FAD1022 L22-L26 — Magnetism — Formula Sheet
Comprehensive formula sheet covering magnetic fields, magnetic forces, Ampere's law, and torque on current loops.
Physical Constants
| Constant | Symbol | Value |
|---|---|---|
| Permeability of free space | $\mu_0$ | $4\pi \times 10^{-7} ; \text{T}\cdot\text{m}\cdot\text{A}^{-1}$ |
| Elementary charge | $e$ | $1.602 \times 10^{-19} ; \text{C}$ |
| Mass of proton | $m_p$ | $1.67 \times 10^{-27} ; \text{kg}$ |
| Mass of electron | $m_e$ | $9.11 \times 10^{-31} ; \text{kg}$ |
Unit Conversions
$$1 , \text{T} = 1 , \frac{\text{N}}{\text{A}\cdot\text{m}}$$
$$1 , \text{G} = 10^{-4} , \text{T}$$
L22 — Magnetic Field
Magnetic Field Due to Current-Carrying Conductors
1. Long Straight Wire
$$B = \frac{\mu_0 I}{2\pi r}$$
- $B$ — magnetic field strength (T)
- $\mu_0$ — permeability of free space
- $I$ — current in the wire (A)
- $r$ — perpendicular distance from the wire (m)
Key relationship: $B \propto \dfrac{1}{r}$ — field strength decreases with distance from the wire.
2. Circular Loop (at Centre)
$$B = \frac{\mu_0 N I}{2r}$$
- $B$ — magnetic field strength at centre (T)
- $N$ — number of turns ($N = 1$ if not specified)
- $I$ — current (A)
- $r$ — radius of the loop (m)
Note: No $\pi$ appears in this formula.
3. Long Solenoid (at Centre)
$$B = \frac{\mu_0 N I}{L} = \mu_0 n I$$
- $B$ — magnetic field inside the solenoid (T)
- $N$ — total number of turns
- $L$ — length of the solenoid (m)
- $n = N/L$ — turns per unit length ($\text{m}^{-1}$)
- $I$ — current (A)
Conditions: Valid when $L \gg r$ (length much greater than radius).
Properties:
- Field inside is nearly uniform and parallel to the axis.
- External field near the centre is approximately zero.
Superposition of Magnetic Fields
For multiple current-carrying wires, the net magnetic field at a point is the vector sum of the fields from each wire:
$$\vec{B}_{\text{net}} = \sum_i \vec{B}_i$$
- Add magnitudes if fields are in the same direction.
- Subtract magnitudes if fields are in opposite directions.
L23 — Force and Motion of Charge in Magnetic Field
Magnetic Force on a Moving Charge (Lorentz Force)
Vector Form
$$\vec{F}_B = q\vec{v} \times \vec{B}$$
- $\vec{F}_B$ — magnetic force (N)
- $q$ — electric charge (C)
- $\vec{v}$ — velocity of the charge ($\text{m}\cdot\text{s}^{-1}$)
- $\vec{B}$ — magnetic field (T)
Magnitude
$$|F_B| = |q|vB\sin\theta$$
- $\theta$ — angle between $\vec{v}$ and $\vec{B}$
Extremes:
- Maximum force: $\theta = 90°$ ($v \perp B$) $\Rightarrow |F_B| = |q|vB$
- Zero force: $\theta = 0°$ ($v \parallel B$) or $v = 0$ (stationary charge)
No magnetic force acts on a stationary charge in a magnetic field.
Motion of a Charged Particle in a Uniform Magnetic Field
When $v \perp B$, the magnetic force provides the centripetal force:
$$F_B = F_C \quad \Rightarrow \quad qvB = \frac{mv^2}{r}$$
Radius of Circular Motion
$$r = \frac{mv}{qB}$$
- $r$ — radius of the circular path (m)
- $m$ — mass of the particle (kg)
- $v$ — speed ($\text{m}\cdot\text{s}^{-1}$)
- $q$ — charge (C)
- $B$ — magnetic field strength (T)
Velocity from Radius
$$v = \frac{rqB}{m}$$
Period of Circular Motion
$$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$$
- $T$ — period of revolution (s)
Period is independent of velocity — all particles with same $q/m$ have the same period regardless of speed.
