class 12 notes

Class 12 Physics Notes Chapter 2 (Electrostatic Potential and Capacitance) – Examplar Problems (English) Book

Philoid

Philoid

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Examplar Problems (English)
Detailed Notes with MCQs of Chapter 2: Electrostatic Potential and Capacitance. This is a crucial chapter, building upon our understanding of electric fields and forces from Chapter 1. Pay close attention, as the concepts here are frequently tested in various government examinations.

Chapter 2: Electrostatic Potential and Capacitance - Detailed Notes

1. Electrostatic Potential Energy (U)

  • Concept: The work done by an external force (against the electrostatic force) in bringing a charge or a system of charges from infinity to their present configuration, without acceleration.
  • Potential Energy of a Two-Charge System: In the absence of an external field, the potential energy of two point charges q₁ and q₂ separated by distance r₁₂ is:
    • U = (1 / 4πε₀) * (q₁q₂ / r₁₂)
    • This is a scalar quantity. It can be positive (like charges) or negative (unlike charges).
  • Potential Energy of a System of N Charges: Sum of potential energy for each distinct pair of charges.
    • U = (1 / 4πε₀) * Σ (qᵢqⱼ / rᵢⱼ) (Sum over all pairs i < j)
  • Potential Energy in an External Electric Field (E):
    • Single Charge (q): U = q * V(r), where V(r) is the potential at the position vector r due to the external field.
    • Electric Dipole (p) in a Uniform External Field (E): U = - p ⋅ E = -pE cosθ, where θ is the angle between the dipole moment p and the electric field E.
      • Stable Equilibrium: θ = 0°, U = -pE (minimum)
      • Unstable Equilibrium: θ = 180°, U = +pE (maximum)

2. Electrostatic Potential (V)

  • Concept: Work done per unit positive test charge by an external force in moving the charge from infinity to a point in the electric field, without acceleration. It represents the electrostatic potential energy per unit charge.
    • V = W<0xE2><0x88><0x9E>→P / q₀ = U / q₀
  • Relation to Electric Field: The electric field is the negative gradient of the potential.
    • E = - dV/dr (in the direction of decreasing potential)
    • In Cartesian coordinates: E = - (∂V/∂x i + ∂V/∂y j + ∂V/∂z k)
    • Potential difference between two points A and B: V<0xE2><0x82><0x8B> - V<0xE2><0x82><0x8A> = - ∫<0xE2><0x82><0x8A><0xE1><0xB5><0x8B> E ⋅ dl
  • Potential due to a Point Charge (q) at distance r:
    • V = (1 / 4πε₀) * (q / r)
  • Potential due to a System of Charges: Scalar sum of potentials due to individual charges (Principle of Superposition).
    • V = V₁ + V₂ + ... + V<0xE2><0x82><0x99> = (1 / 4πε₀) * Σ (qᵢ / rᵢ)
  • Potential due to an Electric Dipole (p):
    • At a point on the axial line (distance r from center): V = (1 / 4πε₀) * (p / r²) (for r >> a)
    • At a point on the equatorial line: V = 0
    • At a general point (r, θ): V = (1 / 4πε₀) * (p cosθ / r²) (for r >> a)
  • SI Unit: Volt (V). 1 V = 1 J/C.
  • Dimension: [ML²T⁻³A⁻¹]

3. Equipotential Surfaces

  • Definition: A surface on which the electrostatic potential is constant at every point.
  • Properties:
    • No work is done in moving a test charge from one point to another on an equipotential surface (ΔV = 0 => W = qΔV = 0).
    • The electric field is always perpendicular (normal) to the equipotential surface at every point. (If E had a component along the surface, work would be done to move a charge along it).
    • Equipotential surfaces are closer together where the electric field is stronger and farther apart where the field is weaker (since E = -dV/dr).
    • Two equipotential surfaces can never intersect. (If they did, there would be two different values of potential at the point of intersection, which is impossible).
  • Examples:
    • For a point charge: Concentric spheres centered on the charge.
    • For a uniform electric field: Planes perpendicular to the field lines.
    • Surface of a charged conductor: Always an equipotential surface.

4. Electrostatics of Conductors

  • Key Properties:
    1. Inside a conductor, electrostatic field (E) is zero. (Free charges redistribute until they cancel any internal field).
    2. At the surface of a charged conductor, the electrostatic field must be normal to the surface at every point. (Otherwise, charges would move along the surface).
    3. The interior of a conductor can have no excess charge in the static situation. (Any excess charge resides only on the surface).
    4. Electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface. (Since E=0 inside, V<0xE2><0x82><0x8B> - V<0xE2><0x82><0x8A> = - ∫<0xE2><0x82><0x8A><0xE1><0xB5><0x8B> E ⋅ dl = 0).
    5. Electric field at the surface of a charged conductor: E = σ / ε₀ n̂, where σ is the surface charge density and n̂ is the unit vector normal to the surface outwards.
    6. Electrostatic Shielding: A cavity inside a conductor remains shielded from outside electric influence (E=0 inside the cavity, regardless of the external field). This is used to protect sensitive instruments.

