Class XII Physics: Electrostatic Potential and Capacitance Study Guide

Introduction to Electrostatic Potential and Conservative Forces

• Conservative Forces: When an external force performs work in moving a body between two points against a force like a spring force or gravitational force, that work is stored as potential energy.     * Upon removal of the external force, the body moves, losing potential energy and gaining an equal amount of kinetic energy.     * The sum of kinetic and potential energies remains conserved in these systems.     * Examples include the spring force, gravitational force, and the Coulomb force between two stationary charges.
• Comparison of Forces: Both gravitational and Coulomb forces exhibit an inverse-square dependence on distance. They differ primarily in their proportionality constants: masses in the gravitational law are replaced by charges in Coulomb’s law.
• Electrostatic Potential Energy: Consider an electrostatic field $\mathbf{E}$ due to a charge configuration. To define potential energy, we imagine bringing a test charge $q$ from a point $R$ to a point $P$ against the repulsive force of a charge $Q$ at the origin.     * Assumptions:         1. The test charge $q$ is sufficiently small that it does not disturb the original configuration of $Q$.         2. An external force $\mathbf{F}_{ext}$ is applied just enough to counter the electric force $\mathbf{F}_E$ (i.e., $\mathbf{F}_{ext} = -\mathbf{F}_E$), meaning the charge moves with infinitesimally slow constant speed (no acceleration).
• Work and Potential Energy Formula: The work done by the external force is the negative of the work done by the electric force:     $\mathbf{W}_{RP} = \int_{R}^{P} \mathbf{F}_{ext} \cdot d\mathbf{r} = -\int_{R}^{P} \mathbf{F}_E \cdot d\mathbf{r}$     * This work is stored as potential energy, specifically the difference between points $R$ and $P$:     $\Delta U = U_P - U_R = W_{RP}$
• Path Independence: A fundamental characteristic of conservative forces is that the work done depends only on the initial and final positions, not on the path taken. This path-independence can be proved using Coulomb’s law.
• Zero Reference Point: Potential energy values are determined to within an additive constant. To define absolute potential energy at a point, a reference point where potential energy is zero must be chosen. In electrostatics, this is conveniently chosen at infinity (r = $\infty$).     * Equation for potential energy at point $P$:     $W_{\infty P} = U_P - U_{\infty} = U_P$

Electrostatic Potential (V)

• Definition: If we divide the potential energy of a test charge by the magnitude of that charge, the resulting quantity is independent of the test charge's magnitude and characteristic only of the electric field configuration. This is the electrostatic potential ($V$).
• Potential Difference: The work done by an external force in bringing a unit positive charge from point $R$ to $P$ is:     $V_P - V_R = \frac{U_P - U_R}{q} = \frac{W_{RP}}{q}$
• Absolute Potential: The electrostatic potential at any point is the work done in bringing a unit positive charge (without acceleration) from infinity to that point.
• Historical Context: Count Alessandro Volta (1745–1827): An Italian physicist who established that the "animal electricity" observed by Luigi Galvani in frog tissues was actually generated when any wet body was placed between dissimilar metals. This led to the development of the first battery, the voltaic pile, made of cardboard disks (electrolyte) between metal disks (electrodes).

Potential due to a Point Charge

• Consider a point charge $Q$ at the origin. To find the potential at a point $P$ with position vector $\mathbf{r}$, we calculate the work done to bring a unit positive charge from infinity.
• Force on unit charge: $\mathbf{F} = \frac{1}{4̀\pi\epsilon_0} \frac{Q}{r'^2} \hat{\mathbf{r}}'$
• Integration for Work:     $W = -\int_{\infty}^{r} \frac{Q}{4\pi\epsilon_0 r'^2} dr' = \frac{Q}{4\pi\epsilon_0 r}$
• Expression for Potential:     $V(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$
• Characteristics:     * For $Q > 0$, $V > 0$ (work done is positive).     * For $Q < 0$, $V < 0$ (work done is negative).     * Variation: Potential scales as $1/r$, whereas the electric field scales as $1/r^2$.

