Quick Revision Notes & Formula Sheet

Class 12 Physics | Chapter 2: Electrostatic Potential and Capacitance

(स्थिरवैद्युत विभव तथा धारिता)

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1. Electrostatic Potential & Potential Difference (विद्युत विभव एवं विभवांतर)

• Electric Potential ($V$): The amount of work done ($W$) in bringing a unit positive test charge from infinity to a specific point in an electric field. $$\mathbf{V = \frac{W}{q_0}}$$
• SI Unit: Volt ($\text{V}$) or Joules per Coulomb ($\text{J/C}$)
• Dimension: $[M^1 L^2 T^{-3} A^{-1}]$
• Nature: Scalar Quantity
• Electric Potential Difference ($\Delta V$): Work done per unit positive charge in moving it from point $A$ to point $B$. $$\mathbf{V_B - V_A = \frac{W_{AB}}{q_0}}$$

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2. Electric Potential Due to Various Charge Configurations

Configuration	Formula	Key Notes
Point Charge ($q$)	$V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$	$V \propto \frac{1}{r}$
System of Charges	$V = \frac{1}{4\pi\varepsilon_0} \sum \frac{q_i}{r_i}$	Superposition Principle (Add with signs)
Electric Dipole (Axial Point / अक्षीय स्थिति)	$V_{axial} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2}$	Valid for short dipole ($r \gg a$)
Electric Dipole (Equatorial Point / निरक्षीय स्थिति)	$V_{eq} = 0$	Potential at any equatorial point is zero
Electric Dipole (General Point $(r, \theta)$)	$V = \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2}$	$\theta$ is angle with dipole axis

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3. Relation Between Electric Field ($E$) and Potential ($V$)

• Potential Gradient (विभव प्रवणता): Rate of change of potential with distance. $$\mathbf{E = -\frac{dV}{dr}}$$
• Important Point: The negative sign indicates that Electric Field ($\vec{E}$) points in the direction of maximum decrease of electric potential.
• For Uniform Electric Field: $E = \frac{V}{d}$ or $V = E \cdot d$

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4. Equipotential Surface (समविभव पृष्ठ)

A surface which has the same electric potential at every point.

• Key Properties:
  1. Work Done = 0: No work is done in moving a test charge between two points on an equipotential surface ($W = q_0 \Delta V = 0$).
  2. Perpendicular Field: Electric field lines are always perpendicular ($\perp$) to the equipotential surface.
  3. No Intersection: Two equipotential surfaces can never intersect each other.
  4. Spacing: Surfaces are closer together in regions of strong electric fields and farther apart in weak fields.

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5. Electrostatic Potential Energy ($U$) (स्थिरवैद्युत स्थितिज ऊर्जा)

• Two Point Charges System ($q_1, q_2$): $$\mathbf{U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}}$$
• Three Point Charges System ($q_1, q_2, q_3$): $$\mathbf{U = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_2 q_3}{r_{23}} + \frac{q_1 q_3}{r_{13}} \right)}$$
• Electric Dipole in Uniform Electric Field ($\vec{E}$):
  • Potential Energy: $U = -\vec{p} \cdot \vec{E} = -p E \cos\theta$
  • Work Done in Rotating Dipole from $\theta_1$ to $\theta_2$: $$\mathbf{W = pE(\cos\theta_1 - \cos\theta_2)}$$
  • Stable Equilibrium: $\theta = 0^\circ \implies U_{min} = -pE$
  • Unstable Equilibrium: $\theta = 180^\circ \implies U_{max} = +pE$
  • Zero Energy State: $\theta = 90^\circ \implies U = 0$

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6. Conductors & Dielectrics (चालक एवं परावैद्युत)

• Inside a Conductor: Electric Field $E = 0$, Charge $Q = 0$, Potential $V = \text{constant}$.
• Dielectric Constant ($K$ or $\varepsilon_r$): $$\mathbf{K = \frac{\varepsilon}{\varepsilon_0} = \frac{C_{medium}}{C_{air}} = \frac{E_0}{E_{net}}}$$
• Polarization ($\vec{P}$): Induced dipole moment per unit volume. $$\vec{P} = \chi_e \vec{E}$$ (where $\chi_e$ = Electric Susceptibility)

