copper), hysteresis in the core (soft iron magnetises and demagnetises easily), and flux leakage (close-wound coils on a continuous core). Real transformers reach 99%+ efficiency.
Why high-voltage transmission: power = VI; for a fixed transmitted power, raising V lowers I, and the cable loss P_loss = I^2R_cable falls as the square - the standard argument with a calculation.
Alternating currents: I = I_0 sin(ωt), V = V_0 sin(ωt). The RMS value is the DC-equivalent: V_rms = V_0/√2. Derivation: power is proportional to V^2; averaging sin^2(ωt) over a cycle gives ½, so the mean power with peak V_0 equals the DC power at V_0/√2 - hence V_rms = V_0/√2 (same for current). UK mains: 230 V rms at 50 Hz means peak V_0 = √2 × 230 = 325 V and period 0.020 s. Oscilloscope: y-gain (V/division) measures peak pd from the trace height; time base (s/division) measures the period (peak-to-peak time) and hence frequency; DC traces are flat lines, AC sinusoids.
A transformer steps 12,000 V down to 230 V with 95% efficiency. The secondary supplies 13 A.
Line 1: Ideal turns ratio: V_s/V_p = N_s/N_p = 230/12000 = 0.0192 (M1 A1) - so N_s/N_p = 1:52.
Line 2: Ideal primary current: I_p = I_s × V_s/V_p = 13 × 0.0192 = 0.25 A (M1: power conservation).
Line 3: With 95% efficiency, input power = output/0.95: I_p = 13 × 230/(0.95 × 12000) = 0.26 A (A1).
Line 4: Losses: eddy currents (lamination), winding resistance, hysteresis (A1).
The standard practical measures the flux density of a magnet (AQA required practical 8; OCR PAG-style; Edexcel practical work on fields): a rigid current-carrying wire (or a wire 'cradle') of measured length L sits on a top-pan balance inside the uniform field of a magnet (or Helmholtz coils), perpendicular to the field. When current flows, the magnetic force F = BIL acts on the wire; by Newton's third law an equal and opposite force acts on the magnet, changing the balance reading.
Method: with the wire clamped rigidly to a stand (not touching the balance), zero the balance with the magnet on it, then vary the current I (ammeter, rheostat, power supply) and record the balance reading at each current. The apparent mass change Δm corresponds to force Δmg = BIL. Plot force (Δmg) against current: a straight line through the origin of gradient BL, so B = gradient/L. Using the gradient across all data averages random error and tests the model (a straight line confirms F ∝ I). Reverse the current to check the force reverses (balance reading moves the other way).
Uncertainty and evaluation: the balance resolution (±0.01 g on a two-decimal-place balance = ±0.0001 N in force) sets the smallest measurable force - use the largest possible current and length (within the field) to increase Δm. Length L measured with a rule (±1 mm); current from an ammeter resolution. Percentage uncertainty in B = Δgradient/gradient + ΔL/L. Systematic errors: the wire not perpendicular to the field (F = BIL sin θ - use a set-square to align), the wire touching the balance (friction corrupts the reading), zero drift of the balance (re-zero between readings), and heating of the wire changing its resistance (take readings promptly). Quote B ± ΔB and compare with the manufacturer's value.
A wire of length 0.050 m on a balance: current 2.0 A gives a reading change of 0.0040 kg; 4.0 A gives 0.0080 kg.
Line 1: Force = Δmg: 0.0040 × 9.81 = 0.039 N; 0.0080 × 9.81 = 0.078 N (M1: force from balance reading).
Line 2: Gradient of force against current = 0.078/4.0 = 0.0196 N/A (M1 A1: straight line through origin).
Line 3: B = gradient/L = 0.0196/0.050 = 0.39 T (A1).
Line 4: With balance resolution ±0.0001 kg, the force uncertainty is ±0.001 N, about 2.5% - repeat and average to reduce it (A1: evaluation).