Electric Potential, Energy & Capacitors – Comprehensive Exam Notes

Electric Potential (V)

  • Conceptual meaning: Ability of a point in an electric field to possess/transfer electrical potential energy.
  • Operational definition:
    • Work W done by an external agent in moving a positive unit test charge slowly (no acceleration) from infinity to the point against the electro-static force.
    • Mathematical form V = \frac{W}{q}.
  • Units & dimensional formula
    • SI unit : 1\;\text{Volt} = 1\;\dfrac{\text{Joule}}{\text{Coulomb}}.
    • Dimension [ML^{2}T^{-3}A^{-1}] (because W is energy [ML^{2}T^{-2}], charge [AT]).
  • Definition of 1 Volt: Potential at a point where 1\;\text{J} of work is needed to bring 1\;\text{C} from infinity.
  • Direction of charge flow
    • Positive charge: from higher V to lower V.
    • Negative charge (electrons): from lower V to higher V.

Potential difference (PD)

  • For two points A,B
    VA-VB = \frac{W{BA}}{q}, where W{BA} is work by external agent in moving charge from B \to A.
  • If one point is at infinity (V\infty = 0), potential at point P is VP = \frac{W{\infty P}}{q} = -\int{\infty}^{P} \vec E\cdot d\vec r.
  • Example conversion: 1\;\text{eV}=1.6\times10^{-19}\;\text{J}.

Electric potential due to discrete charges

  • Single point charge +Q at distance r:
    V=\frac{1}{4\pi\varepsilon_0}\,\frac{Q}{r} (positive if Q>0, negative if Q<0).
  • Group of point charges (scalar superposition):
    V = \sumi \frac{1}{4\pi\varepsilon0}\,\frac{qi}{ri}.
  • Continuous charge distributions
    • Line: V=\dfrac{1}{4\pi\varepsilon_0}\int \dfrac{dq}{r}
    • Surface: V=\dfrac{1}{4\pi\varepsilon_0}\iint \dfrac{dq}{r}
    • Volume: V=\dfrac{1}{4\pi\varepsilon_0}\iiint \dfrac{dq}{r}.

Electric dipole potential

  • Dipole moment \vec p = q\,2\ell\,\hat\imath (assuming two charges \pm q separated by 2\ell).
  • On axial line (\theta = 0, r \gg \ell):
    V=\frac{1}{4\pi\varepsilon_0}\,\frac{2p}{r^{2}} (positive on +ve-charge side, negative opposite).
  • On equatorial line (\theta = 90^\circ): V=0 (no work to move charge along this line).
  • General point (r\gg\ell):
    V=\frac{1}{4\pi\varepsilon0}\,\frac{p\cos\theta}{r^{2}} = \frac{\vec p\cdot \hat r}{4\pi\varepsilon0 r^{2}}.

Relation between \vec E and V

  • For 1-D along x: E_x = -\frac{dV}{dx}.
  • Vector form \vec E = -\nabla V (gradient points from low to high potential; minus sign gives field direction from high to low V).

Equipotential surfaces

  • Definition: locus of points with same V (real or imaginary).
  • Properties:
    1. No work in moving charge on the surface (\Delta V=0).
    2. Surfaces form families parallel to each other.
    3. Electric field everywhere perpendicular (90°) to an equipotential surface.
    4. Two equipotentials never intersect.
    5. Perfect conductors are equipotential volumes; entire surface has single potential.

Electric Potential Energy (EPE)

  • Work–energy theorem: external work in assembling configuration = stored potential energy.

Two-charge system

  • Charges q1,q2 separated by r: U = \frac{1}{4\pi\varepsilon0}\,\frac{q1 q_2}{r}.
    • U>0 (repulsive) for like charges; U<0 (attractive) for unlike.

Three charges / many charges

  • Total energy = sum over all unique pairs:
    U = \sum{i

Dipole in uniform field

  • Torque: \tau = pE\sin\theta.
  • Work done in rotating from \theta1 to \theta2:
    W = pE(\cos\theta1-\cos\theta2).
  • Potential energy U = -pE\cos\theta (minimum when aligned with \vec E).

Capacitors & Dielectrics

Capacitance (C)

  • Charge-voltage relation Q = CV.
  • SI unit 1\;\text{Farad} = 1\;\dfrac{\text{C}}{\text{V}} (very large; practical units: \mu\text{F},\;\text{nF},\;\text{pF}).
  • Capacitance depends only on geometry & medium, not on charge or voltage.

Isolated spherical conductor

  • Radius R: C = 4\pi\varepsilon_0 R (≈ 1\;\text{pF} for R≈9\;\text{m}).
  • Earth (R=6400\;\text{km}): C_{\text{earth}} \approx 710\;\mu\text{F}.

