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: $\vec B = \dfrac{\mu_0}{4\pi} \dfrac{q\,\vec v \times \hat r}{r^2}$ .