Electrostatic Potential and Capacitance Notes: Class 12 Physics Chapter 2
Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance provides concise notes, important formulas, step-by-step derivations, and textbook solutions covering electric potential, equipotential surfaces, dielectrics, capacitance, and energy storage.
Overview of Topics & Subtopics (Click to Jump)
1. Electrostatic Potential & Potential Energy
Short Note:
• Electrostatic Potential (V): The amount of work done by an external force in bringing a unit positive charge from infinity to a specific point without acceleration: V = W / q₀. SI unit is Volt (V) or Joules per Coulomb (J/C).
• Potential Difference (ΔV): Difference in potential between two points B and A: V_B - V_A = W_AB / q₀.
• Electrostatic Potential Energy (U): Work done to assemble a system of charges. For two point charges q₁ and q₂ separated by distance r: U = (1 / 4πε₀) × (q₁ q₂ / r).
• Relation between Field and Potential: Electric field is the negative gradient of potential: E = -dV / dr.
2. Potential due to a Point Charge & Dipole
Short Note:
• Potential due to Point Charge: V = (1 / 4πε₀) × (q / r). Note that V ∝ 1/r (whereas E ∝ 1/r²).
• Potential due to an Electric Dipole: At a distance r making angle θ with dipole moment p: V = (1 / 4πε₀) × (p cos θ / r²).
- On Axial Line (θ = 0°): V_axial = (1 / 4πε₀) × (p / r²)
- On Equatorial Line (θ = 90°): V_eq = 0
• Potential due to System of Charges: Superposition principle applies directly as algebraic sum: V = V₁ + V₂ + V₃ + ...
3. Equipotential Surfaces & Electrostatics of Conductors
Short Note:
• Equipotential Surface: Any surface that has a constant electrostatic potential at every point on it.
- No work is done in moving a test charge over an equipotential surface (W = q ΔV = 0).
- Electric field lines are always perpendicular to equipotential surfaces.
- Equipotential surfaces are closer in regions of strong electric field and farther apart in weak fields.
• Electrostatics of Conductors:
- Electric field inside a conductor is zero (E = 0).
- Electrostatic potential is constant throughout the volume and equal to its value on the surface.
- Excess charges reside entirely on the outer surface of a conductor.
- Electrostatic Shielding: The field inside a cavity of a conductor is always zero, shielding the interior from external electric fields.
4. Capacitors, Capacitance & Dielectrics
Short Note:
• Capacitance (C): Measure of ability to store electric charge: C = Q / V. SI unit is Farad (F).
• Parallel Plate Capacitor: Consists of two conducting plates of area A separated by distance d: C₀ = ε₀ A / d.
• Dielectric Constant (K): Ratio of original field to reduced field: K = E₀ / E = C / C₀.
• Capacitance with Dielectric Slab: When a slab of dielectric constant K fully fills space between plates: C = K × C₀ = (K ε₀ A) / d.
• Dielectric Polarization (P): Induced dipole moment per unit volume inside a dielectric under an external electric field.
5. Energy Stored in a Capacitor & Combinations
Short Note:
• Combinations of Capacitors:
- Series Combination: Charge Q is same on each capacitor, 1 / C_eq = 1 / C₁ + 1 / C₂ + ...
- Parallel Combination: Potential V is same across each capacitor, C_eq = C₁ + C₂ + ...
• Energy Stored in Capacitor (U): Work done in charging a capacitor is stored as electrostatic potential energy:
- U = (1 / 2) Q V = (1 / 2) C V² = Q² / (2 C)
• Energy Density (u): Energy stored per unit volume between capacitor plates: u = (1 / 2) ε₀ E².
Intext & Conceptual Questions
The work done in moving a charge q by distance dr along an equipotential surface is zero because potential remains constant (dW = q dV = 0). Since dW = F · dr = q (E · dr) = q E dr cos θ = 0, and neither q, E, nor dr is zero, cos θ must be 0, which means θ = 90°. Thus, the electric field lines are always perpendicular to the equipotential surface.
• (i) When Battery Remains Connected:
- Potential Difference (V): Remains constant (V = V₀).
- Capacitance (C): Increases by factor K (C = K C₀).
- Charge (Q): Increases by factor K (Q = K Q₀).
- Stored Energy (U): Increases by factor K (U = K U₀).
• (ii) When Battery is Disconnected:
- Charge (Q): Remains constant (Q = Q₀).
- Capacitance (C): Increases by factor K (C = K C₀).
- Potential Difference (V): Decreases by factor K (V = V₀ / K).
- Stored Energy (U): Decreases by factor K (U = U₀ / K).
Textbook Chapter End Exercise Questions
Let q₁ = 5 × 10⁻⁸ C and q₂ = -3 × 10⁻⁸ C placed distance d = 16 cm = 0.16 m apart.
Case 1: Point P lies between the charges at distance x from q₁
V_total = V₁ + V₂ = 0
(1 / 4πε₀) × (q₁ / x) + (1 / 4πε₀) × (q₂ / (d - x)) = 0
q₁ / x = -q₂ / (0.16 - x)
(5 × 10⁻⁸) / x = (3 × 10⁻⁸) / (0.16 - x)
5(0.16 - x) = 3x => 0.80 - 5x = 3x => 8x = 0.80 => x = 0.10 m = 10 cm
Case 2: Point P lies outside the line segment beyond q₂ at distance x from q₁
q₁ / x = -q₂ / (x - 0.16)
(5 × 10⁻⁸) / x = (3 × 10⁻⁸) / (x - 0.16)
5(x - 0.16) = 3x => 5x - 0.80 = 3x => 2x = 0.80 => x = 0.40 m = 40 cm
Therefore, the electric potential is zero at a distance of 10 cm (between the charges) and 40 cm (outside towards the negative charge) from q₁.
| Feature | Series Combination | Parallel Combination |
|---|---|---|
| Charge (Q) | Same charge Q flows through each capacitor. | Total charge splits across individual capacitors (Q = Q₁ + Q₂). |
| Voltage (V) | Total potential divides (V = V₁ + V₂). | Same potential difference V across each capacitor. |
| Equivalent Capacitance | Reciprocal sum: 1/C_eq = 1/C₁ + 1/C₂ + ... | Direct sum: C_eq = C₁ + C₂ + ... |
| Net Capacitance Value | Decreases (C_eq is smaller than smallest individual C). | Increases (C_eq is larger than largest individual C). |