Comprehensive Notes on Electrostatic Potential and Capacitance

Introduction to Electrostatic Potential and Conservative Forces

• Conservative Forces: The notion of potential energy, introduced in earlier studies of mechanics (Class XI, Chapters 5 and 7), applies to forces where the work done in moving a body between two points is independent of the path taken. Examples include spring force and gravitational force.
• Work and Potential Energy: When an external force performs work $W$ in moving a body against a conservative force, that work is stored as potential energy. Upon removal of the external force, the body recovers this energy as kinetic energy, such that the total mechanical energy (sum of kinetic and potential) is conserved.
• Coulomb Force as a Conservative Force: The electrostatic (Coulomb) force between two stationary charges is conservative. This is due to its inverse-square dependence on distance ($F imes \frac{1}{r^2}$), mirroring the law of universal gravitation. The primary difference lies in the constants: masses in gravitation are replaced by charges in electrostatics.
• Electrostatic Potential Energy Definition: For a test charge $q$ moved in an electric field $\mathbf{E}$ produced by a source charge configuration (e.g., a charge $Q$ at the origin), the work done by an external force $\mathbf{F}<em>{ext}$ moving $q$ from point $R$ to point $P$ is given by:     * $W</em>{RP} = \int_R^P \mathbf{F}<em>{ext} \cdot d\mathbf{r} = - \int_R^P \mathbf{F}_E \cdot d\mathbf{r}$     * Here, $\mathbf{F}_E$ is the electrostatic force. For the movement to purely store energy without adding kinetic energy, the external force must exactly counter the electric force ($\mathbf{F}</em>{ext} = -\mathbf{F}_E$), meaning the charge moves with an infinitesimally slow constant speed (no acceleration).

Potential Energy Difference and Absolute Potential

• Potential Energy Difference (\Delta U): The difference in potential energy between points $P$ and $R$ is defined as the work done by the external force to move the charge from $R$ to $P$:     * $\Delta U = U_P - U_R = W_{RP}$     * This depends only on the initial and final positions, a hallmark of conservative forces.
• Reference Point and Arbitrary Constants: Potential energy is defined within an additive constant $\alpha$. Only the difference $U_P - U_R$ is physically significant. To define an absolute value, a reference point for zero potential energy must be chosen. The standard convention is to set potential energy to zero at infinity ($U_{\infty} = 0$).
• Work-Energy Definition at a Point: The potential energy of a charge $q$ at point $P$ is the work done by an external force to bring the charge from infinity to that point:     * $U_P = W_{\infty P}$

Electrostatic Potential (V)

• Definition: Electrostatic potential $V$ is the work done per unit test charge in bringing a unit positive charge from infinity to a specific point in an electric field without acceleration.     * $V_P - V_R = \frac{U_P - U_R}{q} = \frac{W_{RP}}{q}$
• Significant Characteristics:     * It is independent of the test charge $q$ and characterizes the electric field of the source configuration.     * Unit: Volts (V), named after Alessandro Volta.
• Alessandro Volta (1745–1827): An Italian physicist who demonstrated that electricity (previously observed in frog tissues by Luigi Galvani) could be generated by sandwiching wet bodies between dissimilar metals. This led to the creation of the first battery, the voltaic pile.

Potential due to a Point Charge

• Derivation: Consider a charge $Q$ at the origin. To find the potential at point $P$ at distance $r$, we calculate the work to bring a unit positive charge from infinity. Choosing a radial path:     * The force at an intermediate point $r'$ is $\mathbf{F} = \frac{Q}{4\pi\epsilon_0 r'^2} \hat{\mathbf{r}}'$     * Total work $W = -\int_{\infty}^r \frac{Q}{4\pi\epsilon_0 r'^2} dr' = \left[ \frac{Q}{4\pi\epsilon_0 r'} \right]_{\infty}^r = \frac{Q}{4\pi\epsilon_0 r}$
• Formula: The potential $V$ at distance $r$ from a point charge $Q$ is:     * $V(r) = \frac{Q}{4\pi\epsilon_0 r}$
• Sign Dependencies:     * If Q > 0, then V > 0 (work is done against repulsion).     * If Q < 0, then V < 0 (work is done by the field; external work is negative).
• Comparison of E and V: Electric field $E$ varies as $\frac{1}{r^2}$, while potential $V$ varies as $\frac{1}{r}$.

