Comprehensive Notes on Electrostatic Potential and Capacitance

Introduction to Electrostatic Potential and Potential Energy

• Conceptual Foundation: In Class XI (Chapters 5 and 7), the concept of potential energy was established. When an external force performs work to move a body against a force (such as spring force or gravitational force), that work is stored as the potential energy of the body.

• Conservation of Energy: When the external force is removed, the body moves, losing potential energy and gaining an equal amount of kinetic energy, thereby conserving the total energy. Forces following this principle are referred to as conservative forces.

• Examples of Conservative Forces:

  • Spring force.

  • Gravitational force.

  • Coulomb force between two stationary charges.

• Similarity between Gravitational and Coulomb Forces: Both exhibit an inverse-square dependence on distance. They differ primarily in proportionality constants; masses in the gravitational law are replaced by charges in Coulomb’s law. This allows for the definition of electrostatic potential energy of a charge within an electrostatic field.

Electrostatic Potential Energy of a Test Charge

• Scenario: Consider an electrostatic field $E$ created by a charge configuration (e.g., a charge $Q$ at the origin). Imagine bringing a test charge $q$ from point $R$ to point $P$ against the repulsive force of $Q$ (assuming both $Q$ and $q$ are positive/positive or negative/negative).

• Fundamental Assumptions:

  • The test charge $q$ is infinitesimally small so it does not disturb the original charge configuration $Q$.

  • The external force applied ($F_{ext}$) just counters the repulsive electric force ($F_E$), meaning $F_{ext} = -F_E$.

  • The charge is moved with infinitesimally slow constant speed (no net force or acceleration).

• Definition of Work Done: The work done by the external force is the negative of the work done by the electric force. This work gets stored as the potential energy of charge $q$. If the external force is removed, the energy is released as kinetic energy.

• Mathematical Expression for Work (W_{RP}):     $W_{RP} = \int_R^P F_{ext} \cdot dr = -\int_R^P F_E \cdot dr$

• Potential Energy Difference (\Delta U): The work done increases the potential energy by an amount equal to the difference between potentials at points $R$ and $P$:     $\Delta U = U_P - U_R = W_{RP}$

• Path Independence: The work done by an electrostatic field depends only on the initial and final positions of the charge and is independent of the path taken. This is the hallmark of a conservative force.

• Zero Reference Point: Potential energy is undetermined to within an additive constant. Historically and practically, a convenient choice is to set potential energy to zero at infinity. If $R$ is at infinity:     $W_{\infty P} = U_P - U_{\infty} = U_P$

• Definition at a Point: The potential energy of a charge $q$ at any point is the work done by an external force (equal and opposite to the electric force) in bringing that charge from infinity to that point.

Electrostatic Potential

• Definition: Electrostatic potential ($V$) is the potential energy per unit test charge. Since work done is proportional to the charge $q$, the ratio $W/q$ is characteristic only of the electric field configuration.

• Formula for Potential Difference:     $V_P - V_R = \frac{U_P - U_R}{q} = \frac{W_{RP}}{q}$

• Definition of Potential at a Point: The electrostatic potential at any point in a region 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 and professor at Pavia.

  • Clarified that "animal electricity" observed by Luigi Galvani in frog tissues was actually generated when wet bodies were placed between dissimilar metals.

  • Developed the first voltaic pile (battery) using moist cardboard disks (electrolyte) between metal disks (electrodes).

Potential Due to a Point Charge

• Derivation: Consider a point charge $Q$ at the origin. To find the potential at point $P$ at distance $r$, calculate the work done bringing a unit positive test charge from infinity to $P$ along a radial path.

• Electrostatic Force on Unit Charge: At an intermediate point $P'$ at distance $r'$, the force is:     $F = \frac{1}{4\pi\varepsilon_0} \frac{Q}{(r')^2}$

• Integral for Potential:     $V = -\int_{\infty}^r \frac{Q}{4\pi\varepsilon_0 (r')^2} dr' = \frac{Q}{4\pi\varepsilon_0} \int_{\infty}^r -(r')^{-2} dr'$

• Final Formula for Point Charge Potential:     $V(r) = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$

• Key Observations:

  • If Q > 0, then V > 0. If Q < 0, then V < 0.

