Electrostatic Potential and Capacitance

• Electrostatic Potential & Potential Difference

  • Electric Potential ($V$): Work done per unit positive charge to bring it from infinity to a point
    • Formula: $V = \frac{W}{q_0}$
    • SI Unit: Volt (V) ($1\text{ V} = 1\text{ J/C}$)
    • Nature: Scalar Quantity
  • Potential Difference ($\Delta V$): Work done in moving charge between two points ($V_B - V_A = \frac{W_{AB}}{q_0}$)
  • Potential due to Point Charge: $V = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r}$
  • Potential due to Electric Dipole:
    • Axial Position ($\theta = 0^\circ$): $V = \frac{1}{4\pi\varepsilon_0} \cdot \frac{p}{r^2}$
    • Equatorial Position ($\theta = 90^\circ$): $V = 0$
    • General Point: $V = \frac{1}{4\pi\varepsilon_0} \cdot \frac{p \cos\theta}{r^2}$
  • System of Charges: Principle of superposition ($V = V_1 + V_2 + \dots + V_n$)

• Equipotential Surfaces

  • Definition: Surface with the same potential at every point ($V = \text{constant}$)
  • Key Properties:
    • Work Done: Zero work required to move charge along the surface ($W = 0$)
    • Electric Field Orientation: $\vec{E}$ is always perpendicular to the surface
    • Intersection: Two equipotential surfaces never intersect
  • Relation between Field & Potential: $E = -\frac{dV}{dr}$ (Potential Gradient)

• Electrostatic Potential Energy ($U$)

  • System of Two Point Charges: $U = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r}$
  • Dipole in External Field ($\vec{E}$):
    • Potential Energy: $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$
    • Torque: $\tau = \vec{p} \times \vec{E} = pE \sin\theta$
    • Stable Equilibrium: $\theta = 0^\circ$ ($U_{\text{min}} = -pE$)
    • Unstable Equilibrium: $\theta = 180^\circ$ ($U_{\text{max}} = +pE$)

• Conductors and Dielectrics

  • Electrostatics of Conductors:
    • Electric field inside conductor: $E = 0$
    • Net charge inside conductor: $Q = 0$ (resides on outer surface)
    • Potential is constant throughout the volume
  • Dielectrics:
    • Non-conducting materials that transmit electric effects
    • Polar Dielectrics: Permanent dipole moment (e.g., $\text{H}_2\text{O}, \text{HCl}$)
    • Non-Polar Dielectrics: Zero net dipole moment without field (e.g., $\text{O}_2, \text{H}_2$)
    • Dielectric Constant ($K$): $K = \frac{\varepsilon}{\varepsilon_0} = \frac{E_0}{E_{\text{net}}}$
    • Polarization ($P$): Induced dipole moment per unit volume

• Capacitors and Capacitance

  • Capacitance ($C$): Ability to store electric charge ($C = \frac{Q}{V}$)
    • SI Unit: Farad (F)
  • Parallel Plate Capacitor:
    • In Vacuum/Air: $C_0 = \frac{\varepsilon_0 A}{d}$
    • With Dielectric Medium: $C = \frac{K \varepsilon_0 A}{d} = K C_0$
    • With Partial Dielectric Slab (thickness $t < d$): $C = \frac{\varepsilon_0 A}{d - t + \frac{t}{K}}$
    • With Conducting Slab (thickness $t < d$): $C = \frac{\varepsilon_0 A}{d - t}$

• Combination of Capacitors

  • Series Combination:
    • Same Charge ($Q$) across all capacitors
    • Potential Divides: $V = V_1 + V_2 + V_3$
    • Equivalent Capacitance: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$
  • Parallel Combination:
    • Same Potential Difference ($V$) across all capacitors
    • Charge Divides: $Q = Q_1 + Q_2 + Q_3$
    • Equivalent Capacitance: $C_{eq} = C_1 + C_2 + C_3$

• Energy Stored in a Capacitor

  • Energy Formulas:
    • $U = \frac{1}{2} C V^2$
    • $U = \frac{1}{2} Q V$
    • $U = \frac{Q^2}{2C}$
  • Energy Density ($u$): Energy stored per unit volume in electric field
    • Formula: $u = \frac{1}{2} \varepsilon_0 E^2$
  • Redistribution of Charge (Connecting two conductors):
    • Common Potential: $V = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}$
    • Loss of Energy: $\Delta U = \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)}$ (Dissipated as heat/light)