Angular Frequency (Cyclotron Frequency)
$$\omega = \frac{2\pi}{T} = \frac{qB}{m}$$
- $\omega$ — angular frequency ($\text{rad}\cdot\text{s}^{-1}$)
Motion in Combined Electric and Magnetic Fields
Velocity Selector
For a charged particle to pass through undeflected:
$$F_E = F_B \quad \Rightarrow \quad qE = qvB$$
$$v = \frac{E}{B}$$
- $E$ — electric field strength ($\text{N}\cdot\text{C}^{-1}$ or $\text{V}\cdot\text{m}^{-1}$)
- $B$ — magnetic field strength (T)
- $v$ — velocity of selected particles ($\text{m}\cdot\text{s}^{-1}$)
Only particles with this exact velocity pass through the selector slit.
Mass Selector (Mass Spectrometer)
In the second magnetic field $B'$, an ion follows a semicircular path of radius $r$:
$$qvB' = \frac{mv^2}{r}$$
Solving for mass:
$$m = \frac{qB'r}{v}$$
Substituting $v = E/B$ from the velocity selector:
$$m = \frac{qB'B^2r}{E}$$
If the same magnetic field $B$ is used in both regions:
$$m = \frac{qrB^2}{E}$$
- $m$ — mass of the ion (kg)
- $q$ — charge of the ion (C)
- $B'$ — magnetic field in mass selector region (T)
- $B$ — magnetic field in velocity selector region (T)
- $r$ — radius of semicircular path (m)
- $E$ — electric field strength ($\text{V}\cdot\text{m}^{-1}$)
L24 — Magnetic Force between Two Parallel Wires
Force Magnitude
For two parallel wires of length $L$, separated by distance $d$, carrying currents $I_1$ and $I_2$:
$$F_{21} = F_{12} = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$
- $F_{21} = F_{12}$ — magnitude of force on each wire (N) [Newton's third law pair]
- $I_1, I_2$ — currents in the two wires (A)
- $L$ — length of each wire (m)
- $d$ — perpendicular distance between wires (m)
Force Per Unit Length
$$f = \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
- $f$ — force per unit length ($\text{N}\cdot\text{m}^{-1}$)
Direction:
- Same direction currents → attractive force
- Opposite direction currents → repulsive force
L25-26 — Ampere's Law and Torque
Ampere's Law
Integral Form
$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$
- $\vec{B}$ — magnetic field (T)
- $d\vec{l}$ — differential length element of the Amperian loop (m)
- $I_{enc}$ — total current enclosed by the loop (A)
Enclosed Current (Inside a Cylindrical Wire)
For a wire of radius $R$ carrying total current $I$, at a distance $r < R$ from the centre:
$$I_{enc} = \frac{\pi r^2}{\pi R^2} I = \frac{r^2}{R^2} I$$
- $r$ — distance from centre where field is evaluated (m)
- $R$ — radius of the conducting wire (m)
Magnetic Field of a Cylindrical Wire
| Region | Condition | Formula |
|---|---|---|
| At surface | $r = R$ | $B = \dfrac{\mu_0 I}{2\pi R}$ |
| Outside wire | $r > R$ | $B = \dfrac{\mu_0 I}{2\pi r}$ |
| Inside wire | $r < R$ | $B = \dfrac{\mu_0 I r}{2\pi R^2}$ |
- $R$ — radius of the wire (m)
- $r$ — distance from the centre of the wire (m)
- $I$ — total current (A)
Long Solenoid (Ampere's Law Derivation)
$$B = \mu_0 n I = \frac{\mu_0 N I}{L}$$
- $B$ — uniform magnetic field inside the solenoid (T)
- $n = N/L$ — turns per unit length ($\text{m}^{-1}$)
- $N$ — total number of turns
- $L$ — length of solenoid (m)
- $I$ — current (A)
Properties:
- Field outside $\approx 0$
- Field inside is uniform and parallel to the axis
Torque on a Current Loop
$$\tau = N I A B \sin\theta$$
- $\tau$ — torque (N·m)
- $N$ — number of turns (loops)
- $I$ — current (A)
- $A$ — area of the coil ($\text{m}^2$)
- $B$ — magnetic field strength (T)
- $\theta$ — angle between $\vec{B}$ and the normal vector $\vec{A}$ to the coil plane
True for any planar shape of the loop.
