Electrostatics – Study Notes (PSU JOBS – GATE)
Purpose: A concise, self‑contained reference for the Electrostatics section of the GATE physics paper.
Format:
- Simple explanation → Key points → Worked example → Quick revision summary.
- Markdown headings, bullet lists, equations (in LaTeX), and example diagrams (described in text).
1. Simple Explanation
Electrostatics is the study of static electric charges – charges that are at rest. The fundamental ideas are:
| Concept | What it is |
|---|---|
| Electric charge | Quantised, conserved, can be positive or negative. |
| Coulomb’s law | Force between two point charges. |
| Electric field (E) | Force per unit positive test charge. |
| Electric potential (V) | Work done per unit charge to bring a test charge from infinity. |
| Gauss’s law | Relates electric flux through a closed surface to the enclosed charge. |
| Capacitance (C) | Ability of a system to store electric charge for a given potential difference. |
| Dielectrics | Insulating materials that reduce electric field inside them. |
2. Key Points
| Topic | Important Equations / Concepts | Notes |
|---|---|---|
| Coulomb’s Law | (\displaystyle \mathbf{F}_{12}=k,\frac{q_1q_2}{r^2}\hat r) (k=\frac{1}{4\pi\varepsilon_0}) |
(F) attractive if charges opposite, repulsive if same. |
| Electric Field | (\displaystyle \mathbf{E}(\mathbf{r}) = \frac{1}{q_0}\mathbf{F}) For a point charge: (\displaystyle \mathbf{E}=\frac{k,q}{r^2}\hat r) |
Vector: radially outward for (q>0). |
| Superposition | (\displaystyle \mathbf E_{\text{total}}=\sum_i \mathbf E_i) | Linear principle – works for fields and potentials. |
| Electric Potential | (\displaystyle V(\mathbf r)=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}) for point charge. (V(\mathbf r)= -\int_{\infty}^{\mathbf r}\mathbf E\cdot d\mathbf l) |
Scalars, easier to add. |
| Gauss’s Law | (\displaystyle \oint_{\mathcal S}\mathbf E\cdot d\mathbf A = \frac{Q_{\text{enc}}}{\varepsilon_0}) | Use symmetry (spherical, cylindrical, planar) to find (E). |
| Capacitance | (C=\dfrac{Q}{\Delta V}) | For parallel plates: (C=\varepsilon_0,\dfrac{A}{d}). |
| Dielectric | (\displaystyle \mathbf D = \varepsilon_0\mathbf E + \mathbf P) (\displaystyle \varepsilon = \kappa,\varepsilon_0) |
(\kappa) (dielectric constant) > 1. |
| Energy in Electric Field | (U=\frac{1}{2}\int \varepsilon_0 E^2, dV) | Equivalent: (U=\frac{1}{2}CV^2). |
| Boundary Conditions | At conductor surface: (E_{\parallel}=0), (E_{\perp}) discontinuity given by surface charge. |
Common Units
| Quantity | Symbol | SI Unit |
|---|---|---|
| Charge | (q) | coulomb (C) |
| Electric field | (\mathbf{E}) | newton per coulomb (N C(^{-1})) or V m(^{-1}) |
| Potential | (V) | volt (V) |
| Capacitance | (C) | farad (F) |
| Permittivity | (\varepsilon_0) | 8.854 × 10(^{-12}) F m(^{-1}) |
3. Worked Example
Problem:
A solid sphere of radius (R) carries a uniform volume charge density (\rho). Find:
- Electric field (\mathbf{E}) at a distance (r) from the centre for (r<R) and for (r>R).
- Electric potential (V(r)) with reference (V(\infty)=0).
- The force between two identical such spheres (radius (R), density (\rho)) when their centres are separated by (d>2R).
Solution
1. Electric field
Inside the sphere ((r<R))
Apply Gauss’s law to a spherical Gaussian surface of radius (r):
[ Q_{\text{enc}} = \rho , \frac{4}{3}\pi r^{3} ]
[ \oint \mathbf E \cdot d\mathbf A = E(4\pi r^{2}) = \frac{Q_{\text{enc}}}{\varepsilon_0}