Lesson 12 of 12 · 18 min
Chapter review
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Biot-Savart to cyclotron, chapter revision
Must-know facts
18 facts
- 01Magnetic force F = qvB sin θ: zero for a charge at rest or moving along B, largest at right angles.
- 02The magnetic force is perpendicular to v, so it does no work: the speed and kinetic energy never change.
- 03Circle in a uniform field: r = mv/qB. The period T = 2πm/qB does not depend on the speed.
- 04Helix when v has a part along B: pitch p = v∥ × 2πm/qB.
- 05Force on a wire: F = I l × B; zero when the wire lies along B.
- 06Biot–Savart: dB = (μ₀/4π) I dl sin θ/r²; no field along the line of the element.
- 07μ₀ = 4π × 10⁻⁷ T m A⁻¹, and μ₀ε₀ = 1/c².
- 08Centre of a coil: B = μ₀NI/2R. On its axis: B = μ₀NIR²/2(x² + R²)^(3/2).
- 09Long straight wire: B = μ₀I/2πr, field lines are closed circles.
- 10Thick wire with uniform current: B ∝ r inside, B ∝ 1/r outside, largest at the surface.
- 11Long solenoid: B = μ₀nI inside, uniform, independent of radius; about zero outside.
- 12Parallel wires: f = μ₀I₁I₂/2πd per metre; same direction attract, opposite repel.
- 131 A is the current giving 2 × 10⁻⁷ N per metre between wires 1 m apart.
- 14Current loop: m = NIA; τ = m × B; net force zero in a uniform field.
- 15Stable equilibrium when m is along B, unstable when opposite.
- 16Far field of a loop: (μ₀/4π)(2m/x³) on the axis, half that in the plane of the loop.
- 17MCG: kφ = NIAB in a radial field; current sensitivity NAB/k.
- 18Ammeter = galvanometer + small shunt in parallel; voltmeter = galvanometer + large resistance in series.
Common traps
Where marks are lost
Formulas
17 to know
Lorentz force
F = q(E + v × B)
Magnetic part qvB sin θ, perpendicular to v and B.
Force on a straight wire
F = I l × B, |F| = IlB sin θ
l points along the current; B is the external field.
Radius of circular path
r = mv/(qB)
v perpendicular to B; r ∝ momentum.
Cyclotron frequency
ν = qB/(2πm), T = 2πm/(qB)
Independent of speed and radius.
Pitch of the helix
p = v∥T = 2πm v∥/(qB)
v∥ is the velocity component along B.
Biot–Savart law
dB = (μ₀/4π) I dl sin θ/r²
μ₀/4π = 10⁻⁷ T m A⁻¹; direction of dl × r.
Axis of a circular loop
B = μ₀IR²/[2(x² + R²)^(3/2)]
Multiply by N for N turns.
Centre of a circular coil
B = μ₀NI/(2R)
A semicircle gives half of a full loop.
Ampere's circuital law
∮B·dl = μ₀I
Simple form BL = μ₀Iₑ when symmetry allows.
Long straight wire
B = μ₀I/(2πr)
Inside a thick wire with uniform current: B = μ₀Ir/(2πa²).
Long solenoid
B = μ₀nI
n = turns per unit length.
Force between parallel wires
f = μ₀I₁I₂/(2πd)
Per unit length; like currents attract.
Torque on a coil
τ = m × B, τ = NIAB sin θ
m = NIA; θ between m and B.
Dipole field of a loop
B = (μ₀/4π)(2m/x³) axial, (μ₀/4π)(m/x³) in plane
Valid for x ≫ R.
Galvanometer
kφ = NIAB, φ/I = NAB/k, φ/V = NAB/(kR)
k = torsional constant of the spring.
Ammeter shunt
rₛ = I_G R_G/(I − I_G)
Follows from equal voltage across G and the shunt; I_G is the full-scale current of G.
Voltmeter series resistance
R = V/I_G − R_G
V is the full-scale voltage wanted.
Key terms
17 terms
- Magnetic field (B)
- A vector field set up by currents and moving charges, measured by the force it puts on a moving charge; unit tesla.
- Tesla
- The field that exerts 1 N on 1 C moving at 1 m/s at right angles to it; 1 gauss = 10⁻⁴ T.
- Lorentz force
- The total force q(E + v × B) on a charge in electric and magnetic fields.
- Cyclotron frequency
- The rate qB/2πm at which a charge circles in a uniform magnetic field, the same at every speed.
- Pitch
- The distance a charge moves along B during one turn of its helical path.
- Current element
- A tiny length dl of wire with its current, I dl, treated as a vector source of field.
- Permeability of free space (μ₀)
- The constant in the Biot–Savart law, 4π × 10⁻⁷ T m A⁻¹.
- Amperian loop
- A closed path chosen so that B is tangential and constant, or normal, or zero along its parts, making Ampere's law easy to apply.
- Solenoid
- A long, closely wound helical coil whose inside field is uniform and along its axis.
- Ampere (unit)
- The steady current that gives 2 × 10⁻⁷ N per metre between two long parallel wires 1 m apart in vacuum.
- Magnetic moment (m)
- NIA for a current loop, along its area vector; unit A m².
- Magnetic dipole
- A current loop seen from far away; its field has the same form as an electric dipole's.
- Radial field
- A field whose lines point along radii of the coil's axis, so the coil's plane always lies along B.
- Torsional constant (k)
- The restoring torque of the galvanometer spring per unit angle of twist.
- Shunt
- A small resistance put in parallel with a galvanometer so it can measure large currents as an ammeter.
- Current sensitivity
- Deflection per unit current, φ/I = NAB/k.
- Voltage sensitivity
- Deflection per unit voltage, φ/V = NAB/(kR).