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CLASS 12 • PHYSICS
Potential & Capacitance
1. Electric Potential (V)
Point Charge
$V = \frac{kQ}{r}$
Due to Dipole
$V = \frac{kp \cos\theta}{r^2}$
Relation with Field
$E = -\frac{dV}{dr}$
Potential decreases in direction of E
Work Done
$W_{ext} = q(V_B - V_A) = \Delta U$
2. Potential Energy (U)
System of 2 Charges
$U = \frac{k q_1 q_2}{r}$
System of 3 Charges
$U = U_{12} + U_{23} + U_{13}$
Single Charge in Ext. Field
$U = qV(r)$
3. Capacitance
Basic Definition
$C = \frac{Q}{V}$
Parallel Plate Capacitor
$C = \frac{\epsilon_0 A}{d}$
Spherical Conductor
$C = 4\pi\epsilon_0 R$
With Dielectric Slab (t)
$C = \frac{\epsilon_0 A}{d - t(1 - 1/K)}$
4. Combinations
Series Combination
$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2}$
Charge Q is same, V divides
Parallel Combination
$C_p = C_1 + C_2$
Voltage V is same, Q divides
5. Energy Stored
Standard Formulas
$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$
Energy Density
$u = \frac{1}{2}\epsilon_0 E^2$
Common Potential
$V = \frac{C_1V_1 + C_2V_2}{C_1 + C_2}$
Energy Loss (Redistribution)
$\Delta U = \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)}$
6. Effect of Dielectric (K)
| Quantity | Battery Disconnected | Battery Connected |
|---|---|---|
| Capacity (C) | Increases ($KC_0$) | Increases ($KC_0$) |
| Charge (Q) | Constant ($Q_0$) | Increases ($KQ_0$) |
| Voltage (V) | Decreases ($V_0/K$) | Constant ($V_0$) |
| Energy (U) | Decreases ($U_0/K$) | Increases ($KU_0$) |
7. Common Exam Traps
- ⚠️ Resistor Confusion: Capacitor series formula is opposite to Resistors. Don't add them directly! ($1/C_s$ not $C_1+C_2$).
- ⚠️ Work Done Sign: Work done BY field = $-\Delta U$. Work done BY external agent = $+\Delta U$.
- ⚠️ n drops coalescing: If 'n' drops combine, New Potential $V' = n^{2/3}V$. New Capacity $C' = n^{1/3}C$.
8. Golden Theory Rules
- Equipotential Surface: Work done to move charge on it is ZERO. Electric field is always perpendicular to it.
- Conductor Surface: Entire surface of a conductor is an equipotential surface.
- Sharp Points: Charge density is higher at sharp points ($\sigma \propto 1/R$).
