Summary & Formula Sheet

The Formula Marathon

This table contains every formula you need to master this chapter. Use this for your 'last-minute' revision before the exam!

Concept	Formula	Key Notes
Potential (Point Charge)	$V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$	Scalar quantity; use sign of $Q$
Potential (Dipole)	$V = \frac{1}{4\pi\epsilon_0} \frac{p \cos \theta}{r^2}$	$p = q \times 2a$; $V=0$ at equatorial plane
Potential Energy (System)	$U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$	Work done in assembling the system
Dipole Energy in Field	$U = -\vec{p} \cdot \vec{E} = -pE \cos \theta$	Min at $0^{\circ}$ (stable); Max at $180^{\circ}$ (unstable)
Field-Potential Relation	$E = -\frac{dV}{dr}$	Field points towards decreasing potential
Capacitance (General)	$C = \frac{Q}{V}$	SI Unit: Farad (F)
Parallel Plate Capacitor	$C = \frac{\epsilon_0 A}{d}$	With dielectric: $C_{med} = K C_{vac}$
Capacitors in Series	$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$	Charge $Q$ is same; Voltage $V$ divides
Capacitors in Parallel	$C_p = C_1 + C_2 + \dots$	Voltage $V$ is same; Charge $Q$ divides
Energy Stored	$U = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$	Stored in the electric field
Energy Density	$u = \frac{1}{2} \epsilon_0 E^2$	Energy per unit volume
Common Potential	$V = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}$	For connected capacitors
Energy Loss (Sharing)	$\Delta U = \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)}$	Dissipated as heat in wires

Core Concepts Recap

• Conservative Nature: The work done by an electrostatic field is independent of the path; it depends only on initial and final points.
• Equipotential Surfaces: Surfaces where $V$ is constant. No work is done moving a charge on these. $\vec{E}$ is always perpendicular to these surfaces.
• Conductor Properties: $E = 0$ inside; Potential is constant throughout; Excess charge stays on the outer surface; $\vec{E}$ at surface is $\sigma/\epsilon_0$.
• Dielectrics: These are insulators that polarize. Polarization reduces the internal field by factor $K$, thus increasing capacitance.
• Capacitance Factors: $C$ depends only on geometry ($A, d$) and the medium ($K$). It does NOT depend on $Q$ or $V$.

🎯 Exam Success Guide

For Board Exams (CBSE/State):

1. Standard Derivations: Always practice the derivation for Potential due to a Dipole, Capacitance of Parallel Plate (with and without dielectric), and Energy Stored in a capacitor. These are high-weightage 3 or 5-mark questions.
2. Units Matter: Don't forget that $1 \text{ Farad}$ is huge. Use $10^{-6}$ for $\mu F$ and $10^{-12}$ for $pF$ in numericals.
3. Graphing: Be prepared to draw $V$ vs. $r$ and $E$ vs. $r$ on the same axes for a conducting shell.

For JEE Main & NEET:

1. The Battery Scenario: Remember the 'Disconnected battery' ($Q$ constant) vs. 'Connected battery' ($V$ constant) logic for dielectric insertion. This is the most common competitive theme.
2. Energy Loss Shortcut: Don't derive the redistribution energy every time. Memorize the $\Delta U$ formula for sharing charges to save 2 minutes of calculation.
3. Complex Circuits: Master the 'Successive Reduction' method and look for symmetry (Wheatstone Bridge) in capacitor networks.
4. Variable Dielectric: For slabs of different thicknesses, use the series capacitor formula: $\frac{d}{C_{eff}} = \sum \frac{t_i}{C_i}$.