Potential due to an Electric Dipole

• Configuration: Two charges $q$ and $-q$ separated by distance $2a$. Dipole moment $\mathbf{p}$ has magnitude $p = q \times 2a$ and points from $-q$ to $+q$.
• Derivation via Superposition: The potential at a distant point $P$ (where r >> a) is the sum of potentials from both charges:     * $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right)$     * Using geometry and binomial expansion (ignoring terms higher than the first order in $a/r$):     * $\frac{1}{r_1} \approx \frac{1}{r} (1 + \frac{a \cos(\theta)}{r})$ and $\frac{1}{r_2} \approx \frac{1}{r} (1 - \frac{a \cos(\theta)}{r}$
• Final Formula: $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos(\theta)}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2}$
• Key Differences from Point Charge:     * Dipole potential falls off as $\frac{1}{r^2}$ (faster than point charge $\frac{1}{r}$).     * It depends on the angle $\theta$ between $\mathbf{r}$ and $\mathbf{p}$.     * On the dipole axis ($\theta = 0, \pi$): $V = \pm \frac{p}{4\pi\epsilon_0 r^2}$.     * On the equatorial plane ($\theta = \pi/2$): $V = 0$.

Potential due to a System of Charges

• Superposition Principle: The total potential at a point produced by a collection of charges $q_1, q_2, …, q_n$ is the algebraic sum of individual potentials:     * $V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{r_{iP}}$
• Continuous Charge Distribution: For a distribution with density $\rho(\mathbf{r})$, potential is found by integrating:     * $V = \frac{1}{4\pi\epsilon_0} \int \frac{\rho \, dv}{r}$
• Uniformly Charged Spherical Shell:     * Outside ($r \geq R$): $V = \frac{q}{4\pi\epsilon_0 r}$ (behaves like a point charge at the center).     * Inside (r < R): Potential is constant and equal to the value at the surface, $V = \frac{q}{4\pi\epsilon_0 R}$, because the electric field inside is zero.

Equipotential Surfaces

• Definition: A surface where the electrostatic potential is constant at every point.
• Properties:     * No work is required to move a charge between two points on an equipotential surface.     * The electric field $\mathbf{E}$ is always normal to the equipotential surface at every point. This is because if there were a tangential component, work would be required to move a charge along the surface, contradicting the definition.
• Visual Representations:     * Point Charge: Concentric spheres centered on the charge.     * Uniform Electric Field: Parallel planes perpendicular to the field lines.     * Dipole/Two Positive Charges: Distorted closed surfaces (complex shapes).
• Relation Between Field and Potential:     * The magnitude of electric field is the change in potential per unit displacement normal to the equipotential surface: $E = -\frac{\delta V}{\delta l}$.     * The negative sign indicates that the electric field points in the direction of steepest potential decrease.

Potential Energy of a System of Charges

• Two-Charge System: The work done in bringing $q_1$ (work = 0) and then $q_2$ from infinity to separation $r_{12}$:     * $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$
• Three-Charge System: Sum of interactions for all pairs:     * $U = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)$
• Potential Energy in an External Field:     * Single charge $q$ in potential $V(\mathbf{r})$: $U = qV(\mathbf{r})$.     * Two charges $q_1, q_2$ in external field: $U = q_1 V(\mathbf{r}<em>1) + q_2 V(\mathbf{r}_2) + \frac{q_1 q_2}{4\pi\epsilon_0 r</em>{12}}$.
• Electron Volt (eV): The energy gained by an electron accelerated through a potential difference of 1 Volt.     * $1\,eV = 1.6 \times 10^{-19}\,J$.     * Common multiples: $1\,keV = 10^3\,eV$, $1\,MeV = 10^6\,eV$, $1\,GeV = 10^9\,eV$, $1\,TeV = 10^{12}\,eV$.

Potential Energy of a Dipole in an External Field

• Torque: In a uniform field $\mathbf{E}$, a dipole experiences torque $\tau = \mathbf{p} \times \mathbf{E} = pE \sin(\theta)$.
• Work Done to Rotate: Work to rotate from $\theta_0$ to $\theta_1$:     * $W = \int_{\theta_0}^{\theta_1} pE \sin(\theta) d\theta = pE [ \cos(\theta_0) - \cos(\theta_1) ]$
• Potential Energy Formula: By choosing $\theta_0 = \pi/2$ as the reference ($U(\pi/2) = 0$):     * $U(\theta) = -pE \cos(\theta) = -\mathbf{p} \cdot \mathbf{E}$

Electrostatics of Conductors

1. Field Inside: The electrostatic field is zero inside a conductor. Mobile charges redistribute until they cancel the external field.
2. Surface Field Direction: At the surface, the field must be normal. A tangential component would cause surface charges to move, violating the static assumption.
3. Interior Charge: No excess charge exists inside a conductor in statics. By Gauss’s Law, since $E=0$, net flux is zero, meaning net charge is zero. Excess charge resides on the surface.
4. Constant Potential: Potential is constant throughout the volume and surface of a conductor. Since $E=0$, no work is done moving charges within the conductor.
5. Surface Field Magnitude: $E = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$, where $\sigma$ is surface charge density.
6. Electrostatic Shielding: In a cavity within a conductor (with no charges inside), the electric field is zero regardless of external fields or charges on the conductor. This is used to protect sensitive equipment.