Potential due to an Electric Dipole

• An electric dipole consists of charges $q$ and $-q$ separated by a distance $2a$. The dipole moment vector $\mathbf{p}$ has magnitude $q \times 2a$ and points from $-q$ to $q$.
• Superposition Principle: The potential at point $P$ is the sum of potentials from both charges:     $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right)$
• Geometry for r >> a: Using binomial expansion and retaining first-order terms in $a/r$:     $r_1 \cong r(1 - \frac{a}{r}\cos(\theta))$$r_2 \cong r(1 + \frac{a}{r}\cos(\theta))$
• Final Result:     $V = \frac{1}{4\pi\epsilon_0} \frac{p\cos(\theta)}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2}$
• Distinction from Point Charge Potential:     1. Dipole potential depends on the angle $\theta$ between the position vector and the dipole moment.     2. Dipole potential falls off as $1/r^2$ at large distances, faster than the $1/r$ fall-off of a point charge.     3. Potential in the equatorial plane ($\theta = \pi/2$) is zero.

Potential due to a System of Charges and Continuous Distributions

• Superposition: For charges $q_1, q_2, \dots, q_n$ at distances $r_{1P}, r_{2P}, \dots, r_{nP}$, the potential is:     $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1}{r_{1P}} + \frac{q_2}{r_{2P}} + \dots + \frac{q_n}{r_{nP}} \right)$
• Continuous Distribution: For a charge density $\rho(\mathbf{r})$, potential is found by integrating over volume elements:     $V = \frac{1}{4\pi\epsilon_0} \int \frac{\rho dV}{r}$
• Uniformly Charged Spherical Shell:     * Outside (r ≥ R): $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$. The shell behaves as if all charge is concentrated at the center.     * Inside (r < R): The electric field is zero, so no work is done moving a charge inside. Potential is 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 the potential has a constant value at all points.
• Work Done: No work is required to move a charge between two points on an equipotential surface.
• Relationship to Electric Field:     * The electric field is always normal to the equipotential surface at every point.     * If the field were not normal, it would have a tangential component requiring work to move a charge, contradicting the definition.
• Geometric Forms:     * Point charge: Concentric spherical surfaces.     * Uniform field ($E$ along x-axis): Planes parallel to the y-z plane.
• Mathematical Relation: $E = -\frac{\delta V}{\delta l}$.     * Conclusion 1: Electric field points in the direction where potential decreases steepest.     * Conclusion 2: Field magnitude equals the change in potential per unit displacement normal to the surface.

Potential Energy of a System of Charges

• Two Charge System: Work to bring $q_1$ from infinity to $\mathbf{r}_1$ is zero. Potential at $\mathbf{r}_2$ due to $q_1$ is $V_1 = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_{12}}$. Work to bring $q_2$ is $q_2 V_1$.     * Total energy: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.
• Three Charge System: $U = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)$.
• Sign of Energy:     * If $q_1 q_2 > 0$ (like charges), energy is positive (repulsive).     * If $q_1 q_2 < 0$ (unlike charges), energy is negative (attractive).

Potential Energy in an External Field

• Single Charge: Potential energy of charge $q$ at position $\mathbf{r}$ in external potential $V(\mathbf{r})$ is $U = qV(\mathbf{r})$.
• Electron Volt (eV): The energy gained by an electron ($q = 1.6 \times 10^{-19}\,C$) accelerated by $1\,V$.     * $1\,eV = 1.6 \times 10^{-19}\,J$.     * Derived units: $1\,keV = 10^3\,eV$; $1\,MeV = 10^6\,eV$; $1\,GeV = 10^9\,eV$; $1\,TeV = 10^{12}\,eV$.
• Two Charges in External Field:     $U = q_1 V(\mathbf{r}_1) + q_2 V(\mathbf{r}_2) + \frac{q_1 q_2}{4\pi\epsilon_0 r_{12}}$
• Dipole in Uniform External Field:     * Experiences torque \mathbf{̀̀̄\tau} = \mathbf{p} \times \mathbf{E}.     * Work done to rotate from $\theta_0$ to $\theta_1$: $W = pE(\cos(\theta_0) - \cos(\theta_1))$.     * Choosing $\theta_0 = \pi/2$ as the zero-energy reference: $U(\theta) = -pE\cos(\theta) = -\mathbf{p} \cdot \mathbf{E}$.

