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 $A,B$
  $V<em>A-V</em>B = \frac{W<em>{BA}}{q}$, where $W</em>{BA}$ is work by external agent in moving charge from $B \to A$.
• If one point is at infinity ($V<em>\infty = 0$), potential at point $P$ is $V</em>P = \frac{W<em>{\infty P}}{q} = -\int</em>{\infty}^{P} \vec E\cdot d\vec r$.
• 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):
  $V = \sum<em>i \frac{1}{4\pi\varepsilon</em>0}\,\frac{q<em>i}{r</em>i}$.
• 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 ($r\gg\ell$):
  $V=\frac{1}{4\pi\varepsilon<em>0}\,\frac{p\cos\theta}{r^{2}} = \frac{\vec p\cdot \hat r}{4\pi\varepsilon</em>0 r^{2}}$.

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 $q<em>1,q</em>2$ separated by $r$: $U = \frac{1}{4\pi\varepsilon<em>0}\,\frac{q</em>1 q_2}{r}$.
  • U>0 (repulsive) for like charges; U<0 (attractive) for unlike.

Three charges / many charges

• Total energy = sum over all unique pairs:
  $U = \sum<em>{i<j} \frac{1}{4\pi\varepsilon</em>0}\,\frac{q<em>i q</em>j}{r_{ij}}$.

Dipole in uniform field

• Torque: $\tau = pE\sin\theta$.
• Work done in rotating from $\theta<em>1$ to $\theta</em>2$:
  $W = pE(\cos\theta<em>1-\cos\theta</em>2)$.
• 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 $k (\varepsilon<em>r)$ increases capacitance by factor $k$. $C' = k\,C = k\varepsilon</em>0\,\frac{A}{d}$.
• Electric field inside dielectric $E = \dfrac{E_0}{k}$ due to induced charges.
• Electric susceptibility $\chi<em>e = k-1$; polarization $\vec P = \varepsilon</em>0 \chi_e \vec E$.
• 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 $t<em>1,t</em>2,\dots$ with $k<em>1,k</em>2,\dots$ (series of dielectrics):
  $\dfrac{1}{C}=\dfrac{d<em>1}{\varepsilon</em>0 k<em>1 A}+\dfrac{d</em>2}{\varepsilon<em>0 k</em>2 A}+\dots$.
• Equal-area parallel arrangement (different dielectrics side-by-side) behaves like capacitors in parallel:
  $C = \varepsilon<em>0 A\Bigg(\frac{k</em>1}{d}+\frac{k_2}{d}+\dots\Bigg)$.

Energy storage

• Energy $U = \dfrac{1}{2}CV^{2} = \dfrac{Q^{2}}{2C} = \dfrac{1}{2}QV$.
• Energy density in field $u = \dfrac{1}{2}\varepsilon<em>0 E^{2}$ (vacuum); with dielectric $u = \dfrac{1}{2}\varepsilon</em>0 k E^{2}$.
• Force between parallel plates (opposite charges):
  $F = \frac{1}{2}\,\frac{\sigma^{2}A}{\varepsilon<em>0}=\frac{1}{2}\varepsilon</em>0 E^{2}A$.

Combinations of capacitors

Series

• Same charge $Q$ on each, voltages add: $V = V<em>1+V</em>2+\dots$.
• Equivalent capacitance $\dfrac{1}{C<em>{eq}}=\sum\dfrac{1}{C</em>i}$ (less than the smallest).
• Used when high voltage rating required.

Parallel

• Same voltage across each, charges add: $Q = Q<em>1+Q</em>2+\dots$.
• Equivalent capacitance $C<em>{eq}=\sum C</em>i$ (greater than largest).
• Provides large charge storage at given voltage.

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

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

Redistribution / common potential

• Two charged capacitors $C<em>1,C</em>2$ at $V<em>1,V</em>2$ connected together (same polarities):
  • Charge conserved: $Q<em>{tot}=C</em>1V<em>1+C</em>2V_2$.
  • Common potential $V=\dfrac{C<em>1V</em>1+C<em>2V</em>2}{C<em>1+C</em>2}$.
  • New charges $Q<em>1=C</em>1V,\;Q<em>2=C</em>2V$.
  • Energy loss (converted to heat/spark)
    $\Delta U = U<em>i-U</em>f = \frac{C<em>1C</em>2}{2(C<em>1+C</em>2)}(V<em>1-V</em>2)^2$.

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 $\sigma = \varepsilon<em>0 E</em>n$ (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) $=1.602\times10^{-19}\;\text{J}$.

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