Dielectrics and Polarisation

• Nature of Dielectrics: Non-conductors with no mobile charges. An external field reduces the internal field by inducing a dipole moment but does not cancel it entirely.
• Molecules:     * Non-Polar: Centers of positive and negative charges coincide (e.g., $O_2, H_2$). Field induces a dipole moment by displacing charges slightly.     * Polar: Possess permanent dipole moments (e.g., $HCl, H_2O$). Field aligns these pre-existing dipoles despite thermal agitation.
• Polarisation (P): Dipole moment per unit volume. For linear isotropic dielectrics:     * $\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}$, where $\chi_e$ is electric susceptibility.
• Bound Charges: Polarisation creates induced surface charge densities $\pm \sigma_p$ on the dielectric boundaries, which produce an opposing field.

Capacitors and Capacitance

• Capacitor: A system of two conductors separated by an insulator (dielectric). Conductors carry charges $Q$ and $-Q$ with potential difference $V$.
• Capacitance (C): The ratio of charge to potential difference: $C = \frac{Q}{V}$.     * Unit: Farad (F); $1\,F = 1\,C/V$. Common units: $\mu F, nF, pF$.     * $C$ depends on geometry (shape, size, spacing) and the dielectric constant.
• Dielectric Strength: The maximum electric field a dielectric can withstand before breakdown; for air, it is $3 \times 10^6\,V/m$.

Parallel Plate Capacitor

• Construction: Two plates of area $A$ separated by distance $d$.
• Field: Between the plates, $E = \frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}$.
• Potential: $V = E \times d = \frac{Qd}{\epsilon_0 A}$.
• Capacitance (Vacuum): $C_0 = \frac{\epsilon_0 A}{d}$.
• Effect of Dielectric: When a dielectric of constant $K$ fills the space:     * Electric field reduces to $E/K$.     * Potential reduces to $V/K$.     * Capacitance increases: $C = K C_0 = \frac{K \epsilon_0 A}{d}$.
• Permittivity (\epsilon): $\epsilon = \epsilon_0 K$. Dielectric constant $K = \epsilon/\epsilon_0$.

Combination of Capacitors

• Series Combination:     * Charge $Q$ is the same on each capacitor.     * Total potential $V = V_1 + V_2 + … + V_n$.     * Effective capacitance: $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + … + \frac{1}{C_n}$.
• Parallel Combination:     * Potential $V$ is the same across each capacitor.     * Total charge $Q = Q_1 + Q_2 + … + Q_n$.     * Effective capacitance: $C = C_1 + C_2 + … + C_n$.

Energy Stored in a Capacitor

• Derivation: Work done in transferring charge $dq'$ at potential $V' = q'/C$:     * $dW = V' dq' = \frac{q'}{C} dq'$     * Integrating from $0$ to $Q$: $W = \frac{1}{2} \frac{Q^2}{C}$.
• Formulas: $U = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} CV^2 = \frac{1}{2} QV$.
• Energy Density (u): Energy stored per unit volume in an electric field:     * $u = \frac{1}{2} \epsilon_0 E^2$

Questions & Discussion

• Example 2.1: Calculate potential at $P$ from charge $4 \times 10^{-7}\,C$ at $9\,cm$.     * Result: $V = \frac{9 \times 10^9 \times 4 \times 10^{-7}}{0.09} = 4 \times 10^4\,V$. Work to bring $2 \times 10^{-9}\,C$ from infinity: $W = qV = 8 \times 10^{-5}\,J$. Answer is path independent.
• Example 2.2: Zero potential points for charges $3 \times 10^{-8}\,C$ and $-2 \times 10^{-8}\,C$ separated by $15\,cm$.     * Result: Potential is zero at $9\,cm$ from the positive charge (between them) and $45\,cm$ from the positive charge (on the extended line).
• Example 2.3: Field line signs for point charges.     * Key Logic: For positive charge, potential decreases with distance; work by field on positive charge moved against field is negative. Kinetic energy of a negative charge decreases when moved against repulsion.
• Example 2.4: Four charges $\pm q$ at corners of square side $d$.     * Result: Total work $W = -\frac{q^2}{4\pi\epsilon_0 d}(4 - \sqrt{2})$. Potential at center for four equal but alternating charges is zero.
• Example 2.7:     * Bird on wire: No potential difference between feet, so no shock.     * Man on ground: High potential difference between wire and ground leads to fatal current.     * Comb/Hair: Friction charges the comb; paper polarizes; humidity (rain) prevents charging.
• Example 2.10: Sharing charge between capacitors.     * Result: When a charged capacitor is connected to an uncharged identical one, potential halves, and total energy is halved. The missing energy is lost as heat and electromagnetic radiation during the transient current phase.