Electrostatic Potential & Capacitance
SRMJEEE – Physics (Chapter 9)
Goal of these notes – Understand the concepts of electrostatic potential, capacitance, and their inter‑relationships.
How to use – Read the simple explanations first, then study the key points. Work through the example, and finally review the “Quick Revision” section before the exam.
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Electrostatic potential (V) | Energy per unit charge at a point in an electric field. | Allows us to describe the work done on a charge moving in a field. |
| Potential difference (ΔV) | Difference in electric potential between two points. | Drives current in circuits; defines the voltage across a capacitor. |
| Capacitance (C) | Ability of a body to store electric charge per unit potential difference. (\displaystyle C=\frac{Q}{V}) | Quantifies how much charge a capacitor can hold for a given voltage; key for energy storage. |
| Capacitor | Two conductors separated by an insulator (dielectric). | Stores energy, filters signals, smooths power supplies, etc. |
| Dielectric | Insulating material that reduces electric field inside a capacitor and increases its capacitance. | Increases energy storage without increasing leakage. |
| Energy stored in a capacitor | (\displaystyle U=\frac{1}{2}CV^2=\frac{1}{2}QV=\frac{Q^2}{2C}) | Important for power‑supply design and pulse circuits. |
2. Key Points
2.1 Electrostatic Potential
| Symbol | Meaning | Unit |
|---|---|---|
| (V) | Potential at point (P) | Volt (V) = J/C |
| (ΔV = V_B - V_A) | Potential difference between (B) and (A) | Volt |
| (W = qΔV) | Work done on charge (q) (positive) | Joule |
- Potential is a scalar; no direction.
- Potential is defined up to a constant: only differences matter.
2.2 Capacitance
- (C = \dfrac{Q}{V}) (1)
- Units: Farad (F) = C/V
- Typical values: 1 pF = 10⁻¹² F, 1 µF = 10⁻⁶ F, 1 F = 10⁶ pF
2.2.1 Parallel‑Plate Capacitor
(C = \epsilon_0 \epsilon_r \dfrac{A}{d}) (2)
- (A) = area of plates
- (d) = separation
- (\epsilon_0 = 8.85\times10^{-12},{\rm F/m})
- (\epsilon_r) = relative permittivity of dielectric
- (ε_r = 1) for vacuum (or air).
- (ε_r > 1) for dielectrics.
2.2.2 Other Geometries
| Geometry | Formula | Notes |
|---|---|---|
| Cylindrical (coaxial) | (C = \dfrac{2\pi\epsilon_0\epsilon_r L}{\ln(b/a)}) | (a), (b) = radii, (L) = length |
| Spherical | (C = 4\pi\epsilon_0\epsilon_r \dfrac{ab}{b-a}) | For nested spheres of radii (a<b) |
2.3 Dielectrics
| Effect | Formula | Explanation |
|---|---|---|
| Field reduction | (E' = \dfrac{E}{\epsilon_r}) | Polarization reduces effective field. |
| Capacitance increase | (C' = \epsilon_r C_0) | Where (C_0) is capacitance without dielectric. |
| Stored energy | (U' = \frac{1}{2}\epsilon_0 \epsilon_r E^2 V_{\text{dielectric}}) | More energy can be stored. |
2.4 Series & Parallel Combinations
| Connection | Effective Capacitance | Derivation |
|---|---|---|
| Series | (\displaystyle \frac{1}{C_{\text{eq}}} = \sum_i \frac{1}{C_i}) | Charges are equal; potentials add. |
| Parallel | (\displaystyle C_{\text{eq}} = \sum_i C_i) | Potentials are equal; charges add. |
2.5 Energy Stored
| Formulas | Notes |
|---|---|
| (U = \frac{1}{2}CV^2) | When voltage is fixed. |
| (U = \frac{1}{2}QV) | When charge is fixed. |
| (U = \frac{Q^2}{2C}) | Useful when charge is known. |
3. Worked Example
Problem
A parallel‑plate capacitor has plate area (A = 0.02;\text{m}^2) and plate separation (