Electrostatic Potential and Capacitance - Exhaustive Study Notes

Introduction to Electrostatic Potential and Conservative Forces

• Concept of Potential Energy: In mechanics, when an external force does work against a conservative force (like a spring force or gravity), the work is stored as potential energy ($U$). When the external force is removed, the body gains kinetic energy at the expense of potential energy, keeping the sum of the two constant.

• Conservative Forces: The Coulomb force between stationary charges is a conservative force, similar to gravitational force, as both follow an inverse-square dependence on distance. The primary difference lies in the proportionality constants (masses vs. charges).

• Work and Electrostatic Field: If a test charge $q$ is moved from point $R$ to point $P$ in an electrostatic field $\mathbf{E}$ produced by a charge configuration (e.g., a charge $Q$ at the origin):

  • The external force ($\mathbf{F}_{ext}$) applied must be just enough to counter the electric force ($\mathbf{F}_E$), meaning $\mathbf{F}_{ext} = -\mathbf{F}_E$.

  • To avoid acceleration, the charge is moved at an infinitesimally slow constant speed.

  • The work done by the external force is: $W_{RP} = \int_R^P \mathbf{F}_{ext} \cdot d\mathbf{r} = -\int_R^P \mathbf{F}_E \cdot d\mathbf{r}$.

  • This work is stored as the potential energy difference between the two points: $\Delta U = U_P - U_R = W_{RP}$.

• Path Independence: The work done by an electrostatic field depends only on the initial and final positions, not on the path taken. This path-independence is the fundamental characteristic of conservative forces.

• Reference Point for Potential Energy: The actual value of potential energy is not physically significant; only the difference is. It is conventional to choose the potential energy to be zero at infinity. Therefore, the potential energy of a charge $q$ at a point $P$ is the work done by an external force to bring it from infinity to that point.

Electrostatic Potential (V)

• Definition: Electrostatic potential is the work done per unit test charge in bringing a unit positive charge from infinity to a point without acceleration.

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

• Significance: Potential is a characteristic of the electric field specifically, independent of the test charge $q$ used to measure it.

• Historical Note - Count Alessandro Volta (1745–1827): An Italian physicist who established that "animal electricity" (observed by Luigi Galvani) was actually generated by the contact of dissimilar metals with a moist body. He developed the first voltaic pile (battery) using moist cardboard disks (electrolyte) between metal electrodes.

Potential due to a Point Charge

• Derivation: For a point charge $Q$ at the origin, the potential at a point $P$ at distance $r$ is calculated by integrating the work done on a unit positive charge from infinity to $r$.

• Force on unit charge at $r'$: $\mathbf{E} = \frac{Q}{4\pi\epsilon_0 (r')^2} \hat{\mathbf{r}}'$.

• Work Done (Integration): $W = -\int_{\infty}^r \frac{Q}{4\pi\epsilon_0 (r')^2} dr' = \frac{Q}{4\pi\epsilon_0 r}$.

• Potential Formula: $V(r) = \frac{Q}{4\pi\epsilon_0 r}$.

• Observations:

  • If Q > 0, then V > 0 (work done against repulsion is positive).

  • If Q < 0, then V < 0 (work done by external force is negative as the field attracts the charge).

  • $V \propto 1/r$, whereas the electric field $E \propto 1/r^2$.

Potential due to an Electric Dipole

• Dipole Configuration: Two charges $q$ and $-q$ separated by a distance $2a$, with dipole moment $\mathbf{p} = q \times 2\mathbf{a}$.

• Superposition Principle: The potential at a point $P$ is the algebraic sum of the potentials due to individual charges:

  • $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right)$, where $r_1$ and $r_2$ are distances from $q$ and $-q$.

• General Formula (r >> a): At large distances, the potential is given by $V(r) = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2} = \frac{p \cos(\theta)}{4\pi\epsilon_0 r^2}$.

• Special Cases:

  • On the dipole axis ($\theta = 0$ or $\theta = \pi$): $V = \pm \frac{p}{4\pi\epsilon_0 r^2}$.