  • Potential ($V$) varies as $\frac{1}{r}$, whereas the electric field ($E$) varies as $\frac{1}{r^2}$.

Example 2.1: Calculation Practice

• (a) Problem: Calculate potential at point $P$ due to charge $4 \times 10^{-7}\,C$ located $9\,cm$ ($0.09\,m$) away.

  • Solution: $V = \frac{9 \times 10^9 \times 4 \times 10^{-7}}{0.09} = 4 \times 10^4\,V$.

• (b) Problem: Obtain work done bringing $2 \times 10^{-9}\,C$ from infinity to $P$. Does it depend on the path?

  • Solution: $W = qV = (2 \times 10^{-9}\,C)(4 \times 10^4\,V) = 8 \times 10^{-5}\,J$. The work is path-independent because any infinitesimal path can be resolved into components along and perpendicular to the radial direction (work for the latter is zero).

Potential Due to an Electric Dipole

• Dipole Properties: Consists of charges $q$ and $-q$ separated by distance $2a$. Dipole moment $p = q \times 2a$, directed from $-q$ to $q$.

• Potential via Superposition:     $V = \frac{1}{4\pi\varepsilon_0} (\frac{q}{r_1} - \frac{q}{r_2})$

• Approximation for Large Distances ($r \gg a$): Using geometry and Binomial expansion:     $V(r) = \frac{1}{4\pi\varepsilon_0} \frac{p \cdot \hat{r}}{r^2} = \frac{1}{4\pi\varepsilon_0} \frac{p \cos(\theta)}{r^2}$

  • where $\theta$ is the angle between the position vector $r$ and dipole moment $p$.

• Special Cases:

  • On Axis: $\theta = 0$ or $\theta = \pi$, so $V = \pm \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2}$.

  • Equatorial Plane: $\theta = \frac{\pi}{2}$, so $V = 0$.

• Comparison with Single Charge Potential:

  • Dipole potential depends on distance ($r$) and angle ($\theta$); point charge potential depends only on $r$.

  • Dipole potential falls as $\frac{1}{r^2}$; point charge potential falls as $\frac{1}{r}$.

Potential Due to a System of Charges and Continuous Distributions

• Discrete Charges: For charges $q_1, q_2, ..., q_n$ at distances $r_{1P}, r_{2P}, ..., r_{nP}$ from point $P$:     $V = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^n \frac{q_i}{r_{iP}}$

• Continuous Charge Distribution: Defined by charge density $\rho(r)$. Potential is found by integrating over volume elements $dv$ carrying charge $\rho dv$:     $V = \int \frac{\rho dv}{4\pi\varepsilon_0 r}$

• Uniformly Charged Spherical Shell (Radius $R$, Total Charge $q$):

  • Outside ($r \geq R$): $V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$ (acts as if charge is concentrated at center).

  • Inside (r < R): Since $E = 0$ inside, potential is constant and equal to its value at the surface: $V = \frac{1}{4\pi\varepsilon_0} \frac{q}{R}$.

Potential Zero Points: Example 2.2

• Problem: Charges $3 \times 10^{-8}\,C$ and $-2 \times 10^{-8}\,C$ are $15\,cm$ apart. Where is $V = 0$?

• Solution:

  • Between charges: Let $P$ be at distance $x$ from the positive charge. $\frac{3}{x} - \frac{2}{15-x} = 0 \implies x = 9\,cm$.

  • Outside charges: Let $P$ be at distance $x$ on the extended line. $\frac{3}{x} - \frac{2}{x-15} = 0 \implies x = 45\,cm$.

Equipotential Surfaces

• Definition: A surface where the potential has a constant value at all points.

• Examples:

  • Single Point Charge: Concentric spheres centered on the charge.

  • Uniform Electric Field: Planes perpendicular to the field lines.

