Electrostatics deals with electric charges at rest. It forms the foundation for understanding electricity and magnetism.
Overview
graph TD
A[Electrostatics] --> B[Electric Charges]
A --> C[Electric Field]
A --> D[Electric Potential]
A --> E[Capacitance]
B --> B1[Coulomb's Law]
B --> B2[Superposition]
C --> C1[Field Lines]
C --> C2[Gauss's Law]
D --> D1[Potential Energy]
D --> D2[Equipotential Surfaces]
E --> E1[Parallel Plate]
E --> E2[Combinations]Electric Charges
Fundamental Properties
- Conservation of Charge: Total charge in an isolated system remains constant
- Quantization: Charge exists in discrete packets: $q = ne$ where $e = 1.6 \times 10^{-19}$ C
- Additive: Total charge is algebraic sum of individual charges
Types of Charging
- Friction: Transfer of electrons between materials
- Conduction: Direct contact transfer
- Induction: Redistribution of charges without contact
Coulomb’s Law
The force between two point charges $q_1$ and $q_2$ separated by distance $r$:
$$\boxed{\vec{F} = k\frac{q_1 q_2}{r^2}\hat{r} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}\hat{r}}$$where:
- $k = 9 \times 10^9$ N·m²/C²
- $\varepsilon_0 = 8.85 \times 10^{-12}$ C²/N·m²
Vector Form
$$\vec{F}_{12} = k\frac{q_1 q_2}{r_{12}^2}\hat{r}_{12}$$Superposition Principle
For multiple charges, the net force on a charge is the vector sum:
$$\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + ...$$Electric Field
The electric field at a point is the force per unit positive test charge:
$$\vec{E} = \frac{\vec{F}}{q_0} = k\frac{Q}{r^2}\hat{r}$$Electric Field Due to Various Charge Distributions
| Configuration | Electric Field |
|---|---|
| Point charge | $E = \frac{kQ}{r^2}$ |
| Infinite line charge | $E = \frac{\lambda}{2\pi\varepsilon_0 r}$ |
| Infinite plane sheet | $E = \frac{\sigma}{2\varepsilon_0}$ |
| Uniformly charged sphere (outside) | $E = \frac{kQ}{r^2}$ |
| Uniformly charged sphere (inside) | $E = \frac{kQr}{R^3}$ |
Electric Field Lines
Properties:
- Start from positive charges, end at negative charges
- Never cross each other
- Perpendicular to equipotential surfaces
- Density indicates field strength
Interactive Demo: Electric Field Lines
Click the buttons to change the charges and see how electric field lines are affected:
Electric Dipole
An electric dipole consists of two equal and opposite charges $±q$ separated by distance $2a$.
Dipole Moment
$$\vec{p} = q \cdot 2a \cdot \hat{p}$$Direction: From negative to positive charge
Electric Field Due to Dipole
On axial line (at distance $r$ from center):
$$E_{axial} = \frac{2kp}{r^3} \quad (r >> a)$$On equatorial line:
$$E_{eq} = \frac{kp}{r^3} \quad (r >> a)$$Torque on Dipole in Uniform Field
$$\boxed{\vec{\tau} = \vec{p} \times \vec{E}}$$ $$\tau = pE\sin\theta$$Gauss’s Law
The electric flux through a closed surface equals the enclosed charge divided by $\varepsilon_0$:
$$\boxed{\oint \vec{E} \cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_0}}$$Applications
graph LR
A[Gauss's Law] --> B[Infinite Wire]
A --> C[Infinite Plane]
A --> D[Spherical Shell]
A --> E[Solid Sphere]Infinite Line Charge ($\lambda$ = linear charge density):
$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$Infinite Plane Sheet ($\sigma$ = surface charge density):
$$E = \frac{\sigma}{2\varepsilon_0}$$Uniformly Charged Spherical Shell:
- Outside: $E = \frac{kQ}{r^2}$
- Inside: $E = 0$
Electric Potential
Electric potential at a point is the work done per unit charge to bring a test charge from infinity:
$$V = \frac{W_{\infty \to P}}{q_0} = \frac{kQ}{r}$$Relation with Electric Field
$$\vec{E} = -\nabla V = -\frac{dV}{dr}\hat{r}$$For uniform field: $V = -\int \vec{E} \cdot d\vec{r}$
Potential Due to Various Systems
| Configuration | Potential |
|---|---|
| Point charge | $V = \frac{kQ}{r}$ |
| Electric dipole (axial) | $V = \frac{kp\cos\theta}{r^2}$ |
| System of charges | $V = k\sum\frac{q_i}{r_i}$ |
| Uniformly charged sphere (surface) | $V = \frac{kQ}{R}$ |
Equipotential Surfaces
- Surfaces with constant potential
- $\vec{E}$ is perpendicular to equipotential surfaces
- Work done along equipotential surface = 0
Electric Potential Energy
Two Point Charges
$$U = k\frac{q_1 q_2}{r}$$System of Charges
$$U = k\sum_{iCapacitors
A capacitor stores electric charge and energy.
Capacitance
$$C = \frac{Q}{V}$$Unit: Farad (F)
Parallel Plate Capacitor
$$\boxed{C = \frac{\varepsilon_0 A}{d}}$$With dielectric: $C = \frac{K\varepsilon_0 A}{d}$
Combinations
Series:
$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$$Parallel:
$$C_{eq} = C_1 + C_2 + ...$$Energy Stored
$$\boxed{U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}}$$Energy density: $u = \frac{1}{2}\varepsilon_0 E^2$
Practice Problems
Two charges +4μC and -4μC are placed 20 cm apart. Find the electric field at the midpoint.
A charge of 5μC is placed in a uniform field of 2000 N/C. Find the force on it.
Three capacitors of 2μF, 3μF, and 6μF are connected in series. Find equivalent capacitance.
Find the potential at a point 30 cm from a charge of 4nC.