Formula Sheet: Electrostatics
Comprehensive formula sheet extracted from FAD1022 Tutorial 1 — Electrostatics
1. Physical Constants
| Symbol | Value | Description |
|---|---|---|
| $k$ | $9.0 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$ | Coulomb constant |
| $q_e$ | $1.609 \times 10^{-19} \text{ C}$ | Elementary charge (magnitude) |
| $m_e$ | $9.11 \times 10^{-31} \text{ kg}$ | Electron mass |
2. Coulomb's Law
2.1 Force Between Two Point Charges
$$F = k \frac{|q_1 q_2|}{r^2}$$
| Variable | Description | SI Unit |
|---|---|---|
| $F$ | Magnitude of electrostatic force | $\text{N}$ (Newtons) |
| $k$ | Coulomb constant | $\text{N m}^2 \text{ C}^{-2}$ |
| $q_1, q_2$ | Point charges | $\text{C}$ (Coulombs) |
| $r$ | Distance between charges | $\text{m}$ (meters) |
Direction: Like charges repel; opposite charges attract.
2.2 Coulomb Constant in Terms of Permittivity
$$k = \frac{1}{4\pi\varepsilon_0}$$
| Variable | Description | SI Unit |
|---|---|---|
| $\varepsilon_0$ | Permittivity of free space | $8.85 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}$ (or $\text{F m}^{-1}$) |
3. Electric Field
3.1 Electric Field Definition
$$\vec{E} = \frac{\vec{F}}{q_{test}}$$
| Variable | Description | SI Unit |
|---|---|---|
| $\vec{E}$ | Electric field vector | $\text{N C}^{-1}$ or $\text{V m}^{-1}$ |
| $\vec{F}$ | Force on test charge | $\text{N}$ |
| $q_{test}$ | Small positive test charge | $\text{C}$ |
3.2 Electric Field Due to a Point Charge
$$E = k \frac{|q|}{r^2}$$
| Variable | Description | SI Unit |
|---|---|---|
| $E$ | Magnitude of electric field | $\text{N C}^{-1}$ |
| $q$ | Source point charge | $\text{C}$ |
| $r$ | Distance from charge to point of interest | $\text{m}$ |
Direction: Radially outward for positive $q$; radially inward for negative $q$.
3.3 Force on a Charge in an Electric Field
$$\vec{F} = q\vec{E}$$
| Variable | Description | SI Unit |
|---|---|---|
| $\vec{F}$ | Force on charge $q$ | $\text{N}$ |
| $q$ | Charge placed in field | $\text{C}$ |
| $\vec{E}$ | Electric field | $\text{N C}^{-1}$ |
4. Superposition Principle
4.1 Net Force from Multiple Charges
$$\vec{F}{net} = \sum{i} \vec{F}_i = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots$$
4.2 Net Electric Field from Multiple Charges
$$\vec{E}{net} = \sum{i} \vec{E}_i = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \dots$$
The total force (or field) is the vector sum of individual contributions.
4.3 Component Resolution
$$F_x = F \cos\theta, \quad F_y = F \sin\theta$$
$$E_x = E \cos\theta, \quad E_y = E \sin\theta$$
| Variable | Description |
|---|---|
| $\theta$ | Angle between force/field vector and positive x-axis |
5. Charged Particle Motion in Uniform Electric Field
5.1 Force and Acceleration
$$F = |q|E$$
$$a = \frac{F}{m} = \frac{|q|E}{m}$$
| Variable | Description | SI Unit |
|---|---|---|
| $a$ | Acceleration of particle | $\text{m s}^{-2}$ |
| $m$ | Mass of particle | $\text{kg}$ |
| $E$ | Uniform electric field strength | $\text{N C}^{-1}$ |
For an electron: $a = \dfrac{eE}{m_e}$
5.2 Kinematics (Projectile Motion Analogy)
Assuming initial velocity $v_0$ is perpendicular to $\vec{E}$ (horizontal entry into vertical field):
Horizontal motion (no acceleration):
$$x = v_0 t$$
$$v_x = v_0 = \text{constant}$$
Vertical motion (constant acceleration $a_y = \dfrac{qE}{m}$):
$$v_y = a_y t = \frac{qE}{m} t$$
$$y = \frac{1}{2} a_y t^2 = \frac{1}{2} \frac{qE}{m} t^2$$
Deflection angle on exit:
$$\tan\theta = \frac{v_y}{v_x}$$
$$\theta = \arctan\left(\frac{v_y}{v_x}\right)$$
| Variable | Description | SI Unit |
|---|---|---|
| $v_0$ | Initial horizontal velocity | $\text{m s}^{-1}$ |
| $v_x$ | Horizontal velocity component | $\text{m s}^{-1}$ |
| $v_y$ | Vertical velocity component | $\text{m s}^{-1}$ |
| $x$ | Horizontal displacement | $\text{m}$ |
| $y$ | Vertical displacement | $\text{m}$ |
| $t$ | Time of flight between plates | $\text{s}$ |
| $\theta$ | Angle of deflection | radians or degrees |
5.3 Time of Flight Between Plates
$$t = \frac{L}{v_0}$$
| Variable | Description | SI Unit |
|---|---|---|
| $L$ | Length of plates | $\text{m}$ |
6. Summary of Key Relationships
| Concept | Formula |
|---|---|
| Coulomb Force | $F = k \dfrac{q_1 q_2}{r^2}$ |
| Electric Field (point charge) | $E = k \dfrac{q}{r^2}$ |
| Electric Field (definition) | $E = \dfrac{F}{q}$ |
| Force in E-field | $F = qE$ |
| Electron acceleration | $a = \dfrac{eE}{m_e}$ |
Related Concepts
- Electrostatics
- Coulomb's Law
- Electric Field
- Electric Force
- Point Charge
- Superposition Principle
- Electron Motion in Electric Field