Class 12 · Chapter 2
Electrostatic Potential & Capacitance
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
Electrostatic Potential & Capacitance Short Notes
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Condensed revision points for Electrostatic Potential & Capacitance — for quick recall before exams, not a substitute for the full notes.
Capacitance — Basics
- C = Q/V; SI unit farad (F) = C/V. Depends only on geometry & medium, never on Q or V.
- Bringing an earthed conductor near a charged one lowers its potential ⟹ raises its capacitance — the working principle of every capacitor.
- Charges on the two plates of a capacitor are always equal and opposite; net charge on the system is zero.
Energy Stored
- U = Q²/2C = ½CV² = ½QV.
- Work done by battery while charging W = QV; only half is stored — the other half is lost as heat in the wires, regardless of resistance value.
- Heat produced on discharging through any resistor = ½CV² (independent of R).
- Earthed conductor: V = 0 always ⟹ effectively C = ∞.
Spherical Conductor / Capacitor
- Isolated sphere: C = 4πε₀R (air), C = 4πε₀ε_rR (medium). Earth's capacitance ≈ 711 µF (R ≈ 6400 km).
- Spherical capacitor, outer sphere earthed: C = 4πε₀R₁R₂/(R₂−R₁).
- Spherical capacitor, inner sphere earthed: C = 4πε₀R₁R₂/(R₂−R₁) + 4πε₀R₂.
Parallel Plate Capacitor
- E = σ/ε₀ = σ/(ε₀ε_r) between the plates; C₀ = ε₀A/d (air), C = ε₀ε_rA/d (medium) = KC₀.
- C independent of plate thickness/material; depends only on facing area, separation, medium.
- Sliding plates apart sideways reduces overlap area ⟹ reduces C proportionally.
- Edge effect (non-uniform field at plate edges) ignored when d ≪ plate dimensions.
Force, Pressure & Energy Density
- Force between plates F = Q²/2ε₀A = QE/2 = CV²/2d (only half the naive QE — each plate feels only the other plate's field).
- Electrostatic pressure P = σ²/2ε₀ = ½ε₀E². Energy density u = ½ε₀E² — same formula as pressure.
- Charged soap bubble equilibrium: P_T + P_outside = P_inside + P_electrostatic.
Dielectrics
- Polar (H₂O, HCl, NH₃): permanent dipoles, randomly oriented, align under field. Non-polar (N₂, O₂, CH₄): induced dipole only under field.
- Polarisation P⃗ = np⃗ = induced surface charge density σ_i.
- Dielectric strength = max field a dielectric withstands before breakdown; unit V/m.
- K = ε_r = E₀/E = V₀/V = C/C₀ ≥ 1 always (K=1 for vacuum/air).
- Q_i = Q(1−1/K); σ_i = σ(1−1/K) — applicable to parallel plate capacitor only.
Dielectric Effect — Battery On vs Off
- Battery disconnected (Q constant), dielectric inserted: V↓, C↑, U↓.
- Battery connected (V constant), dielectric inserted: Q↑, C↑, U↑.
- Partial slab (thickness t<d, constant K): C = ε₀A/[d−t(1−1/K)]. Full fill (t=d): C = KC₀.
- Conducting slab (K→∞), thickness t<d: C = ε₀A/(d−t) — slab thickness simply removed from the gap.
Combination of Capacitors
- Series: 1/C_S = 1/C₁+1/C₂+…; same Q on each; V divides inversely with C. C_S < smallest C.
- Parallel: C_P = C₁+C₂+…; same V on each; Q divides directly with C. C_P > largest C.
- N identical capacitors C: series → C/N; parallel → NC.
- Maximum energy storage at fixed V ⟹ connect in parallel (largest C_eq).
- Balanced bridge-type capacitor networks: the 'bridge' branch carries zero charge and can be deleted.
Combination of Dielectric Slabs (Same Capacitor)
- Distance-wise (stacked) division ≡ series: Ce = [2K₁K₂/(K₁+K₂)]C for equal thickness split (harmonic mean).
- Area-wise (side by side) division ≡ parallel: Ce = [(K₁+K₂)/2]C for equal area split (arithmetic mean).
- Arithmetic mean ≥ harmonic mean ⟹ area-wise division always gives larger C than distance-wise, for the same K₁, K₂.
Charging & Discharging (RC Circuit)
- Charging: Q(t) = Q₀(1−e^(−t/RC)); at t = RC (time constant τ), charge ≈ 63% of maximum.
- Discharging: Q(t) = Q₀e^(−t/RC); at t = τ, charge ≈ 37% of initial value.
- Steady state (t ≫ RC): fully charged capacitor branch acts as an open circuit — no current flows through it.
- Just after switching on (t = 0), an uncharged capacitor acts as a plain (zero-resistance) connecting wire.
Van de Graaff Generator
- Builds potentials of the order of 10⁷ V using corona discharge + an insulating belt carrying charge to a hollow terminal.
- Charge given to a hollow conductor moves entirely to its outer surface — lets new charge keep being added even as terminal potential rises.
- Process stops being effective once the field at the terminal surface nears air's breakdown value, ≈ 3 × 10⁶ V/m.
- Used to accelerate charged particles to high energies for nuclear/particle physics experiments.
Common Exam Traps
- Force between plates is Q²/2ε₀A, not QE — the factor of ½ is dropped easily; each plate is pulled only by the other plate's field, not its own.
- K = ε_r is always ≥ 1; a 'dielectric constant less than 1' is not physically meaningful for a real dielectric.
- Battery connected vs disconnected flips the direction of every Q/V/U change when a dielectric is inserted — check which one applies before answering.
- C_S (series) is smaller than the smallest C; C_P (parallel) is larger than the largest C — the opposite pattern from how resistors combine.
- Q_i = Q(1−1/K) gives the induced charge, not the new free charge on the plates — for battery-connected cases the free charge itself also changes.
- Energy lost in charge redistribution depends only on (V₁−V₂)², not on the individual charges — equal initial potentials mean zero loss even if charges differ.
- At t = RC, charge is at 63% of maximum while charging, but only 37% of initial value while discharging — these are not the same number and are easy to swap.
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