FAD1022 L1-L3 — Electrostatics — Formula Sheet
Comprehensive formula sheet extracted from FAD1022 L1-L3 — Electrostatics.
1. Fundamental Constants & Relationships
| Symbol | Value / Expression | Units | Description |
|---|---|---|---|
| $k$ | $9.0 \times 10^{9}$ | $\text{N m}^{2}\text{ C}^{-2}$ | Coulomb's constant |
| $k$ | $\dfrac{1}{4\pi\varepsilon_{0}}$ | $\text{N m}^{2}\text{ C}^{-2}$ | Definition of Coulomb's constant |
| $\varepsilon_{0}$ | — | $\text{C}^{2}\text{ N}^{-1}\text{ m}^{-2}$ (or $\text{F m}^{-1}$) | Permittivity of free space |
| $e$ | — | $\text{C}$ | Elementary charge (magnitude of electron/proton charge) |
Charge Quantization (L1 concept): Electric charge is quantized in multiples of the elementary charge $e$. $$Q = ne \quad (n \in \mathbb{Z})$$
2. Coulomb's Law
The electrostatic force between two point charges:
$$F = \frac{Qq}{4\pi\varepsilon_{0} r^{2}}$$
Using Coulomb's constant $k$:
$$\boxed{F = \frac{kQq}{r^{2}}}$$
| Variable | Meaning | Units |
|---|---|---|
| $F$ | Magnitude of the electrostatic force | $\text{N}$ |
| $Q, q$ | Magnitudes of the point charges | $\text{C}$ |
| $r$ | Separation distance between the charges | $\text{m}$ |
| $k$ | Coulomb's constant | $\text{N m}^{2}\text{ C}^{-2}$ |
| $\varepsilon_{0}$ | Permittivity of free space | $\text{C}^{2}\text{ N}^{-1}\text{ m}^{-2}$ |
- Direction: Like charges repel; opposite charges attract.
- The force acts along the line joining the two charges.
3. Electric Field Strength
3.1 Definition
The electric field strength at a point is the electric force per unit positive test charge:
$$\boxed{E = \frac{F}{q_{0}}}$$
| Variable | Meaning | Units |
|---|---|---|
| $E$ | Magnitude of electric field strength | $\text{N C}^{-1}$ or $\text{V m}^{-1}$ |
| $F$ | Magnitude of the electric force on the test charge | $\text{N}$ |
| $q_{0}$ | Magnitude of the test charge | $\text{C}$ |
- Vector quantity: Direction depends on the sign of the source charge.
- Units: $\text{N C}^{-1}$ (newtons per coulomb) or $\text{V m}^{-1}$ (volts per metre).
3.2 Electric Field due to a Point Charge
From Coulomb's law, the electric field at distance $r$ from a point charge $Q$:
$$E = \frac{F}{q} = \frac{Q}{4\pi\varepsilon_{0} r^{2}}$$
In terms of $k$:
$$\boxed{E = \frac{kQ}{r^{2}}}$$
| Variable | Meaning | Units |
|---|---|---|
| $E$ | Electric field strength at distance $r$ | $\text{N C}^{-1}$ |
| $Q$ | Source point charge | $\text{C}$ |
| $r$ | Distance from the point charge | $\text{m}$ |
| $k$ | Coulomb's constant | $\text{N m}^{2}\text{ C}^{-2}$ |
- Positive charge ($+Q$): $\vec{E}$ points radially outward.
- Negative charge ($-Q$): $\vec{E}$ points radially inward.
3.3 Principle of Superposition (Vector Addition)
For a system of multiple point charges, the resultant electric field at a point is the vector sum of the fields due to each individual charge:
$$\vec{E}{\text{net}} = \sum{i} \vec{E}{i} = \vec{E}{1} + \vec{E}{2} + \vec{E}{3} + \dots$$
- Resolve each $\vec{E}_{i}$ into components ($x$, $y$) if necessary.
