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Electrostatic Potential and Capacitance – Class 12 Physics Chapter 2 Notes & NCERT PDF

Class 12PhysicsChapter 2NCERT book: Physics Part I - Class 12

Notes and a simple summary of Chapter 2 of the NCERT Class 12 Physics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 2: Electrostatic Potential and Capacitance

Introduction

Chapter 2 connects the force/field picture of Chapter 1 with the energy picture. Because the electrostatic force is conservative, we can define a potential energy and an electrostatic potential at every point. The second half of the chapter uses these ideas for conductors, dielectrics and capacitors — devices that store charge and energy.

Key Concepts

2.2 Electrostatic potential

Work done by an external force in moving a charge slowly (no acceleration) between two points is independent of path. The potential at a point is the work done per unit positive test charge in bringing it from infinity to that point: V = W/q. SI unit: volt (V) = J C⁻¹. Only potential differences are physically meaningful; V(∞) = 0 is a convenient choice.

2.3–2.5 Potential due to charges

  • Point charge: V = kq/r (scalar, falls as 1/r).
  • Dipole (r ≫ a): V = k p cos θ / r² — zero everywhere on the equatorial plane, falls as 1/r².
  • System of charges: V = Σ kqᵢ/rᵢ (simple algebraic superposition, since potential is a scalar).
  • Charged shell: V = kq/r outside and kq/R (constant) inside.

    • 2.6 Equipotential surfaces

    Surfaces of constant potential. No work is needed to move a charge along them, so the field is always perpendicular to an equipotential surface. For a point charge they are concentric spheres; for a uniform field they are parallel planes. E = −dV/dl: the field points in the direction of steepest decrease of potential, and E is large where equipotentials are close together.

    2.7–2.8 Potential energy

  • Two charges: U = k q₁q₂ / r₁₂.
  • Three charges: U = k(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃).
  • Single charge in an external potential: U = qV(r).
  • Dipole in a uniform field: U(θ) = −pE cos θ; minimum (stable) at θ = 0, maximum at θ = π.

    • 2.9 Electrostatics of conductors

  • Inside a conductor, E = 0.
  • Just outside, E is normal to the surface.
  • Inside, net charge is zero; excess charge resides on the surface.
  • The whole conductor is at one potential.
  • Field just outside: E = σ/ε₀ (normal).
  • Electrostatic shielding: the field inside a cavity of a conductor is zero — the reason a car is a safe place in a thunderstorm.

    • 2.10 Dielectrics and polarisation

    Non-conducting materials in which an external field induces (or aligns existing) dipole moments. Polarisation P = χₑE; the induced surface charges reduce the field inside. Polar molecules (e.g. H₂O) have permanent dipoles; non-polar molecules (e.g. O₂) acquire induced dipoles.

    2.11–2.13 Capacitors

    A capacitor is two conductors separated by an insulator. C = Q/V; SI unit farad (F); practical units µF, pF.

  • Parallel-plate capacitor: C = ε₀A/d.
  • With a dielectric of constant K filling the gap: C = Kε₀A/d — capacitance increases K times. K is also called the relative permittivity εᵣ.

    • 2.14 Combination of capacitors

  • Series: 1/C = 1/C₁ + 1/C₂ + …; same charge on each.
  • Parallel: C = C₁ + C₂ + …; same voltage across each.

    • 2.15 Energy stored

    U = Q²/2C = ½CV² = ½QV. The energy can be seen as stored in the field, with energy density u = ½ε₀E².

    Worked Examples

  • Example 2.1: potential 9 cm from 4 × 10⁻⁷ C: V = 9 × 10⁹ × 4 × 10⁻⁷/0.09 = 4 × 10⁴ V. Work to bring 2 × 10⁻⁹ C from infinity: W = qV = 8 × 10⁻⁵ J (path-independent).
  • Example 2.2: 3 × 10⁻⁸ C and −2 × 10⁻⁸ C, 15 cm apart. With the positive charge at the origin, V = 0 where 3/x = 2/(15 − x) → x = 9 cm (between them), and where 3/x = 2/(x − 15) → x = 45 cm (beyond the negative charge).
  • Example 2.5: 7 µC and −2 µC placed 18 cm apart: U = 9 × 10⁹ × 7 × 10⁻⁶ × (−2 × 10⁻⁶)/0.18 = −0.7 J. The negative sign means work must be supplied to separate them.
  • Example 2.9: three 10 µF capacitors in series give 10/3 µF; in parallel with a fourth 10 µF, C = 13.3 µF. On 500 V, the lone capacitor holds 10 µF × 500 V = 5 × 10⁻³ C; each series capacitor holds (10/3) µF × 500 V ≈ 1.7 × 10⁻³ C.
  • Example 2.10: 900 pF at 100 V stores ½CV² = 4.5 × 10⁻⁶ J. Connected to an identical uncharged capacitor, charge is shared, the common voltage is 50 V and total energy becomes 2.25 × 10⁻⁶ J — half is lost as heat and radiation during charge flow.

