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Class 12 Physics Notes Chapter 2 (Electrostatic potential and capacitance) – Physics Part-I Book

Philoid

Philoid

— 8 min read

Physics Part-I
Alright students, let's focus on Chapter 2: Electrostatic Potential and Capacitance from your NCERT Physics Part-I textbook. This is a crucial chapter for competitive exams, building directly upon the concepts of electric fields we learned in Chapter 1. We'll break down the key ideas and formulas you need to master.

Chapter 2: Electrostatic Potential and Capacitance - Detailed Notes

1. Electrostatic Potential Energy and Potential Difference

  • Concept: When a test charge q₀ is moved against an electrostatic field E from point A to point B, work is done by the external force. This work gets stored as potential energy.
  • Potential Energy Difference (ΔU): The work done W_AB by an external force in moving a test charge q₀ from A to B without acceleration is W_AB = U_B - U_A = ΔU.
    • W_AB = - ∫_A^B F_e ⋅ dr = ∫_A^B F_{ext} ⋅ dr = q₀ ∫_A^B E ⋅ dr (Note: F_e is electrostatic force, F_{ext} = -F_e)
  • Electrostatic Potential Difference (ΔV): It is the work done per unit test charge by an external force in moving it from point A to point B without acceleration.
    • V_B - V_A = ΔV = W_AB / q₀ = ΔU / q₀
    • Unit: Volt (V). 1 V = 1 Joule/Coulomb (J/C).
    • Dimension: [ML²T⁻³A⁻¹]
    • It's a scalar quantity.

2. Electrostatic Potential (V)

  • Definition: Electrostatic potential at a point P in an electric field is the work done per unit test charge in bringing it from infinity (where potential is taken as zero) to that point P without acceleration.
    • V_P = W_{∞→P} / q₀ = - ∫_∞^P E ⋅ dr
  • Potential due to a Point Charge (q): At a distance r from a point charge q.
    • V = (1 / 4πε₀) * (q / r)
    • If q > 0, V is positive. If q < 0, V is negative.
  • Potential due to a System of Charges: The potential at a point due to a system of charges (q₁, q₂, q₃,...) is the algebraic sum of the potentials due to individual charges at that point (Superposition Principle).
    • V = V₁ + V₂ + V₃ + ... = (1 / 4πε₀) * Σ (qᵢ / rᵢ) where rᵢ is the distance of charge qᵢ from the point.
  • Potential due to an Electric Dipole:
    • At a point on the axial line (distance r from the center): V = (1 / 4πε₀) * (p / r²) (for r >> a, where 2a is dipole length, p = q * 2a is dipole moment)
    • At a point on the equatorial line: V = 0
    • At a general point (r, θ): V = (1 / 4πε₀) * (p cosθ / r²)

3. Equipotential Surfaces

  • Definition: A surface over which the electrostatic potential is constant at every point.
  • Properties:
    1. No work is done in moving a test charge over an equipotential surface (W = q₀ΔV = q₀ * 0 = 0).
    2. The electric field (E) is always perpendicular to the equipotential surface at every point.
    3. Equipotential surfaces indicate regions of strong or weak electric fields. They are closer together where the field is strong and farther apart where the field is weak (E = -dV/dr).
    4. Two equipotential surfaces can never intersect. (If they did, there would be two values of potential at the intersection point, which is impossible).
  • Examples:
    • For a point charge: Concentric spheres centered at the charge.
    • For a uniform electric field: Planes perpendicular to the field lines.
    • For a dipole: Surfaces are complex but perpendicular to field lines.

4. Relation between Electric Field (E) and Potential (V)

  • E = - dV/dr (The electric field is the negative gradient of the potential). The negative sign indicates that the direction of E is the direction in which the potential decreases most rapidly.
  • In Cartesian coordinates: E_x = -∂V/∂x, E_y = -∂V/∂y, E_z = -∂V/∂z. So, E = -(î ∂V/∂x + ĵ ∂V/∂y + k̂ ∂V/∂z) = -∇V (where ∇ is the gradient operator).
  • Potential difference can be found by integrating the field: V_B - V_A = - ∫_A^B E ⋅ dr