  • Dipole/Identical Charges: Surfaces appear as distorted ellipsoids or separate lobes.

• Normal Property: The electric field is always normal to the equipotential surface at every point.

  • Proof: If the field had a tangential component, work would be required to move a charge along the surface, contradicting the definition that $\Delta V = 0$.

• Relation to Field Strength: Equipotential surfaces are closer together in regions of strong electric fields.

Relation Between Field (E) and Potential (V)

• Mathematical Relation: Considering two surfaces $A$ and $B$ with potentials $V$ and $V + \delta V$ separated by distance $\delta l$:     $|E| = -\frac{\delta V}{\delta l}$

• Key Conclusions:

  1. Electric field points in the direction where potential decreases most steeply.

  2. The magnitude of the field is equal to the change in potential per unit displacement normal to the equipotential surface.

Potential Energy of a System of Charges

• Two-Charge System: Work done to bring $q_1$ from infinity (zero work) plus work to bring $q_2$ against the field of $q_1$:     $U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}}$

• Three-Charge System: Total work is the sum of interaction pairs:     $U = \frac{1}{4\pi\varepsilon_0} (\frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}})$

• Example 2.4: Four charges at square corners ($d$):

  • Total Work = $\frac{q^2}{4\pi\varepsilon_0 d} (-4 + \sqrt{2})$.

  • Work to bring charge $q_0$ to center: Zero, because the potential at the geometric center of such a symmetric arrangement is zero.

Potential Energy in an External Field

• Single Charge: If the external potential is $V(r)$, the potential energy of charge $q$ is:     $U = qV(r)$

• The Electron Volt (eV): The energy gained by an electron ($1.6 \times 10^{-19}\,C$) accelerated by $1\,Volt$.

  • $1\,eV = 1.6 \times 10^{-19}\,J$.

  • $1\,keV = 10^3\,eV$, $1\,MeV = 10^6\,eV$, $1\,GeV = 10^9\,eV$, $1\,TeV = 10^{12}\,eV$.

• Two Charges in External Field: Sum of their individual potential energies and their mutual interaction energy:     $U = q_1 V(r_1) + q_2 V(r_2) + \frac{q_1 q_2}{4\pi\varepsilon_0 r_{12}}$

Potential Energy of a Dipole in a Uniform External Field

• Torque: $\tau = p \times E$.

• Work/Potential Energy: Work done in rotating the dipole from $\theta_0$ to $\theta_1$:     $W = pE (\cos(\theta_0) - \cos(\theta_1))$

• Standard Definition ($\theta_0 = \frac{\pi}{2}$):     $U(\theta) = -p \cdot E = -pE \cos(\theta)$

• Alignment Heat Release (Example 2.6): 1 mole ($6 \times 10^{23}$ molecules) with dipole moment $10^{-29}\,C\cdot m$ in field $10^6\,V/m$ rotated $60^{\circ}$. Heat released = $3\,J$.

Electrostatics of Conductors

1. Inside a conductor, E = 0: Mobile electrons drift until they create an internal field that perfectly cancels the external field.

2. E is normal to the surface: Tangential components would cause surface charges to move, violating the static condition.

3. No excess charge inside: Follows from Gauss's Law; flux through any internal surface is zero, thus net interior charge is zero. Excess charge resides only on the surface.

4. Potential is constant throughout (and on surface): Since $E=0$ inside, no work is done moving a charge between any two points within or on the conductor.

5. Surface Electric Field Magnitude:     $E = \frac{\sigma}{\varepsilon_0} \hat{n}$

   • where $\sigma$ is surface charge density and $\hat{n}$ is the outward normal vector.

6. Electrostatic Shielding: In a cavity within a conductor, the electric field is zero regardless of outside charges or fields. This protects sensitive instruments.

Dielectrics and Polarisation

• Dielectrics: Non-conductors with negligible charge carriers. They do not cancel external fields but reduce them through molecular polarisation.

• Molecular Types:

  • Non-polar: Centers of positive and negative charges coincide (e.g., $O_2, H_2$). They develop an induced dipole moment in external fields.