- Add components separately: $E_{\text{net},x} = \sum E_{i,x}$ and $E_{\text{net},y} = \sum E_{i,y}$.
- Magnitude: $E_{\text{net}} = \sqrt{E_{\text{net},x}^{2} + E_{\text{net},y}^{2}}$.
- Direction: $\theta = \tan^{-1}!\left(\dfrac{E_{\text{net},y}}{E_{\text{net},x}}\right)$.
3.4 Neutral Point
A neutral point is a location in space where the resultant electric field is zero:
$$\vec{E}_{\text{net}} = 0$$
- Occurs between two like charges (e.g., $+q$ and $+q$).
- The point lies on the line joining the charges, closer to the smaller charge.
4. Electric Force on a Charge in an Electric Field
4.1 General Relationship
$$\boxed{\vec{F} = q\vec{E}}$$
| Variable | Meaning | Units |
|---|---|---|
| $\vec{F}$ | Electric force on the charge | $\text{N}$ |
| $q$ | Charge of the particle | $\text{C}$ |
| $\vec{E}$ | Electric field vector | $\text{N C}^{-1}$ |
4.2 Direction Rules
| Charge | Direction of $\vec{F}$ |
|---|---|
| Positive ($q > 0$) | Same direction as $\vec{E}$ |
| Negative ($q < 0$) | Opposite direction to $\vec{E}$ |
4.3 Specific Particles
| Particle | Charge | Force Magnitude | Direction |
|---|---|---|---|
| Electron | $-e$ | $F = Ee$ | Opposite to $\vec{E}$ |
| Proton | $+e$ | $F = Ee$ | Same as $\vec{E}$ |
| Alpha particle ($^{4}_{2}\text{He}$) | $+2e$ | $F = E(2e)$ | Same as $\vec{E}$ |
5. Charge Motion in a Uniform Electric Field
A uniform electric field exists between two broad, parallel, oppositely charged plates (central region, neglecting edge effects).
5.1 Dynamic Equilibrium (Horizontal Motion)
A charge moves horizontally at constant velocity when the electrostatic force balances its weight:
$$\boxed{F_{E} = W \quad \Rightarrow \quad qE = mg}$$
| Variable | Meaning | Units |
|---|---|---|
| $q$ | Charge of the particle | $\text{C}$ |
| $E$ | Electric field strength | $\text{N C}^{-1}$ |
| $m$ | Mass of the particle | $\text{kg}$ |
| $g$ | Acceleration due to gravity | $\text{m s}^{-2}$ |
5.2 Charge Moving Perpendicular to $\vec{E}$ (Parabolic Trajectory)
An electron (charge $q_{0}$, mass $m$) enters with initial velocity $v_{0}$ perpendicular to a uniform field $\vec{E}$.
Vertical acceleration: $$\boxed{a_{y} = \frac{q_{0} E}{m}}$$
Horizontal velocity (constant): $$\boxed{v_{x} = v_{0}}$$
Time to traverse plates (plate length $x$): $$\boxed{t = \frac{x}{v_{0}}}$$
Vertical velocity on exit: $$\boxed{v_{y} = \frac{q_{0} E x}{m v_{0}}}$$
Resultant velocity on exit: $$\boxed{v = \sqrt{v_{x}^{2} + v_{y}^{2}}}$$
Direction of velocity (angle to horizontal): $$\boxed{\theta = \tan^{-1}!\left(\frac{v_{y}}{v_{x}}\right)}$$
Vertical displacement (taking downward as negative): $$\boxed{s_{y} = -\frac{1}{2} a_{y} t^{2}}$$
| Variable | Meaning | Units |
|---|---|---|
| $a_{y}$ | Vertical acceleration | $\text{m s}^{-2}$ |
| $v_{0}$ | Initial horizontal velocity | $\text{m s}^{-1}$ |
| $v_{x}$ | Horizontal velocity component | $\text{m s}^{-1}$ |
| $v_{y}$ | Vertical velocity component | $\text{m s}^{-1}$ |
| $v$ | Resultant speed | $\text{m s}^{-1}$ |
| $\theta$ | Deflection angle | degrees or radians |
| $s_{y}$ | Vertical displacement | $\text{m}$ |
| $x$ | Horizontal distance / plate length | $\text{m}$ |
| $t$ | Time of flight between plates | $\text{s}$ |
Trajectory: Parabolic curve (similar to projectile motion under gravity).
