Electrostatic Potential and Capacitance

INTRODUCTION TO CONSERVATIVE FORCES

• Concept of Potential Energy: As established in earlier physics concepts, when an external force performs work to move a body against a restoring force (like a spring force or gravitational force), the work is stored as potential energy ($U$).

• Work-Energy Conservation: Upon removal of the external force, the body moves, converting potential energy into kinetic energy ($K$). The total mechanical energy ($K + U$) remains conserved.

• Defining Conservative Forces: Forces like spring force, gravitational force, and the Coulomb force between stationary charges are termed conservative forces.

• Coulomb Force Basis: The Coulomb force shares the inverse-square dependence on distance ($r$) seen in gravity. The primary difference is the replacement of mass in gravity with charge in electrostatic laws.

ELECTROSTATIC POTENTIAL ENERGY

• Fundamental Scenario: Consider a field $\mathbf{E}$ due to a charge $Q$ at the origin. To move a test charge $q$ from point $R$ to point $P$ against the repulsive force (Q, q > 0), an external force $\mathbf{F}_{ext}$ is applied.

• Assumptions for Calculation:

  • The test charge $q$ is infinitesimal so as not to disturb the source charge $Q$.

  • $\mathbf{F}_{ext} = -\mathbf{F}_E$ at every point. This ensures no net acceleration (infinitesimally slow movement).

• Work and Potential Energy Formula: The work done by the external force ($W_{RP}$) is the negative of the work done by the electric field and is stored as potential energy difference:          $W_{RP} = \int_R^P \mathbf{F}_{ext} \cdot d\mathbf{r} = - \int_R^P \mathbf{F}_E \cdot d\mathbf{r}$

• Potential Energy Difference ($\Delta U$):          $\Delta U = U_P - U_R = W_{RP}$

• Important Properties:

  • Path Independence: The work done depends only on the initial ($R$) and final ($P$) positions, not the path taken.

  • Additive Constant: Potential energy is undetermined to within an additive constant (represented as $\alpha$). Only the difference is physically significant.

  • Zero Point Reference: It is conventional to define potential energy as zero at infinity. If point $R$ is at infinity, the potential energy at $P$ is the work required to bring charge $q$ from infinity to $P$:          $U_P = W_{\infty P}$

ELECTROSTATIC POTENTIAL ($V$)

• Definition: Work done per unit test charge by an external force to bring a unit positive charge from infinity to a specific point ($P$) in an electrostatic field.

• Mathematical Expression:          $V_P - V_R = \frac{U_P - U_R}{q} = \frac{W_{RP}}{q}$

• The Volt and Alessandro Volta: The unit is named after Count Alessandro Volta (1745–1827), an Italian physicist who developed the first voltaic pile (battery) using metal disks and moist cardboard electrolyte.

• Potential Due to a Point Charge: For a charge $Q$ at the origin, the potential at a point at distance $r$ is calculated by integrating the work from infinity down to $r$:          $V(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

• Comparison with Electric Field: While field intensity decays as $1/r^2$, potential decays more slowly as $1/r$.

POTENTIAL DUE TO AN ELECTRIC DIPOLE

• Superposition Principle: The total potential at point $P$ is the algebraic sum of potentials due to charges $q$ and $-q$:          $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right)$

• General Formula for Large Distances ($r \gg a$):          $V(r) = \frac{1}{4\pi\epsilon_0} \frac{p \cos(\theta)}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2}$

• Special Cases:

  • Axial Points: Potential is maximum positive at $\theta = 0$ ($V = \frac{p}{4\pi\epsilon_0 r^2}$) and maximum negative at $\theta = \pi$.

  • Equatorial Plane: Potential is zero everywhere on the plane perpendicular to the dipole axis ($\theta = \pi/2$).

• Distinction from Point Charges: Dipole potential depends on angle $\theta$ and falls off as $1/r^2$ rather than $1/r$.

