Formula Sheet — Magnetism Part 1
Source: FAD1022 Tutorial 8 — Magnetism Part 1
1. Physical Constants
Permeability of Free Space
$$\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$$
| Symbol | Meaning | Value | Units |
|---|---|---|---|
| $\mu_0$ | Permeability of free space | $4\pi \times 10^{-7}$ | T m A$^{-1}$ or H m$^{-1}$ |
Electron Properties
$$m_e = 9.1 \times 10^{-31} \text{ kg}$$
$$e = 1.6 \times 10^{-19} \text{ C}$$
| Symbol | Meaning | Value | Units |
|---|---|---|---|
| $m_e$ | Mass of electron | $9.1 \times 10^{-31}$ | kg |
| $e$ | Charge of electron (magnitude) | $1.6 \times 10^{-19}$ | C |
2. Magnetic Field Due to Current-Carrying Wires
Straight Infinite Wire (Biot-Savart Law)
$$B = \frac{\mu_0 I}{2\pi r}$$
| Symbol | Meaning | Units |
|---|---|---|
| $B$ | Magnetic field magnitude | T (tesla) |
| $\mu_0$ | Permeability of free space | T m A$^{-1}$ |
| $I$ | Current in the wire | A (amperes) |
| $r$ | Perpendicular distance from wire | m (meters) |
Direction: Given by the right-hand rule (thumb along current, fingers curl in direction of $B$).
Circular Coil (Center)
$$B = \frac{\mu_0 N I}{2r}$$
| Symbol | Meaning | Units |
|---|---|---|
| $B$ | Magnetic field at center of coil | T (tesla) |
| $\mu_0$ | Permeability of free space | T m A$^{-1}$ |
| $N$ | Number of turns | dimensionless |
| $I$ | Current in the coil | A (amperes) |
| $r$ | Radius of the coil | m (meters) |
Direction: Given by the right-hand rule (fingers curl with current, thumb points along $B$).
Superposition Principle
$$\vec{B}_{\text{resultant}} = \sum_i \vec{B}_i$$
| Symbol | Meaning | Units |
|---|---|---|
| $\vec{B}_{\text{resultant}}$ | Resultant magnetic field | T (tesla) |
| $\vec{B}_i$ | Magnetic field due to source $i$ | T (tesla) |
3. Magnetic Force on Moving Charges (Lorentz Force)
Vector Form
$$\vec{F} = q\vec{v} \times \vec{B}$$
Magnitude
$$F = |q|vB\sin\theta$$
| Symbol | Meaning | Units |
|---|---|---|
| $\vec{F}$ | Magnetic force | N (newtons) |
| $q$ | Electric charge | C (coulombs) |
| $\vec{v}$ | Velocity of the charge | m/s |
| $\vec{B}$ | Magnetic field | T (tesla) |
| $\theta$ | Angle between $\vec{v}$ and $\vec{B}$ | degrees (°) or radians (rad) |
Note: For an electron, $q = -e$.
4. Magnetic Force Between Parallel Current-Carrying Wires
Force per Unit Length
$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\dfrac{F}{L}$ | Force per unit length | N m$^{-1}$ |
| $\mu_0$ | Permeability of free space | T m A$^{-1}$ |
| $I_1$ | Current in wire 1 | A (amperes) |
| $I_2$ | Current in wire 2 | A (amperes) |
| $d$ | Distance between wires | m (meters) |
| $L$ | Length of wire segment | m (meters) |
Direction: Parallel currents attract; antiparallel currents repel.
Total Force on a Wire Segment
$$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$
| Symbol | Meaning | Units |
|---|---|---|
| $F$ | Total magnetic force | N (newtons) |
5. Velocity Selector
Condition for Zero Deflection
$$v = \frac{E}{B}$$
| Symbol | Meaning | Units |
|---|---|---|
| $v$ | Speed of charged particle | m/s |
| $E$ | Electric field magnitude | V m$^{-1}$ or N C$^{-1}$ |
| $B$ | Magnetic field magnitude | T (tesla) |
Derivation: When electric and magnetic forces balance: $$qE = qvB \implies v = \frac{E}{B}$$
6. Circular Motion of Charged Particle in Magnetic Field
Radius of Circular Path
$$r = \frac{mv}{|q|B}$$
| Symbol | Meaning | Units |
|---|---|---|
| $r$ | Radius of circular path | m (meters) |
| $m$ | Mass of particle | kg |
| $v$ | Speed of particle | m/s |
| $ | q | $ |
| $B$ | Magnetic field magnitude | T (tesla) |
Derivation: Centripetal force equals magnetic force: $$\frac{mv^2}{r} = |q|vB$$
Period of Circular Motion
$$T = \frac{2\pi m}{|q|B}$$
| Symbol | Meaning | Units |
|---|---|---|
| $T$ | Period of motion | s (seconds) |
| $m$ | Mass of particle | kg |
| $ | q | $ |
| $B$ | Magnetic field magnitude | T (tesla) |
Note: The period is independent of velocity and radius.
Angular Frequency (Cyclotron Frequency)
$$\omega = \frac{|q|B}{m} = \frac{2\pi}{T}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\omega$ | Angular frequency | rad/s |
7. Summary Table
| Topic | Formula | Variables |
|---|---|---|
| Straight wire field | $B = \dfrac{\mu_0 I}{2\pi r}$ | $I$ = current, $r$ = distance |
| Coil center field | $B = \dfrac{\mu_0 N I}{2r}$ | $N$ = turns, $I$ = current, $r$ = radius |
| Lorentz force | $\vec{F} = q\vec{v} \times \vec{B}$ | $q$ = charge, $\vec{v}$ = velocity, $\vec{B}$ = field |
| Force magnitude | $F = | q |
| Force between wires | $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ | $d$ = separation, $L$ = length |
| Velocity selector | $v = \dfrac{E}{B}$ | $E$ = electric field, $B$ = magnetic field |
| Circular path radius | $r = \dfrac{mv}{ | q |
| Period of motion | $T = \dfrac{2\pi m}{ | q |
Related Concepts
- Magnetism
- Magnetic Field
- Biot-Savart Law
- Ampere's Law
- Magnetic Field of Straight Wire
- Magnetic Field of Circular Coil
- Lorentz Force
- Magnetic Force on Moving Charge
- Magnetic Force Between Wires
- Velocity Selector
- Circular Motion in Magnetic Field