Electrostatic Potential (Class 12 Physics)

Introduction

• Presented by Roshni from Learnohub (free learning platform).
• This session covers electrostatic potential.
• The next session will cover capacitance.
• Aims to make the concept of electrostatic potential crystal clear.

Recap of Previous Session

• Previous session topics: Electric charges and the electric field.
• Focus of this session: Behavior of a charge in an electric field.

Fundamentals of Electrostatic Potential

• A charge in an electric field experiences a force causing displacement, leading to work done and potential energy storage.
• Electrostatic potential energy is analogous to gravitational potential energy.
• Gravitational force:
  • Formula: $F = G \frac{m_1 m_2}{r^2}$
  • Conservative force.
• Electrostatic force:
  • Formula: $F = k \frac{q_1 q_2}{r^2}$
  • Conservative force.

Electrostatic Potential Concept

• When a charge moves from one point to another in an electric field, work is done, which is stored as potential energy.
• Formula for work done: $W = q(V_P - V_R)$.
• Potential energy difference: $U_P - U_R = W_{RP}$.
• Electric potential energy difference involves work done in moving charge from one point to another.

Definitions

• Electrostatic Potential Energy (U): Work done to bring a charge from infinity to a point without acceleration.
  • $U_P = W_{\infty P} = Q(V_P - V_\infty)$.
  • If $V_\infty = 0$: $U_P = QV_P$.
• Electrostatic Potential (V): Work done per unit charge to move a charge from infinity to a point.
  • Formula: $V = \frac{kQ}{r}$.
  • Scalar quantity, SI unit: Volt (V).

Mathematical Expressions

• Electrostatic Potential due to a point charge: $V(r) = \frac{kQ}{r}$.
• Electrostatic Potential Energy due to a point charge: $U(r) = \frac{kQ_1 Q_2}{r}$.

Potential Difference vs. Electrostatic Potential

• Potential difference is more physically significant as work is done when there is a potential difference.
• Both have the same unit: Volt.
• At infinity, potential is zero.

Potential Due to a System of Charges

• System of charges: Total potential at a point is the sum of potentials due to individual charges.
  • Superposition Principle: $V_P = V_{P1} + V_{P2} + V_{P3} + \ldots$
  • $V_P = k \left ( \frac{Q1}{R1} + \frac{Q2}{R2} + \frac{Q3}{R3} + \ldots \right )$
• Electric Dipole:
  • Axial Point: $V = - \frac{kP}{x^2}$ (if x >> a).
  • Equatorial Point: $V = 0$.
  • General Case: $V(P) = \frac{kP \cos \theta}{R^2}$.

Electric Potential Due to a System

• Example Setup: A system of charges with potential at the center of a hexagon.
• Each vertex has a charge, calculate potential at the center.
  • $P_{net} = 6 \times \frac{kQ}{r}$.

Equipotential Surfaces

• Surfaces at which the electric potential is the same at every point.
• Types: Electrostatic Equipotential Surfaces:
  • Point Charge: Spherical surfaces.
  • Line Charge: Cylindrical surfaces.
  • Uniform Electric Field: Perpendicular planes.
• Properties:
  • Work done to move a charge on equipotential surface is zero.
  • Electric field is always perpendicular to equipotential surfaces.
  • No intersecting equipotential surfaces.
• Example Problems: Calculate potentials and potential energy in static and external fields.

Potential in External Electric Fields

• Single Charge: $U_{ext} = QV(r)$.
• System of 2 Charges: $U_1 + U_2 = Q_1 V(R_1) + Q_2 (V(R_2) + \frac{KQ_1Q_2}{R_{12}})$.
• Potential Energy in Dipole:
  • Dipole in an electric field experiences torque: $P \text{cross } E$.
  • Potential Energy: $U = - P \text{dot } E$.

Example Problems

• Detailed solutions covering concepts of potentials and potential energy.

Conclusion

• Upcoming video will cover capacitance.
• Feedback and questions are encouraged.