Electrostatic Potential and Capacitance Study Guide

Electrostatic Potential

• Definition: The electrostatic potential at a point in an electric field is defined as the work done by an external force in bringing a unit positive test charge from infinity to that point without any acceleration.
• Mathematical Expression: $V = \frac{\text{Work done}(W)}{\text{Charge}(q_0)}$
• Unit: The SI unit of electrostatic potential is the volt ($V$), where $1\,V = 1\,J/C$.
• Dimensional Formula: The dimensional formula for potential is $[ML^2T^{-3}A^{-1}]$.
• Nature: It is a scalar quantity.
• Path Independence: Electrostatic potential is a path-independent (state) function. This is because electrostatic forces are conservative forces.
• Potential Difference:
  • Defined as the amount of work done in moving a unit positive test charge from one point to another against the electrostatic force without any acceleration.
  • Formula: $V_B - V_A = \frac{W_{AB}}{q_0} = -\int_A^B \mathbf{E} \cdot d\mathbf{l}$
  • Scorify Note: The negative line integral of the electric field from the initial position to the final position gives the potential difference.

Potential due to Specific Charge Distributions

• Potential due to a Point Charge:
  • At any point $P$ lying at a distance $r$ from a point charge $q$, the potential is given by $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$.
  • Potential due to a positive charge is positive ($V > 0$).
  • Potential due to a negative charge is negative ($V < 0$).
• Potential due to an Electric Dipole:
  • For a dipole with dipole moment $p$, the potential at a distance $r$ from the center of the dipole at an angle $\theta$ is given by $V = \frac{1}{4\pi\epsilon_0} \frac{p \cos(\theta)}{r^2}$.
  • Alternatively, expressed using the dot product: $V = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \mathbf{\hat{r}}}{r^2}$.
  • This formula is strictly valid for short dipoles where $r \gg a$ (where $2a$ is the dipole length).
• Potential due to a System of Charges:
  • According to the superposition principle, the electrostatic potential at any point $P$ due to $n$ point charges is the algebraic sum of the individual potentials.
  • Formula: $V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{|\mathbf{r} - \mathbf{r}_i|}$

Equipotential Surfaces

• Definition: A surface which has the same electrostatic potential at every point is known as an equipotential surface.
• Shapes for Different Fields:
  • Point Charge: The equipotential surfaces are concentric spherical surfaces.
  • Uniform Field: The equipotential surfaces are parallel planes perpendicular to the field lines.
• Key Properties:
  • Work Done: No work is done in moving a test charge on an equipotential surface ($W = q\Delta V = 0$).
  • Electric Field Direction: The electric field is always normal to the equipotential surface at every point.
  • Intersection: Two equipotential surfaces can never intersect.
• Relation Between Electric Field and Potential:
  • The relationship is given by $E = -\frac{dV}{dr}$ or $\mathbf{E} = -\nabla V$.
  • The negative sign indicates that the electric field acts in the direction of the steepest decrease in potential.
  • Drawing Tip: When asked to draw equipotential surfaces, always ensure the electric field lines are drawn perpendicular to the surfaces.

Energy in an External Field

• Single Point Charge: In an external field $V(\mathbf{r})$, the energy for a single charge is $U = qV(\mathbf{r})$.
• Two-Charge System: The total potential energy consists of the energy due to the external field and the mutual interaction energy:     $U = q_1V(\mathbf{r}_1) + q_2V(\mathbf{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r_{12}}$
• Note on Potential Energy of Two Charges (No external field): The total potential energy is simply the work done to bring them from infinity to their locations: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r_{12}}$.

Dipole in an External Field

• Potential Energy of a Dipole: At an angle $\theta$ in a uniform electric field $E$, the potential energy is $U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos(\theta)$.
• Work Done in Rotating an Electric Dipole:
  • The work done ($W$) in rotating a dipole of moment $p$ in a uniform field $E$ from an initial angle $\theta_1$ to a final angle $\theta_2$ is $W = pE(\cos(\theta_1) - \cos(\theta_2))$.
• Special Conditions:
  • Case I: Rotation from stable to unstable equilibrium: From $\theta_1 = 0^{\circ}$ to $\theta_2 = 180^{\circ}$.         $W = pE(\cos(0^{\circ}) - \cos(180^{\circ})) = pE[1 - (-1)] = 2pE$
  • Case II: Rotation to a position perpendicular to the field: From $\theta_1 = 0^{\circ}$ to $\theta_2 = 90^{\circ}$.         $W = pE(\cos(0^{\circ}) - \cos(90^{\circ})) = pE(1 - 0) = pE$
• Board Tip on Equilibrium states:
  • Stable Equilibrium ($\theta = 0^{\circ}$): The dipole is parallel to $E$ with minimum energy ($U = -pE$).
  • Unstable Equilibrium ($\theta = 180^{\circ}$): The dipole is anti-parallel to $E$ with maximum energy ($U = +pE$).