  • Polar: Possess permanent dipole moments (e.g., $H_2O, HCl$). In an external field, these dipoles tend to align.

• Polarisation ($P$): Total dipole moment per unit volume. For linear isotropic dielectrics:     $P = \varepsilon_0 \chi_e E$

  • where $\chi_e$ is electric susceptibility.

• Modified Field: Polarisation induces surface charge densities ($\pm \sigma_p$) that produce an internal field opposing the external field, resulting in a reduced net field.

Capacitors and Capacitance

• Capacitor: A system of two conductors separated by an insulator. One conductor has charge $Q$, the other $-Q$, with potential difference $V$.

• Capacitance ($C$): The constant ratio of charge to potential difference:     $C = \frac{Q}{V}$

• Units: Farad ($F$). $1\,F = 1\,C/V$. Common units are $\mu F (10^{-6}), nF (10^{-9}), pF (10^{-12})$.

• Dielectric Strength: The maximum electric field a medium can withstand before breaking down (insulating property fails). For air, this is approx $3 \times 10^6\,V/m$.

The Parallel Plate Capacitor

• Structure: Two plates of area $A$ separated by distance $d$.

• Field between plates ($d^2 \ll A$): $E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$.

• Potential Difference: $V = Ed = \frac{Qd}{\varepsilon_0 A}$.

• Capacitance (Vacuum):     $C_0 = \frac{\varepsilon_0 A}{d}$

• Dielectric Effect: If a dielectric with constant $K$ fills the space, potential drops and capacitance increases:     $C = K C_0 = \frac{\varepsilon K A}{d} = \frac{\varepsilon_0 K A}{d}$

• Dielectric Constant ($K$): Ratio $C/C_0$. Always > 1 for a dielectric.

Combination of Capacitors

• Series Combination: The charge $Q$ is the same on all capacitors. Total potential is sum of individuals.     $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$

• Parallel Combination: The potential $V$ is the same across all capacitors. Total charge is sum of individuals.     $C_{eq} = C_1 + C_2 + ... + C_n$

Energy Stored in a Capacitor

• Work of Charging: transferring charge bit-by-bit from one plate to another requires work against the growing potential difference.

• Energy Formulae:     $U = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$

• Energy Density ($u$): Energy stored per unit volume ($Ad$) in the electric field:     $u = \frac{1}{2} \varepsilon_0 E^2$

• Example 2.11: Charging a $600\, pF$ capacitor to $200\,V$ then sharing charge with another uncharged $600\, pF$ capacitor results in a $50\%$ loss of electrostatic energy, dissipated as heat or electromagnetic radiation during the transient current flow.

Physical Quantities Summary Table

• Potential (V): Dimension $[M^1 L^2 T^{-3} A^{-1}]$, Unit: $V$. (Difference is physically significant).

• Capacitance (C): Dimension $[M^{-1} L^{-2} T^{-4} A^2]$, Unit: $F$.

• Polarisation (P): Dimension $[L^{-2} AT]$, Unit: $C\,m^{-2}$. (Dipole moment per unit volume).

• Dielectric Constant (K): Dimensionless.

Points to Ponder

• Electrostatic potential at a charge's own location is infinite and ill-defined.

• Electric field is discontinuous across a shell surface ($0$ to $\sigma/\varepsilon_0$), but potential is continuous ($Q/4\pi\varepsilon_0 R$).

• Capacitors are designed to confine field lines to a small volume of space.

• Shielding works from the outside in; charges inside a cavity can still influence the exterior.

Questions & Discussion

• Q: Why does a comb run through dry hair attract paper?

  • A: Friction charges the comb; the comb polarizes paper molecules, creating attraction. Wet hair reduces friction, preventing the charge.

• Q: Why are aircraft tires slightly conducting?

  • A: To bleed off static electricity accumulated during flight/friction, preventing sparks and fires.

• Q: Why can a bird perch on a high power line safely?

  • A: There is no potential difference across the bird's body. A man on the ground creates a path to a lower potential (ground), causing a fatal current.