5. Electrostatic Potential Energy

  • Potential Energy of a charge q in an external potential V: U = qV
  • Potential Energy of a System of Two Charges (q₁, q₂): The work done to bring these charges from infinity to their respective positions (r₁₂ apart).
    • U = (1 / 4πε₀) * (q₁q₂ / r₁₂)
  • Potential Energy of a System of N Charges: Sum of potential energies for all possible pairs of charges.
    • U = (1 / 4πε₀) * Σ_{all pairs i<j} (qᵢqⱼ / rᵢⱼ)
  • Potential Energy of a Dipole in a Uniform External Field E:
    • U = -pE cosθ = - p ⋅ E
    • θ is the angle between the dipole moment p and the electric field E.
    • Stable Equilibrium: θ = 0°, U = -pE (minimum potential energy).
    • Unstable Equilibrium: θ = 180°, U = +pE (maximum potential energy).
    • Zero Potential Energy: θ = 90°, U = 0.
    • Torque on Dipole: τ = p × E (Magnitude τ = pE sinθ)
    • Work done in rotating dipole: W = ΔU = pE(cosθ₁ - cosθ₂)

6. Electrostatics of Conductors

  • Inside a conductor, electrostatic field is zero (E = 0).
  • At the surface of a charged conductor, the electrostatic field must be normal (perpendicular) to the surface at every point.
  • The interior of a conductor can have no excess charge in the static situation. Any excess charge resides only on the surface.
  • Electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface (V = constant).
  • Electric field at the surface of a charged conductor: E = σ / ε₀, where σ is the surface charge density. (Vector form: E = (σ / ε₀) n̂, where n̂ is the unit vector normal to the surface).
  • Electrostatic Shielding: The phenomenon of protecting a certain region of space from an external electric field by enclosing it within a conducting surface. The field inside the cavity of a conductor is always zero.

7. Dielectrics and Polarization

  • Dielectrics: Insulating materials that transmit electric effects without conducting. They can be polarized.
  • Polarization (P): When a dielectric is placed in an external electric field (E₀), dipoles are induced or aligned within the material. This creates an internal electric field (E_p) opposing the external field.
    • The net field inside the dielectric is E = E₀ - E_p.
  • Dielectric Constant (K) or Relative Permittivity (ε_r): The factor by which the electric field inside the dielectric is reduced compared to the external field.
    • K = ε_r = E₀ / E
    • K ≥ 1 (K=1 for vacuum, K ≈ 1 for air, K > 1 for other dielectrics, K → ∞ for conductors).
  • Electric Susceptibility (χ): A measure of how easily a dielectric polarizes. P = ε₀ χ E.
  • Relation: K = 1 + χ

8. Capacitance

  • Capacitor: A device consisting of two conductors separated by an insulating medium, used to store electric charge and energy.
  • Capacitance (C): The ratio of the magnitude of charge (Q) on either conductor to the potential difference (V) between them.
    • C = Q / V
    • It depends on the geometrical configuration (shape, size, separation) of the conductors and the nature of the insulating medium between them. It does not depend on Q or V.
    • Unit: Farad (F). 1 F = 1 Coulomb/Volt (C/V). Practical units: µF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).
    • Dimension: [M⁻¹L⁻²T⁴A²]
  • Parallel Plate Capacitor: Two large plane parallel conducting plates of area A separated by a small distance d.
    • Capacitance (vacuum/air): C₀ = ε₀ A / d
    • Capacitance (with dielectric medium of dielectric constant K filling the space): C = K ε₀ A / d = K C₀
    • Capacitance (with a dielectric slab of thickness t < d): C = ε₀ A / (d - t + t/K)
    • Capacitance (with a conducting slab of thickness t < d): C = ε₀ A / (d - t) (as K → ∞ for conductor)
  • Spherical Capacitor: Capacitance of an isolated spherical conductor of radius R: C = 4πε₀ R.

9. Combination of Capacitors

  • Capacitors in Series:
    • Charge (Q) on each capacitor is the same.
    • Potential difference (V) across the combination is the sum of individual potential differences (V = V₁ + V₂ + ...).
    • Equivalent Capacitance (C_s): 1/C_s = 1/C₁ + 1/C₂ + 1/C₃ + ...
    • The equivalent capacitance is smaller than the smallest individual capacitance.
  • Capacitors in Parallel:
    • Potential difference (V) across each capacitor is the same.
    • Total charge (Q) is the sum of charges on individual capacitors (Q = Q₁ + Q₂ + ...).
    • Equivalent Capacitance (C_p): C_p = C₁ + C₂ + C₃ + ...
    • The equivalent capacitance is larger than the largest individual capacitance.