Parallel-plate capacitor

  • Plate area A, separation d, vacuum between:
    C = \varepsilon_0\,\frac{A}{d}.
  • Electric field E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{V}{d}.

Dielectric insertion

  • Dielectric constant k (\varepsilonr) increases capacitance by factor k. C' = k\,C = k\varepsilon0\,\frac{A}{d}.
  • Electric field inside dielectric E = \dfrac{E_0}{k} due to induced charges.
  • Electric susceptibility \chie = k-1; polarization \vec P = \varepsilon0 \chi_e \vec E.
  • Dielectric strength: max sustainable field before breakdown (e.g.
    air \approx 3\times10^{6}\;\text{V·m}^{-1}, mica \approx 3{-}6\;\text{kV·mm}^{-1}).

Partially filled capacitor

  • Slab thickness t (k) within gap d: C = \frac{\varepsilon_0 A}{d-t+\dfrac{t}{k}}.
    • Limits:
    • t=0 ⇒ vacuum value;
    • t=d ⇒ completely filled C=k\varepsilon_0 \dfrac{A}{d};
    • Metallic insert (k\to\infty): C=\varepsilon_0 A/(d-t) (plates effectively closer).
  • Multiple layers t1,t2,\dots with k1,k2,\dots (series of dielectrics):
    \dfrac{1}{C}=\dfrac{d1}{\varepsilon0 k1 A}+\dfrac{d2}{\varepsilon0 k2 A}+\dots.
  • Equal-area parallel arrangement (different dielectrics side-by-side) behaves like capacitors in parallel:
    C = \varepsilon0 A\Bigg(\frac{k1}{d}+\frac{k_2}{d}+\dots\Bigg).

Energy storage

  • Energy U = \dfrac{1}{2}CV^{2} = \dfrac{Q^{2}}{2C} = \dfrac{1}{2}QV.
  • Energy density in field u = \dfrac{1}{2}\varepsilon0 E^{2} (vacuum); with dielectric u = \dfrac{1}{2}\varepsilon0 k E^{2}.
  • Force between parallel plates (opposite charges):
    F = \frac{1}{2}\,\frac{\sigma^{2}A}{\varepsilon0}=\frac{1}{2}\varepsilon0 E^{2}A.

Combinations of capacitors

Series

  • Same charge Q on each, voltages add: V = V1+V2+\dots.
  • Equivalent capacitance \dfrac{1}{C{eq}}=\sum\dfrac{1}{Ci} (less than the smallest).
  • Used when high voltage rating required.

Parallel

  • Same voltage across each, charges add: Q = Q1+Q2+\dots.
  • Equivalent capacitance C{eq}=\sum Ci (greater than largest).
  • Provides large charge storage at given voltage.

Mixed example (3 capacitors 2 µF each in series across 300 V)

  • C{eq}=\dfrac{2}{3}\;\mu\text{F}, charge Q=C{eq}V=2\times10^{-6}\,\tfrac{300}{3}=2\times10^{-4}\;\text{C}, voltage on each =100\;\text{V}.

Redistribution / common potential

  • Two charged capacitors C1,C2 at V1,V2 connected together (same polarities):
    • Charge conserved: Q{tot}=C1V1+C2V_2.
    • Common potential V=\dfrac{C1V1+C2V2}{C1+C2}.
    • New charges Q1=C1V,\;Q2=C2V.
    • Energy loss (converted to heat/spark)
      \Delta U = Ui-Uf = \frac{C1C2}{2(C1+C2)}(V1-V2)^2.

Van-de-Graaff Generator (application)

  • Uses mechanical transport of charge (insulating belt, metal combs) to accumulate large charge on a hollow spherical conductor.
  • Potential V=\dfrac{1}{4\pi\varepsilon_0}\,\dfrac{Q}{R}; with radius several metres and continual charging, voltages of 10^6 V achieved (limited by air breakdown).
  • Employed as high-energy particle accelerator (electrostatic) and for X-ray generation.

Surface electric field & corona

  • Surface charge density on conductor \sigma = \varepsilon0 En (normal component).
  • Sharper curvature (smaller radius R) ⇒ larger E and \sigma; leads to corona discharge (ionisation of surrounding air) causing leakage and sparks.
  • Lightning rods and spherical terminals minimise sharp points to control discharges.

Quick Reference / Numbers

  • \varepsilon_0 = 8.854\times10^{-12}\;\text{C}^2\text{N}^{-1}\text{m}^{-2}.
  • Dielectric constants: air ≈ 1.0006, water ≈80, mica ≈3{-}6.
  • Dielectric strengths: air ≈3\times10^{6}\,\text{V·m}^{-1}, mica ≈100\,\text{MV·m}^{-1}.
  • 1 electron-volt (eV) =1.602\times10^{-19}\;\text{J}.

These organised points capture every key definition, formula, example, and practical implication from the transcript, providing a full study replacement for the original notes.