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7. Electrical Capacitance (विद्युत धारिता)

• Definition: Ability of a conductor to store charge. $$\mathbf{Q = CV \implies C = \frac{Q}{V}}$$
• SI Unit: Farad ($\text{F}$) or Coulomb per Volt ($\text{C/V}$)
• Dimension: $[M^{-1} L^{-2} T^4 A^2]$
• Capacitance of Isolated Spherical Conductor: $$\mathbf{C = 4\pi\varepsilon_0 R}$$

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8. Parallel Plate Capacitor (समांतर प्लेट संधारित्र)

• Air/Vacuum between plates: $$\mathbf{C_0 = \frac{\varepsilon_0 A}{d}}$$
• With Medium of Dielectric Constant $K$ fully filled: $$\mathbf{C = \frac{K \varepsilon_0 A}{d} = K \cdot C_0}$$
• With Dielectric Slab of Thickness $t$ ($t < d$): $$\mathbf{C = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}}$$
• With Conducting Slab of Thickness $t$ ($t < d$): $$\mathbf{C = \frac{\varepsilon_0 A}{d - t}}$$

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9. Combination of Capacitors (संधारित्रों का संयोजन)

A. Series Combination (श्रेणीक्रम संयोजन)

• Charge ($Q$): Same across all capacitors.
• Potential ($V$): Divides ($V = V_1 + V_2 + V_3$).
• Equivalent Capacitance ($C_{eq}$): $$\mathbf{\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}}$$
• For two capacitors: $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$

B. Parallel Combination (समांतरक्रम/पार्र्श्वक्रम संयोजन)

• Potential ($V$): Same across all capacitors.
• Charge ($Q$): Divides ($Q = Q_1 + Q_2 + Q_3$).
• Equivalent Capacitance ($C_{eq}$): $$\mathbf{C_{eq} = C_1 + C_2 + C_3}$$

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10. Energy Stored in a Capacitor (संधारित्र में संचित ऊर्जा)

• Formulas: $$\mathbf{U = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{Q^2}{2C}}$$
• Energy Density ($u$) (Energy per unit volume in Electric Field): $$\mathbf{u = \frac{1}{2} \varepsilon_0 E^2}$$

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11. Redistribution of Charges & Energy Loss (आवेशों का पुनर्वितरण एवं ऊर्जा हानि)

When two charged conductors of capacitances $C_1, C_2$ at potentials $V_1, V_2$ are connected:

• Common Potential (उभयनिष्ठ विभव): $$\mathbf{V = \frac{Q_1 + Q_2}{C_1 + C_2} = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}}$$
• Loss of Energy during sharing of charges ($\Delta U$): $$\mathbf{\Delta U = \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)}}$$ (This energy is lost in the form of heat and spark in connecting wires).

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12. Effect of Dielectric on Various Parameters

Parameter	Battery Remains Connected	Battery Disconnected
Charge ($Q$)	Increases ($Q = K Q_0$)	Constant ($Q = Q_0$)
Capacitance ($C$)	Increases ($C = K C_0$)	Increases ($C = K C_0$)
Potential ($V$)	Constant ($V = V_0$)	Decreases ($V = V_0 / K$)
Electric Field ($E$)	Constant ($E = E_0$)	Decreases ($E = E_0 / K$)
Energy Stored ($U$)	Increases ($U = K U_0$)	Decreases ($U = U_0 / K$)

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💡 Quick MP Board Exam Tips

• Derivation Check: Always practice derivations for:
  1. Potential due to an electric dipole in axial position.
  2. Capacitance of a parallel plate capacitor (with and without a dielectric slab).
  3. Energy stored in a parallel plate capacitor.
• Vector vs Scalar: Remember that Electric Field ($\vec{E}$) is a vector quantity, while Electric Potential ($V$) is a scalar quantity. Include proper $+/-$ algebraic signs for potential calculations.
• Constants Values:
  • $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \text{ N m}^2/\text{C}^2$
  • $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/\text{N m}^2$ or $\text{F/m}$