Magnetic Fields & Forces – Comprehensive Bullet‐Point Notes

Magnetism Fundamentals

  • Magnetism originates from moving electric charges (currents) and the intrinsic motion (spin) of sub-atomic particles.
    • Unlike electrostatics, magnetostatic forces act only on moving charges.
  • Two-stage interaction analogy:
    • Stage 1 (Electric): A stationary charge creates an electric field \vec E; another charge feels \vec F_E=q\vec E.
    • Stage 1 (Magnetic): A moving charge/current creates a magnetic field \vec B.
    • Stage 2: A second moving charge/current inside that \vec B experiences a magnetic force.
  • Permanent magnets contain micro-currents (electron circulation/spin) that mimic macroscopic currents.
  • Magnetic poles come in pairs (no isolated magnetic ‘charges’ found ⇒ \nabla!\cdot\vec B=0).
  • Earth behaves like a huge bar magnet; the geographic North Pole is near a magnetic south pole.
    • Field reversals occur irregularly (every 10^4–10^7 yrs).

Magnetic Force on a Single Charge

  • Vector form: \boxed{\vec F=q\,\vec v\times\vec B}.
    • |\vec F|=|q|vB\sin\theta ((\theta) = angle between \vec v and \vec B).
    • SI unit of \vec B: tesla (T) where 1\,\text{T}=1\,\text{N}/(\text{A·m}).
  • Direction via right-hand rule (RHR):
    • Positive charge: fingers \vec v\rightarrow \vec B, thumb gives \vec F.
    • Negative charge: force is opposite thumb.
  • Special orientations:
    • \theta=0° or 180°: \vec F=0 (parallel/anti-parallel motion).
    • \theta=90°: |\vec F|=|q|vB (maximum).

Example 7.1 – Proton in uniform \vec B

  • Data: q=+1.6\times10^{-19}\,\text{C},\ v=3.0\times10^{5}\,\text{m/s},\ B=2.0\,\text{T},\ \theta=30°.
  • |\vec F|=(1.6\times10^{-19})(3.0\times10^{5})(2.0)\sin30°=4.8\times10^{-14}\,\text{N}.
  • RHR → force points in -\hat y; sign flips for electrons.

Magnetic Field Lines & Flux

  • Field line properties:
    • Tangent to \vec B at every point.
    • Density ∝ |\vec B|.
    • Emerge from N-poles, enter S-poles, and form closed loops (Gauss’s law for magnetism \displaystyle \oint_{\text{closed}}\vec B\cdot d\vec A=0).
  • Magnetic flux through surface A: \boxed{\PhiB=\int!\vec B\cdot d\vec A=B{\perp}A = BA\cos\theta}.
    • Unit: weber (Wb) where 1\,\text{Wb}=1\,\text{T·m}^2.

Example 7.2 – Flat plate

  • A=3.0\,\text{cm}^2=3.0\times10^{-4}\,\text{m}^2,\ \Phi_B=+0.90\,\text{mWb}.
  • B=\dfrac{\Phi_B}{A\cos60°}=6.0\,\text{T}; area vector makes 60° with \vec B (not 120° because flux given +ve).

Motion of a Charged Particle in Uniform \vec B

  • \vec F ⟂ \vec v → speed is constant; work done = 0.
  • Pure perpendicular entry (cyclotron motion):
    • Radius \displaystyle R=\frac{mv}{|q|B}.
    • Angular speed \omega=\dfrac{|q|B}{m}; cyclotron frequency f=\omega/2\pi (mass-spectrometry & cyclotrons).
  • Oblique entry: helical path (parallel component unchanged).

Example 7.3 – Magnetron (microwave oven)

  • Required B so electrons orbit at f=2450\,\text{MHz}.
  • \omega=2\pi f=1.54\times10^{10}\,\text{s}^{-1}.
  • For electron me=9.11\times10^{-31}\,\text{kg},\ q=1.60\times10^{-19}\,\text{C}: B=\dfrac{me\omega}{|q|}=0.0877\,\text{T} (easy with permanent magnet).

Magnetic Force on Current-Carrying Conductors

  • For straight segment length \ell carrying current I:
    \boxed{\vec F=I\,\vec \ell\times\vec B} ((\vec \ell) points with current).
  • Magnitude F=I\ell B\sin\theta; RHR as before.

Example 7.5 – Copper rod between electromagnet poles

  • I=50.0\,\text{A}\ (\text{west}→\text{east}),\ B=1.20\,\text{T} toward NE (45°).
    • (a) F=I\ell B\sin45°=42.4\,\text{N} upward.
    • (b) Max when rod rotated so \theta=90° ⇒ F_{\max}=60.0\,\text{N} (enables magnetic levitation if weight ≤60 N).

Magnetic Torque on Current Loops & Coils

  • Single planar loop:
    • Magnetic moment (dipole): \boxed{\vec \mu = I\,\vec A} ((\vec A) ⟂ plane by RHR).
    • Torque: \boxed{\vec \tau = \vec \mu \times \vec B}; magnitude \tau = \mu B\sin\theta = IAB\sin\theta.
    • Potential energy: U=-\vec \mu\cdot\vec B=-\mu B\cos\theta.
  • Coil with N tightly packed turns: \mu=NIA,\ \tau=N IAB\sin\theta,\ U=-NIA B\cos\theta.

