Electrostatics – Comprehensive Study Notes
(IIT‑JEE / Competitive Exam Focus)
Objective – Provide a rigorous, exam‑oriented exposition of the core concepts of electrostatics, complete with derivations, key formulae, subtle points, and a challenging practice problem.
1. Fundamental Concepts
| Topic | Key Idea | Symbol | Units (SI) |
|---|---|---|---|
| Charge | Quantised, conservative | (q) | coulomb (C) |
| Electric Field | Force per unit charge | (\mathbf{E}) | V m⁻¹ |
| Potential | Work per unit charge | (V) | volt (V) |
| Capacitance | Ability to store charge per potential difference | (C) | farad (F) |
| Dielectric | Insulating medium that polarises | (ε), (ε_r) | farad m⁻¹ |
2. Coulomb’s Law (Vector Form)
[ \boxed{\mathbf{F}{12}=k_e,\frac{q_1q_2}{r^2};\hat{\mathbf{r}}{12}} ]
- (k_e=\frac{1}{4\pi\varepsilon_0})
- (\hat{\mathbf{r}}_{12}) points from (q_1) to (q_2).
- Derivation (Brief) – From empirical measurement of force between two point charges; linear superposition confirms that the field is (\mathbf{E}=\frac{k_e q}{r^2}\hat{\mathbf{r}}).
Tricky Points
- Direction: field points away from positive charge, towards negative.
- Sign of (k_e): always positive; sign is carried by the charges.
- Units consistency: (k_e) in N m² C⁻² → (E) in N C⁻¹ = V m⁻¹.
3. Electric Field of Continuous Charge Distributions
-
Linear charge density (\lambda)
[ \mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\lambda,\mathrm{d}\ell}{|\mathbf{r}-\mathbf{r}'|^2},\hat{\mathbf{R}} ] -
Surface charge density (\sigma)
[ \mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\sigma,\mathrm{d}a}{|\mathbf{r}-\mathbf{r}'|^2},\hat{\mathbf{R}} ] -
Volume charge density (\rho)
[ \mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho,\mathrm{d}v}{|\mathbf{r}-\mathbf{r}'|^2},\hat{\mathbf{R}} ]
Symmetry shortcut – Use Gauss’s law to avoid integrals when symmetry is high (spherical, cylindrical, planar).
4. Gauss’s Law (Integral & Differential Forms)
Integral Form
[ \boxed{\oint_{\mathcal{S}}\mathbf{E}\cdot\mathrm{d}\mathbf{A}= \frac{Q_{\text{enc}}}{\varepsilon_0}} ]
- (Q_{\text{enc}}): total free charge inside the closed surface (\mathcal{S}).
- Application: Compute (\mathbf{E}) for symmetric charge distributions.
Differential Form
[ \boxed{\nabla!\cdot!\mathbf{E}=\frac{\rho}{\varepsilon_0}} ]
- Useful for deriving Poisson’s and Laplace’s equations.
Tricky Points
- Gauss’s law counts only free charge – bound charges in dielectrics are handled via (\mathbf{D}).
- Choice of Gaussian surface: must exploit symmetry; otherwise it becomes a computational nightmare.
- Units: (\mathbf{E}) in N C⁻¹, (\mathrm{d}\mathbf{A}) in m² → left side in N C⁻¹·m² = N m² C⁻¹, matches right side.
5. Electric Potential
Definition
[ V(\mathbf{r}) \equiv -\int_{\infty}^{\mathbf{r}} \mathbf{E}!\cdot!\mathrm{d}\mathbf{l} ]
- Reference at infinity where (V=0).
- For point charge (q): [ \boxed{V(r)=\frac{1}{4\pi\varepsilon_0},\frac{q}{r}} ]
Relation to Field
[ \boxed{\mathbf{E}=-\nabla V} \