5. Dielectrics and Polarization

  • Dielectrics: Insulating materials that transmit electric effects without conducting. They get polarized when placed in an external electric field.
  • Polarization (P): When an external field E₀ is applied, the dipoles (induced or permanent) in the dielectric align partially, creating an internal electric field E<0xE1><0xB5><0x96> that opposes the external field. The net field inside the dielectric is E = E₀ - E<0xE1><0xB5><0x96>.
  • Dielectric Constant (K) or Relative Permittivity (ε<0xE1><0xB5><0xA3>): The factor by which the net electric field inside the dielectric is reduced compared to the external field.
    • K = E₀ / E (K > 1 for dielectrics, K ≈ ∞ for conductors, K = 1 for vacuum)
  • Electric Susceptibility (χ<0xE2><0x82><0x91>): A measure of how easily a dielectric polarizes. P = ε₀ χ<0xE2><0x82><0x91> E.
  • Relation: K = 1 + χ<0xE2><0x82><0x91>

6. Capacitors and Capacitance

  • Capacitor: A device consisting of two conductors separated by an insulating medium (dielectric), 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 the conductors.
    • C = Q / V
    • It depends on the geometrical configuration (shape, size, separation) of the conductors and the nature of the dielectric medium between them. It does not depend on Q or V.
  • SI Unit: Farad (F). 1 F = 1 C/V. Practical units: μF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).
  • Parallel Plate Capacitor:
    • Area of plates: A, Separation: d, Dielectric: vacuum/air (ε₀)
    • C₀ = ε₀ A / d
    • With a dielectric medium (dielectric constant K or permittivity ε = Kε₀) completely filling the space:
    • C = K ε₀ A / d = K C₀
    • Capacitance increases by a factor K when a dielectric is introduced.
  • Combination of Capacitors:
    • Series Combination: Charge (Q) is the same on each capacitor. Potential difference (V) adds up. Equivalent capacitance (C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0>) is given by:
      • 1 / C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0> = 1 / C₁ + 1 / C₂ + 1 / C₃ + ...
      • The equivalent capacitance is less than the smallest individual capacitance.
    • Parallel Combination: Potential difference (V) is the same across each capacitor. Charge (Q) adds up. Equivalent capacitance (C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0>) is given by:
      • C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0> = C₁ + C₂ + C₃ + ...
      • The equivalent capacitance is greater than the largest individual capacitance.

7. Energy Stored in a Capacitor

  • Concept: Work done in charging the capacitor is stored as electrostatic potential energy in the electric field between the plates.
  • Formulas:
    • U = (1/2) QV
    • U = (1/2) CV²
    • U = Q² / (2C)
  • Energy Density (u): Energy stored per unit volume in the electric field.
    • For a parallel plate capacitor (in vacuum): Volume = Ad
    • u = U / (Ad) = (1/2) ε₀ E²
    • This formula is generally valid for energy density in any electric field.

8. Van de Graaff Generator (Principle is important)

  • Principle: Based on:
    1. Action of sharp points (corona discharge): Charge density is high at sharp points, leading to ionization of surrounding air and charge leakage or spraying.
    2. Property that charge given to a hollow conductor resides on its outer surface, and the potential inside is constant.
  • Use: To build up very high potentials (millions of volts), used to accelerate charged particles for nuclear experiments.

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 energy
    c) Electric potential
    d) Electric flux

  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. A parallel plate capacitor has capacitance C. If the distance between the plates is halved and the area of the plates is doubled, the new capacitance will be:
    a) C/2
    b) C
    c) 2C
    d) 4C

  4. When a dielectric slab (K > 1) is introduced between the plates of an isolated charged parallel plate capacitor (charge Q remains constant), the:
    a) Electric field between the plates increases
    b) Potential difference between the plates increases
    c) Capacitance decreases
    d) Potential difference between the plates decreases

  5. Three capacitors of capacitances 2μF, 3μF, and 6μF are connected in series. The equivalent capacitance of the combination is:
    a) 11 μF
    b) 1 μF
    c) 0.5 μF
    d) 6 μF

  6. The energy stored in a capacitor of capacitance C charged to a potential V is given by:
    a) (1/2) C V
    b) (1/2) C² V
    c) (1/2) C V²
    d) C V²

  7. If the potential difference across a capacitor is doubled, the energy stored in it becomes:
    a) Half
    b) Double
    c) Four times
    d) Remains same

  8. Inside a hollow charged spherical conductor, the electric potential is:
    a) Zero
    b) Constant and same as on the surface
    c) Varies inversely with distance from the center
    d) Varies directly with distance from the center

  9. The SI unit of electric potential energy is:
    a) Volt (V)
    b) Farad (F)
    c) Joule (J)
    d) Newton per Coulomb (N/C)

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

Answers to MCQs:

  1. (c) Electric potential
  2. (c) Planes parallel to the yz-plane (perpendicular to the E field along x-axis)
  3. (d) 4C (C' = ε₀(2A)/(d/2) = 4 ε₀A/d = 4C)
  4. (d) Potential difference between the plates decreases (V = Q/C; C increases to KC₀, so V decreases to V₀/K)
  5. (b) 1 μF (1/C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0> = 1/2 + 1/3 + 1/6 = (3+2+1)/6 = 6/6 = 1; C<0xE2><0x82><0x9B><0xE1><0xB5><0xA0> = 1 μF)
  6. (c) (1/2) C V²
  7. (c) Four times (U = (1/2)CV²; U' = (1/2)C(2V)² = 4 * (1/2)CV² = 4U)
  8. (b) Constant and same as on the surface (Since E=0 inside)
  9. (c) Joule (J)
  10. (a) 0° (U = -pE cosθ; minimum when cosθ = 1)

Revise these notes thoroughly. Focus on understanding the definitions, the relationships between quantities (like E and V), the properties of conductors and dielectrics, and the formulas for capacitance and energy storage. Practice problems involving combinations of capacitors and the effect of dielectrics. Good luck!

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