  • On the equatorial plane ($\theta = \pi/2$): $V = 0$.

• Comparison with Point Charge: Dipole potential falls off as $1/r^2$, compared to $1/r$ for a point charge. It also depends on the angle $\theta$ but is axially symmetric about $\mathbf{p}$.

Potential due to a System of Charges

• Calculation: For charges $q_1, q_2, \dots, q_n$ at distances $r_{1P}, r_{2P}, \dots, r_{nP}$ from point $P$, the total potential is: $V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{r_{iP}}$.

• Continuous Charge Distribution: For a distribution with density $\rho(\mathbf{r})$, the potential is found by integrating over the volume: $V = \int \frac{\rho \, dv}{4\pi\epsilon_0 r}$.

• Uniformly Charged Spherical Shell:

  • Outside the shell ($r \geq R$): $V = \frac{q}{4\pi\epsilon_0 r}$.

  • Inside the shell (r < R): Potential is constant and equals the value at the surface, $V = \frac{q}{4\pi\epsilon_0 R}$, since $E = 0$ inside.

Equipotential Surfaces

• Definition: A surface where the potential has a constant value at every point.

• Examples:

  • Single point charge: Concentric spherical surfaces.

  • Uniform electric field: Planes normal to the field lines.

• Key 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; otherwise, there would be a tangential component, requiring work to be done.

  • Relation between Field and Potential: $E = -\frac{\delta V}{\delta l}$.

  • Conclusion: The electric field points in the direction where the potential decreases most steeply, and its magnitude is the change in potential per unit displacement normal to the surface.

Potential Energy of a System of Charges

• Two-Charge System: The work done to bring $q_1$ to $\mathbf{r}_1$ is zero. Then, the work done to bring $q_2$ to $\mathbf{r}_2$ in the field of $q_1$ is $q_2 V_1(\mathbf{r}_2)$.

  • $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.

• Three-Charge System: Sum of work for each pair added successively:

  • $U = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)$.

Potential Energy in an External Field

• Single Charge: If an external potential $V(\mathbf{r})$ is specified (from unknown sources), the potential energy of charge $q$ is $U = qV(\mathbf{r})$.

  • Electron Volt (eV): The energy gained by an electron ($q = 1.6 \times 10^{-19}\,C$) when accelerated by a potential difference of $1\,V$. $1\,eV = 1.6 \times 10^{-19}\,J$.

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

• Two-Charge System 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:

  • Torque: $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$.

  • Potential Energy: $U(\theta) = -\mathbf{p} \cdot \mathbf{E} = -pE \cos(\theta)$.

  • This is minimum ($-pE$) when $\mathbf{p}$ is parallel to $\mathbf{E}$ and maximum ($+pE$) when anti-parallel.

Electrostatics of Conductors

1. Field Inside: Electrostatic field is zero inside a conductor ($E = 0$).

2. Field at Surface: Electrostatic field must be normal to the surface at every point. Any tangential component would cause charges to move.

3. Net Charge: There is no net charge in the interior; excess charge resides only on the outer surface.

4. Constant Potential: The electrostatic potential is constant throughout the volume and surface of a conductor.

5. Surface Field Magnitude: The electric field at the surface of a charged conductor is $\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{\mathbf{n}}$.

6. Electrostatic Shielding: The field inside a cavity within a conductor is zero, regardless of the size, shape, or external field. This is used to protect sensitive instruments (Faraday cage effect).

Dielectrics and Polarisation

• Dielectrics: Non-conductors where an external field induces dipole moments rather than free charge movement.

  • Non-polar molecules: Centers of positive and negative charges coincide (e.g., $O_2, H_2$). An external field induces a dipole moment by displacing the charges.

  • Polar molecules: Have a permanent dipole moment due to charge separation (e.g., $H_2O, HCl$). An external field tends to align these random dipoles.

• Polarisation ($\mathbf{P}$): Defined as the dipole moment per unit volume. For linear isotropic dielectrics, $\mathbf{P} = \chi_e \mathbf{E}$, where $\chi_e$ is the electric susceptibility.