Electrostatics of Conductors

• Definition: Conductors contain a large number of free charge carriers to conduct electricity (e.g., metals, graphite).
• Properties of Conductors:
  1. Internal Field: Inside a conductor, the electrostatic field is zero ($E = 0$).
  2. Surface Field: At the surface, the electrostatic field must be normal to the surface, given by $E = \frac{\sigma}{\epsilon_0}$.
  3. Excess Charge: No excess charge can exist in the interior in a static situation; it resides solely on the outer surface.
  4. Potential: Electrostatic potential is constant throughout the volume of the conductor and exactly equals the value on its surface.
  5. Cavity: The electric field is zero in the cavity of a hollow charged conductor.

Electrostatic Shielding

• Concept: This is the phenomenon of making a region free from any electric field.
• Principle: The electric field inside a cavity of any conductor is zero, regardless of the external fields or charges placed on the outer surface.
• Real-world Example (Scorify): During a thunderstorm, a car is safer because its metallic body acts as a Faraday cage, keeping the internal electric field nearly zero.

Dielectrics and Polarisation

• Dielectric: A non-conducting substance that has no free charge carriers.
• Behavior in External Field ($E_0$):
  • Conductor: Free charge carriers move to the surface to create an induced field that exactly cancels the external field ($E_{net} = 0$).
  • Dielectric: External fields induce a net dipole moment by stretching or reorienting molecules. This creates an opposing internal field ($E_p$) that reduces the external field but does not cancel it.
• Net Electric Field: The net field ($E$) inside a dielectric is $E = E_0 - E_p$. Consequently, $E < E_0$.
• Polarisation ($P$): The induced dipole moment per unit volume is called polarisation.
• Molecular Behavior:
  • Non-polar molecules: Centers of positive and negative charges coincide. The external field stretches them, inducing a dipole.
  • Polar molecules: They have permanent dipole moments. The external field aligns these chaotic dipoles in one direction.
• Linear Isotropic Dielectrics: $\mathbf{P} = \chi_e \mathbf{E}$, where $\chi_e$ is electric susceptibility.
• Dielectric Constant ($K$): The relationship is $K = 1 + \chi_e$.

Capacitor and Capacitance

• Capacitor: An arrangement of two conductors separated by an insulating medium used to store electric charge and electric energy.
• Capacitance ($C$): If charge $Q$ is given to a conductor leading to a potential increase $V$, then $C = \frac{Q}{V}$.
• Dependence: Capacitance depends entirely on the geometrical configuration (shape, size, separation) of conductors and the nature of the dielectric medium between them.
• SI Unit: The farad ($F$).
• Parallel Plate Capacitor: Consists of two large parallel conducting plates separated by distance $d$.
  • Capacitance in vacuum/air: $C_0 = \frac{\epsilon_0 A}{d}$.
• Effect of Dielectric Medium:
  • When a dielectric with constant $K$ is inserted, capacitance increases: $C = \frac{K \epsilon_0 A}{d} = KC_0$.
• Partially Filled Dielectric Slab: For a slab of thickness $t$ and constant $K$, the capacitance is $C = \frac{\epsilon_0 A}{d - t + (t/K)}$.
• Combination of Capacitors:
  • Series: Charge ($Q$) is identical across all capacitors. Equivalent capacitance is $\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$
  • Parallel: Potential difference ($V$) is identical across all capacitors. Equivalent capacitance is $C_p = C_1 + C_2 + C_3 + \dots$

Energy Stored in a Capacitor

• Electrostatic Energy ($U$): The energy stored is given by the formulas:     $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$
• Energy Density ($u$): The energy stored per unit volume in an electric field $E$ is $u = \frac{1}{2}\epsilon_0 E^2$.
• Scorify Note: When two charged capacitors are connected, charge flows until their potentials equalize, attaining a common potential.