5.3 Charge Moving Parallel to $\vec{E}$ (Linear Acceleration)
General acceleration: $$\boxed{a = \frac{qE}{m}}$$
Case 4(a) — Positive charge:
- Moves towards the negative plate.
- Acceleration $a = \dfrac{qE}{m}$ is in the same direction as $\vec{E}$.
- Speed increases.
Case 4(b) — Negative charge:
- Moves towards the positive plate.
- Acceleration $a = \dfrac{qE}{m}$ is in the opposite direction to $\vec{E}$.
- Speed increases (magnitude of velocity increases).
5.4 Kinematic Equations (Applied in L3)
The lecture applies standard SUVAT kinematic equations alongside $F = qE$:
$$v = u + at$$ $$s = ut + \frac{1}{2}at^{2}$$ $$v^{2} = u^{2} + 2as$$
| Variable | Meaning | Units |
|---|---|---|
| $u$ | Initial velocity | $\text{m s}^{-1}$ |
| $v$ | Final velocity | $\text{m s}^{-1}$ |
| $a$ | Acceleration ($= qE/m$) | $\text{m s}^{-2}$ |
| $s$ | Displacement | $\text{m}$ |
| $t$ | Time | $\text{s}$ |
6. Electric Field Line Properties (Conceptual Relationships)
| Property | Relationship |
|---|---|
| Direction | $\vec{E}$ is tangent to the field line at every point. |
| Magnitude | $ |
| Source/Sink | Lines start on positive charges and end on negative charges. |
| Proportionality | Number of lines $\propto$ magnitude of charge. |
| Uniqueness | Field lines never cross (electric field has a unique value at each point). |
7. Quick Reference: Units & Conversions
| Quantity | Symbol | SI Unit | Equivalent |
|---|---|---|---|
| Charge | $Q, q, q_{0}$ | Coulomb ($\text{C}$) | — |
| Force | $F$ | Newton ($\text{N}$) | $\text{kg m s}^{-2}$ |
| Electric Field | $E$ | $\text{N C}^{-1}$ | $\text{V m}^{-1}$ |
| Distance | $r, x, s$ | metre ($\text{m}$) | $1\ \text{cm} = 10^{-2}\ \text{m}$; $1\ \text{mm} = 10^{-3}\ \text{m}$ |
| Velocity | $v, u$ | $\text{m s}^{-1}$ | — |
| Acceleration | $a$ | $\text{m s}^{-2}$ | — |
| Time | $t$ | second ($\text{s}$) | — |
| Mass | $m$ | kilogram ($\text{kg}$) | — |
| Angle | $\theta$ | degrees or radians | — |
8. Summary of Key Formulas
| Topic | Formula |
|---|---|
| Coulomb's Law | $F = \dfrac{kQq}{r^{2}}$ |
| Electric Field (definition) | $E = \dfrac{F}{q_{0}}$ |
| Electric Field (point charge) | $E = \dfrac{kQ}{r^{2}}$ |
| Force on charge in field | $F = qE$ |
| Equilibrium ($qE = mg$) | $qE = mg$ |
| Perpendicular acceleration | $a_{y} = \dfrac{q_{0}E}{m}$ |
| Parallel acceleration | $a = \dfrac{qE}{m}$ |
| Resultant velocity | $v = \sqrt{v_{x}^{2} + v_{y}^{2}}$ |
| Deflection angle | $\theta = \tan^{-1}!\left(\dfrac{v_{y}}{v_{x}}\right)$ |
Extracted from: FAD1022 L1-L3 — Electrostatics