Electrostatics of Conductors

• Mobile Charge Carriers: In metals, these are free electrons forming a "gas." In electrolytes, they are positive and negative ions.
• Six Key Properties of Conductors:     1. Field inside is zero: Free charges redistribute until internal field cancels any external or induced field.     2. Field at surface is normal: A tangential component would cause charges to move on the surface.     3. No excess charge in the interior: Excess charge resides only on the surface (derived from Gauss's law where field inside is zero).     4. Constant Potential: Since $E = 0$ inside, no work is done moving charges within the conductor; thus potential is constant throughout the volume.     5. Surface Field Magnitude: $E = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$.     6. Electrostatic Shielding: The field inside a cavity in a conductor is zero, regardless of external charges or shell shape. This protects sensitive instruments.

Dielectrics and Polarisation

• Dielectrics: Non-conducting substances without free charge carriers.
• Polar vs. Non-polar Molecules:     * Non-polar: Centers of positive and negative charges coincide (e.g., $O_2$, $H_2$). No permanent dipole moment.     * Polar: Charges are separated even without a field (e.g., $HCl$, $H_2O$). Permanent dipole moment exists.
• Polarisation ($P$): The net dipole moment per unit volume developed in an external field.     * Linear isotropic dielectrics follow: $\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}$, where $\chi_e$ is electric susceptibility.
• Effect on Field: Induced surface charges create an opposing field which reduces the net electric field inside the dielectric.

Capacitors and Capacitance

• Definition: A system of two conductors separated by an insulator.
• Formula: $Q = CV$ where $C$ is capacitance.
• Units: Farad ($F$). $1\,F = 1\,C V^{-1}$. Common units: $\mu F$, $nF$, $pF$.
• Dielectric Strength: The maximum electric field a medium can withstand without breakdown. For air, this is $3 \times 10^6\,V m^{-1}$.
• Parallel Plate Capacitor:     * Inner field: $E = \frac{\sigma}{\epsilon_0} = \frac{Q}{\epsilon_0 A}$.     * Potential difference: $V = Ed = \frac{Qd}{\epsilon_0 A}$.     * Capacitance: $C = \frac{\epsilon_0 A}{d}$.
• Effect of Dielectric on Capacitance: Inserting a dielectric with dielectric constant $K$ reduces the electric field to $E/K$ and increases capacitance:     $C = K C_0 = \frac{\epsilon A}{d}$     where $\epsilon = \epsilon_0 K$ is the permittivity of the medium.

Combinations of Capacitors

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

Energy Stored in a Capacitor

• Work Done: Work is done to transfer charge from one plate to another.
• Formula:     $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$
• Energy Density ($u$): Energy stored per unit volume of the electric field.     * $u = \frac{1}{2}\epsilon_0 E^2$
• Energy Loss: When two capacitors are connected, energy is lost as heat or electromagnetic radiation during the transient current period while potential equalizes.

Numerical Examples and Discussion

• Example 2.1: A charge of $4 \times 10^{-7}\,C$ at $9\,cm$ gives potential $\frac{9 \times 10^9 \times 4 \times 10^{-7}}{0.09} = 4 \times 10^4\,V$. Work for $2 \times 10^{-9}\,C$ is $8 \times 10^{-5}\,J$.
• Example 2.4: Four charges $\pm q$ on corners of a square (side $d$). Total work to assemble: $W = \frac{-q^2}{4\pi\epsilon_0 d}(4 - \sqrt{2})$. Potential at the center is zero.
• Example 2.9: Network of four $10\,\mu F$ capacitors. Three in series ($10/3\,\mu F$) in parallel with one ($10\,\mu F$). Equivalent capacitance = $13.3\,\mu F$.
• Questions & Discussion:     * Comb and Paper bits: Combs get charged by friction, polarizing paper molecules. Wet hair reduces friction, thus preventing charging.     * Aircraft tires: Made slightly conducting to dissipate static charge buildup from friction during landing/take-off to prevent sparks.     * Bird on line: A bird doesn't get shocked because there is no potential difference between its feet. A human on the ground creates a path for current due to the potential difference between the line and the Earth.

Points to Ponder

• Confined Field: Capacitors are configured to confine electric field lines to small regions.
• Self-Potential: Potential of a charge at its own location is undefined (infinite).
• Discontinuity: Electric field is discontinuous across a shell surface ($0$ inside, $\sigma/\epsilon_0$ outside), but potential is continuous.
• Damping: Torque on a dipole causes oscillation; it only aligns with the field if dissipative/damping mechanisms are present.