10. Energy Stored in a Capacitor

  • Work done in charging a capacitor gets stored as electrostatic potential energy (U) in the electric field between the plates.
  • U = (1/2) QV = (1/2) CV² = (1/2) Q²/C
  • Energy Density (u): Energy stored per unit volume in the electric field between the plates.
    • u = U / Volume = U / (Ad) = (1/2) ε₀ E² (for vacuum)
    • u = (1/2) K ε₀ E² = (1/2) ε E² (for dielectric medium)
  • Loss of Energy on Sharing Charges: When two charged capacitors are connected, charge flows until they reach a common potential. There is generally a loss of energy in the form of heat and electromagnetic radiation during this redistribution.
    • Common Potential: V = (C₁V₁ + C₂V₂) / (C₁ + C₂) (if connected positive to positive)
    • Energy Loss: ΔU = U_{initial} - U_{final} = (1/2) * (C₁C₂ / (C₁ + C₂)) * (V₁ - V₂)² (Always positive or zero).

11. Van de Graaff Generator

  • Principle: Based on (a) charge residing on the outer surface of a conductor, and (b) the phenomenon of corona discharge (charge leakage from sharp points).
  • Use: To build up very high potentials (millions of volts), used to accelerate charged particles.

Multiple Choice Questions (MCQs)

  1. The work done in moving a unit positive charge from infinity to a point in an electric field is called:
    a) Electric field intensity
    b) Electric potential difference
    c) Electric potential
    d) Potential energy

  2. Equipotential surfaces associated with a uniform electric field along the positive x-axis are:
    a) Planes parallel to the xy-plane
    b) Planes parallel to the xz-plane
    c) Planes parallel to the yz-plane
    d) Coaxial cylinders with axis along the x-axis

  3. If a dielectric slab is inserted between the plates of a parallel plate capacitor while the battery remains connected, the:
    a) Charge on the plates decreases
    b) Potential difference across the plates increases
    c) Capacitance decreases
    d) Charge on the plates increases

  4. Three capacitors each of capacitance C are connected in series. The resultant capacitance will be:
    a) 3C
    b) C/3
    c) 3/C
    d) C

  5. The energy stored in a capacitor of capacitance C having charge Q is given by:
    a) Q²/ (2C)
    b) QC
    c) 2Q²/C
    d) Q/ (2C)

  6. The relation between electric field E and electric potential V is:
    a) E = -dV/dr
    b) V = -dE/dr
    c) E = dV/dr
    d) V = ∫ E dr (definite integral for potential difference)

  7. A hollow metallic sphere of radius 10 cm is charged such that the potential on its surface is 80 V. The potential at the center of the sphere is:
    a) 0 V
    b) 8 V
    c) 80 V
    d) 800 V

  8. When air is replaced by a dielectric medium of dielectric constant K, the maximum force of attraction between two charges separated by a distance:
    a) Increases K times
    b) Remains unchanged
    c) Decreases K times
    d) Decreases K² times

  9. The potential energy of an electric dipole in a uniform electric field E is minimum when the angle between p and E is:
    a) 0°
    b) 90°
    c) 180°
    d) 270°

  10. Two capacitors of capacitances C₁ and C₂ are connected in parallel. A charge Q is given to the combination. The ratio of charge on C₁ to charge on C₂ will be:
    a) C₂ / C₁
    b) C₁ / C₂
    c) √(C₁ / C₂)
    d) √(C₂ / C₁)

Answers to MCQs:

  1. (c) Electric potential
  2. (c) Planes parallel to the yz-plane (Since E is along x, potential is constant on planes perpendicular to x)
  3. (d) Charge on the plates increases (V is constant, C increases as C = KC₀, so Q = CV increases)
  4. (b) C/3 (1/C_s = 1/C + 1/C + 1/C = 3/C => C_s = C/3)
  5. (a) Q²/ (2C) (Other forms are ½ CV² and ½ QV)
  6. (a) E = -dV/dr
  7. (c) 80 V (Potential inside a conductor is constant and equal to the potential on its surface)
  8. (c) Decreases K times (Force F = (1/4πε₀K) * (q₁q₂/r²), so F_medium = F_air / K)
  9. (a) 0° (U = -pE cosθ; minimum when cosθ = 1)
  10. (b) C₁ / C₂ (In parallel, V is same. Q₁ = C₁V, Q₂ = C₂V. So Q₁/Q₂ = C₁/C₂)

Make sure you understand the definitions, formulas, and the underlying concepts thoroughly. Practice numerical problems based on these formulas, especially capacitor combinations and energy calculations. Good luck with your preparation!

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