Example 7.6–7 (30-turn coil)

  • r=0.0500\,\text{m},\ I=5.00\,\text{A},\ B=1.20\,\text{T},\ N=30 (coil horizontal, \theta=90° initially).
    • \mu=NIA=1.18\,\text{A·m}^2.
    • \tau=\mu B=1.41\,\text{N·m}.
    • If coil rotates to align with \vec B\ (\theta=0°): \Delta U=-1.41\,\text{J} (energy released).

Electric Motors (DC)

  • Rotor = current loop; stator field exerts torque \vec \tau=\vec \mu\times\vec B → mechanical rotation.
  • Commutator reverses current every half-turn to maintain unidirectional torque.
  • Back-emf \varepsilon induced by rotating rotor opposes supply (Lenz’s law).
  • For series motor: V_{ab}=\varepsilon + I r.
  • Power terms:
    • Input P_{in}=VI.
    • Resistive loss P_R=I^2 r.
    • Mechanical output P{mech}=P{in}-P_R=\varepsilon I.
    • Efficiency \eta=P{mech}/P{in}=\varepsilon/V.

Example 7.8 – 120-V motor, r=2.00\,\Omega,\ I=4.00\,\text{A} at full load

  • (a) \varepsilon=V-Ir=112\,\text{V}.
  • (b) P_{in}=480\,\text{W}.
  • (c) P_R=I^2 r=32\,\text{W}.
  • (d) P_{mech}=448\,\text{W}.
  • (e) \eta=93\%.
  • (f) If rotor jams ⇒ \varepsilon→0, current I=V/r=60\,\text{A}, losses P_R=7200\,\text{W} ⇒ catastrophic heating (fuses/breakers trip).

The Hall Effect

  • Charge carriers in conductor subject to \vec B (perpendicular to current) experience magnetic deflection → transverse electric field Ez builds until qEz = qv_d B.
  • Hall voltage VH = Ez d (thickness d of slab).
  • Carrier concentration:
    \boxed{n = \dfrac{Jx By}{q Ez}} where Jx=I/A.
  • Sign of V_H distinguishes electron vs hole conduction.

Example 7.9 – Copper strip

  • Dimensions: thickness d=2.0\,\text{mm},\ w=1.50\,\text{cm}; B=0.40\,\text{T},\ I=75\,\text{A},\ V_H=0.81\,\mu\text{V}.
  • Jx=2.5\times10^{6}\,\text{A/m}^2,\ Ez=5.4\times10^{-5}\,\text{V/m}.
  • n\approx1.16\times10^{29}\,\text{m}^{-3} (ideal free-electron model gives 8.5\times10^{28}\,\text{m}^{-3}).

Magnetic Field of a Moving Point Charge

  • Biot–Savart analogue for a single charge (steady velocity): \boxed{\vec B = \dfrac{\mu_0}{4\pi}\,\dfrac{q\,\vec v \times \hat r}{r^2}}.
    • \hat r = unit vector from charge to field point, r = separation.
    • Field circles around direction of motion (RHR #2: thumb = \vec v, fingers give \vec B).
  • Superposition applies for multiple charges/currents.

Example 7.10 – Two protons moving oppositely along x

  • Electric repulsion: FE=\dfrac{1}{4\pi\varepsilon0}\dfrac{q^2}{r^2} upward on top proton.
  • Magnetic interaction: lower proton’s \vec B points +z at upper proton; force magnitude
    FB=q v B=qv\left(\dfrac{\mu0}{4\pi}\dfrac{qv}{r^2}\right)=\dfrac{\mu_0 q^2 v^2}{4\pi r^2} downward (attractive) because currents oppose.
  • Ratio \displaystyle \frac{FB}{FE}=\frac{\mu0\varepsilon0 v^2}{1}=\left(\frac{v}{c}\right)^2 (tiny unless v\approx c).
    • Demonstrates why magnetic effects are relativistic corrections to Coulomb force.

Practical & Conceptual Connections

  • Magnetic levitation, MRI, mass spectrometers, cyclotrons, microwave magnetrons, Hall-effect sensors.
  • Energy perspective: torque tendencies, potential minima (stable) vs maxima (unstable) for dipoles.
  • Safety note: Motors draw huge stall currents; protective devices essential.
  • Symmetry: Maxwell equation \nabla!\cdot\vec B=0 implies closed field lines; no magnetic monopoles observed.
  • Relativity link: Magnetic force can be viewed as electrostatic force in a different inertial frame (length contraction explains F_B scaling).

Quick Reference – Key Equations

  • Force on charge: \vec F=q(\vec v\times\vec B).
  • Force on wire: \vec F=I\vec \ell\times\vec B.
  • Radius of circular motion: R=\dfrac{mv}{|q|B}.
  • Magnetic moment: \vec \mu = N I \vec A.
  • Torque: \vec \tau = \vec \mu \times \vec B.
  • Potential energy: U=-\vec \mu\cdot\vec B.
  • Hall carrier density: n=\dfrac{J B}{q E_H}.
  • Moving charge field: \vec B = \dfrac{\mu_0}{4\pi} \dfrac{q\,\vec v \times \hat r}{r^2}.