• Reduction of Field: Induced surface charges ($\pm\sigma_p$) create an opposing field that reduces the net electric field inside the dielectric.

• Electric Displacement (D): Defined as $\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}$. This vector is directly related to the free charge density $\sigma$.

Capacitors and Capacitance

• Capacitor: A system of two conductors separated by an insulator. It stores charge and energy.

• Capacitance (C): The ratio of charge $Q$ on one plate to the potential difference $V$: $C = Q/V$.

  • Unit: Farad ($F$). $1\,F = 1\,C/V$. Sub-multiples: $1\,\mu F = 10^{-6}\,F$, $1\,nF = 10^{-9}\,F$, $1\,pF = 10^{-12}\,F$.

• Parallel Plate Capacitor (Vacuum): $C_0 = \frac{\epsilon_0 A}{d}$ where $A$ is plate area and $d$ is separation.

• Dielectric Strength: The maximum electric field a dielectric can withstand before breakdown ($3 \times 10^6\,V m^{-1}$ for air).

Effect of Dielectrics and Combinations

• Dielectric Constant (K): When a dielectric fills the space, capacitance increases by factor $K$: $C = K C_0$. Here, $K = \frac{\epsilon}{\epsilon_0}$, where $\epsilon$ is the permittivity of the medium.

• Capacitors in Series:

  • Charges ($Q$) are the same across all capacitors.

  • $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$.

• Capacitors in Parallel:

  • Potential difference ($V$) is the same across all capacitors.

  • $C_{eq} = C_1 + C_2 + \dots + C_n$.

Energy Stored in a Capacitor

• Work Done in Charging: The total work required to charge a capacitor to $Q$ is stored as electrostatic potential energy:

  • $U = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} CV^2 = \frac{1}{2} QV$.

• Energy Density ($u$): Energy stored per unit volume in a parallel plate capacitor:

  • $u = \frac{1}{2} \epsilon_0 E^2$.

• Energy Loss in Sharing: When a charged capacitor is connected to an uncharged one, some energy is lost as heat and electromagnetic radiation due to transient charging currents, even if no total charge is lost.

Example Summaries

• Example 2.1: Calculating potential from a $4 \times 10^{-7}\,C$ charge at $9\,cm$ distance gives $4 \times 10^4\,V$. Moving a $2 \times 10^{-9}\,C$ charge to this point results in $8 \times 10^{-5}\,J$ of work, which is path-independent.

• Example 2.2: For charges of $3 \times 10^{-8}\,C$ and $-2 \times 10^{-8}\,C$ located $15\,cm$ apart, potential is zero at $9\,cm$ and $45\,cm$ from the positive charge along the line.

• Example 2.4: Work needed to assemble four charges \pm q on corners of a square of side $d$ is $U = \frac{-q^2}{4\pi\epsilon_0 d} (4 - \sqrt{2})$. Potential at the center is zero.

• Example 2.5: Interaction energy of two charges $7\,\mu C$ and $-2\,\mu C$ at $18\,cm$ separation is $-0.7\,J$. In an external field $E = A/r^2$, the total energy must account for the separate interactions of each charge with the external potential.

• Example 2.10: A $900\,pF$ capacitor charged to $100\,V$ stores $4.5 \times 10^{-6}\,J$. When connected to an identical uncharged capacitor, the shared potential becomes $50\,V$, and the total system energy drops to $2.25 \times 10^{-6}\,J$ (50% loss).

Practical Conceptual Applications

• Dry vs. Wet Hair: A comb run through dry hair attracts paper because of frictional charging. On rainy days or with wet hair, friction is reduced, and charge leaks away through moist air.

• Aircraft Tyres: Made slightly conducting to dissipate static electricity accumulated during friction with the runway to prevent sparks and fire.

• Fuel Trucks: Use metallic chains/ropes touching the ground to discharge accumulated static charge during motion.

• Birds on High Power Lines: A bird sitting on one line is at the same potential as the wire. No potential difference means no current flows through the bird. A man on the ground completes a circuit between the high potential line and zero potential ground, causing a shock.