    • Formula / Key-Facts Sheet

  • V = W/q; point charge V = kq/r; dipole V = kp cos θ/r²
  • E = −dV/dl; field ⟂ equipotentials
  • U(two charges) = kq₁q₂/r; dipole U = −pE cos θ
  • C = Q/V; parallel plate C = ε₀A/d; with dielectric C = Kε₀A/d
  • Series 1/C = Σ1/Cᵢ; parallel C = ΣCᵢ
  • U = ½CV² = Q²/2C = ½QV; u = ½ε₀E²

    • Board Exam Focus

  • Numericals: point where potential is zero, work to assemble charges (square arrangement, Example 2.4), capacitor networks, energy before and after connecting capacitors.
  • Derivations: potential of a dipole; potential energy of a dipole in a field; capacitance of a parallel-plate capacitor with and without a dielectric; energy stored in a capacitor.
  • Answer outline — energy stored: charge the capacitor in small steps dq against potential q/C → dW = (q/C)dq → integrate 0 to Q → W = Q²/2C = ½CV².
  • Answer outline — why E = 0 inside a conductor: free electrons move until the internal field is cancelled; in electrostatic equilibrium no net force acts on them.
  • Conceptual: why equipotentials are perpendicular to field lines; electrostatic shielding.

    • Key Terms

    Electrostatic potential (स्थिरवैद्युत विभव), Potential difference (विभवांतर), Equipotential surface (समविभव पृष्ठ), Potential energy (स्थितिज ऊर्जा), Conservative force (संरक्षी बल), Conductor (चालक), Electrostatic shielding (स्थिरवैद्युत परिरक्षण), Dielectric (परावैद्युत), Polarisation (ध्रुवण), Dielectric constant (परावैद्युतांक), Capacitor (संधारित्र), Capacitance (धारिता), Farad (फैरड), Series combination (श्रेणीक्रम संयोजन), Parallel combination (पार्श्वक्रम संयोजन)

    Common Mistakes

  • Adding potentials as vectors — potential is a scalar; just add with signs.
  • Assuming V = 0 means E = 0 (on a dipole's equatorial plane V = 0 but E ≠ 0).
  • Mixing up series and parallel rules for capacitors (they are the reverse of resistors).
  • Forgetting that energy is lost when two capacitors share charge.
  • Using σ/2ε₀ for the field just outside a conductor (correct is σ/ε₀).

    • 💡 Key Learning Points

    • ✓Potential and potential difference as work per unit charge; potential is a scalar
    • ✓Potential of a point charge, a dipole and a system of charges; equipotential surfaces
    • ✓Potential energy of charge systems and of a dipole in a uniform field
    • ✓Electrostatic properties of conductors, dielectrics and polarisation
    • ✓Capacitance, parallel-plate capacitor, dielectric effect, combinations and stored energy

    👨‍🏫 Teaching Tips

    • →Use a hill-and-valley analogy for potential and potential energy
    • →Sketch equipotentials for a point charge, a dipole and a uniform field
    • →Emphasise scalar addition of potentials with a two-charge zero-potential problem
    • →Demonstrate shielding by placing a radio inside a metal box
    • →Build capacitor networks on the board, reducing series/parallel step by step
    • →Derive U = ½CV² by integrating dq, then discuss energy density
    • →Discuss the 'lost energy' paradox of connecting two capacitors

    📋 Assessment Questions

    1. Can students find points of zero potential for two charges?
    2. Can they list and justify the six properties of conductors?
    3. Can they derive C = Kε₀A/d and U = ½CV²?
    4. Can they reduce a mixed capacitor network and find charges?
    5. Can they relate E and V and sketch equipotentials?

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    Keep learning

    All chapters in Physics Part I - Class 12

    1. 1. Electric Charges and Fields
    2. 2. Electrostatic Potential and Capacitance
    3. 3. Current Electricity
    4. 4. Moving Charges and Magnetism
    5. 5. Magnetism and Matter
    6. 6. Electromagnetic Induction
    7. 7. Alternating Current
    8. 8. Electromagnetic Waves