POTENTIAL DUE TO A SYSTEM OF CHARGES

• Superposition Calculation: For charges $q_1, q_2, \dots, q_n$:          $V = V_1 + V_2 + \dots + V_n = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{r_i}$

• Uniformly Charged Spherical Shell:

  • Outside ($r \ge R$): $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$

  • Inside (r < R): Electric field is zero, so potential remains constant and equal to its value at the surface: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{R}$

EQUIPOTENTIAL SURFACES

• Definition: A surface where potential remains constant ($V$ is constant).

• Key Properties:

  • Work Done: No work is required to move a charge between any two points on an equipotential surface.

  • Normal Field: The electric field is always normal to the equipotential surface at every point. (If it weren't, a tangential component would perform work, contradicting the definition).

• Physical Shapes:

  • Point Charge: Concentric spheres.

  • Uniform Field: Parallel planes normal to the field lines.

• Relation Between Field and Potential:          $|\mathbf{E}| = \frac{-\delta V}{\delta l}$     

  • Electric field points in the direction of the steepest potential decrease.

  • Field magnitude is the change in potential per unit displacement normal to the surface.

POTENTIAL ENERGY OF CHARGE SYSTEMS

• Two-Charge System ($q_1, q_2$): Work done to assemble charges brought from infinity:          $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$

• Three-Charge System: Sum of the pairwise interactions:          $U = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)$

• External Field Interaction:

  • Single charge in external field: $U = qV(\mathbf{r})$

  • Two charges in external field: $U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$

• Units of Energy (Electron Volt): An electron accelerated through 1 Volt gains energy:          $1\,eV = 1.6 \times 10^{-19}\,J$

DIPOLE IN AN EXTERNAL UNIFORM FIELD

• Torque: $\tau = \mathbf{p} \times \mathbf{E}$ tends to align the dipole with the field.

• Potential Energy of Dipole: The work done to rotate the dipole from $\theta_0 = \pi/2$ to $\theta$:          $U(\theta) = -\mathbf{p} \cdot \mathbf{E} = -p E \cos(\theta)$

ELECTROSTATICS OF CONDUCTORS

1. Field inside is zero: Static charges distribute to cancel any internal field.

2. Field at surface is normal: Tangential field would cause surface charge movement.

3. No excess charge inside: By Gauss's Law, excess charge resides entirely on the surface.

4. Constant potential: Since $\mathbf{E} = 0$ inside, there is no potential difference within the conductor.

5. Surface Field Magnitude:          $\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$

6. Electrostatic Shielding: The field inside a cavity of any conductor is zero, regardless of outside charges/fields. This protects sensitive instruments.

DIELECTRICS AND POLARISATION

• Dielectrics: Non-conducting substances without free charge carriers. An external field induces a dipole moment by stretching molecules.

• Polar vs. Non-Polar Molecules:

  • Non-Polar: Centers of positive/negative charges coincide (e.g., $O_2, H_2$). Field induces a dipole moment.

  • Polar: Permanent dipole moment exists (e.g., $HCl, H_2O$). Field aligns existing dipoles.

• Polarisation ($\mathbf{P}$): Dipole moment per unit volume. For linear isotropic dielectrics:          $\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}$          where $\chi_e$ is electric susceptibility.

CAPACITORS AND CAPACITANCE

• Definition: A system of two conductors separated by an insulator. Capacitance ($C$) is the ratio of charge ($Q$) to potential difference ($V$):          $C = \frac{Q}{V}$

• Units: Farad ($F$). Common sub-multiples: $\mu F, nF, pF$.

• Parallel Plate Capacitor: For area $A$ and separation $d$:          $C = \frac{\epsilon_0 A}{d}$

• Dielectric Effect: Inserting a dielectric with constant $K$ increases capacitance:          $C = K C_0 = \frac{\epsilon A}{d}$

• Dielectric Strength: The maximum electric field a medium handles without breakdown. Air is roughly $3 \times 10^6\,V/m$.

COMBINATIONS AND ENERGY

• Series Combination ($n$ capacitors):          $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$

• Parallel Combination ($n$ capacitors):          $C = C_1 + C_2 + \dots + C_n$

• Energy Stored in a Capacitor ($U$):          $U = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$

• Energy Density ($u$): Energy per unit volume in a field:          $u = \frac{1}{2} \epsilon_0 E^2$