Extremes:
- Maximum torque: $\theta = 90°$ → plane of coil is parallel to $\vec{B}$ $$\tau_{\max} = N I A B$$
- Zero torque: $\theta = 0°$ → plane of coil is perpendicular to $\vec{B}$ $$\tau = 0$$
Summary Table of All Formulas
| Topic | Formula | Variables |
|---|---|---|
| Straight wire field | $B = \dfrac{\mu_0 I}{2\pi r}$ | $I$ = current, $r$ = distance |
| Circular loop field | $B = \dfrac{\mu_0 N I}{2r}$ | $N$ = turns, $r$ = radius |
| Solenoid field | $B = \mu_0 n I = \dfrac{\mu_0 N I}{L}$ | $n$ = turns/length |
| Lorentz force (vector) | $\vec{F}_B = q\vec{v} \times \vec{B}$ | $q$ = charge, $\vec{v}$ = velocity |
| Lorentz force (magnitude) | $ | F_B |
| Circular motion radius | $r = \dfrac{mv}{qB}$ | $m$ = mass |
| Period of motion | $T = \dfrac{2\pi m}{qB}$ | independent of $v$ |
| Angular frequency | $\omega = \dfrac{qB}{m}$ | cyclotron frequency |
| Velocity selector | $v = \dfrac{E}{B}$ | selected velocity |
| Mass spectrometer | $m = \dfrac{qB'B^2r}{E}$ | $B'$ = selector field |
| Force between wires | $F = \dfrac{\mu_0 I_1 I_2 L}{2\pi d}$ | $d$ = separation |
| Force per unit length | $f = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ | $f = F/L$ |
| Ampere's Law | $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ | $I_{enc}$ = enclosed current |
| Inside wire | $B = \dfrac{\mu_0 I r}{2\pi R^2}$ | $r < R$ |
| Torque on loop | $\tau = N I A B \sin\theta$ | $\theta$ = angle with normal |
Direction Rules Reference
| Rule | Application | Method |
|---|---|---|
| Right Hand Grip Rule (RHR) | Field around wire | Thumb = current, fingers = $\vec{B}$ |
| Right Hand Rule (cross product) | Force on $+q$ | Fingers = $\vec{v}$, curl to $\vec{B}$, thumb = $\vec{F}$ |
| Fleming's Left Hand Rule | Force on $+q$ | First = $\vec{B}$, second = $\vec{v}$, thumb = $\vec{F}$ |
| Negative charge | Force on $-q$ | Reverse the $+q$ force direction |
2D Vector Notation:
- $\odot$ = out of page (arrow tip toward you)
- $\otimes$ = into page (arrow tail away)
Quick Reference: Motion of Charged Particles
| Condition | Motion Type |
|---|---|
| $v \perp B$ | Circular motion |
| $v \parallel B$ | Straight line (no deflection) |
| $v$ at angle to $B$ | Helical (spiral) motion |
| $v = E/B$ in crossed $E$ and $B$ | Straight line (velocity selector) |
Formula sheet compiled from FAD1022 L22-L26 lectures. Good luck with finals!