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We Will Learn :

1. Introduction
2. Electrostatic Potential Energy
3. Electrostatic Potential
4. Electrostatic Forces are Conservative
5. Electrostatic Potential Due to a Point Charge
6. Potential at a point due to group of electric charges (Updating soon)

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Introduction

• Earlier, we studied electric field (E) due to a point charge and charge distribution.
• Electric field → a vector quantity called electric field intensity (E).
• Now, we learn about a scalar quantity (V) called electrostatic potential.
• Electrostatic potential (V):
  • Represents the degree of electrification of a body.
  • Decides the direction of charge flow between two bodies in contact.
  • Charge flows from higher potential → lower potential until both are equal.
• Along with Coulomb’s law (for E), we also use Gauss’s Law:
  • Useful for calculating electric field of symmetric charge distributions.
  • Helps in solving complex electrostatics problems.

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Electrostatic Potential Energy

• Similar to gravitational potential energy, a charge in an electric field also possesses electrostatic potential energy.
• Consider a charge +Q at the origin and a small test charge +q brought from point A → B.
  • +q is assumed so small that it does not disturb +Q.
  • An external force just balances the repulsive electric force, so net force = 0 and +q moves without acceleration.

• Work done by external force (Wext):
  • Negative of work done by electric force.
  • This work gets stored as potential energy of the test charge.
• If external force is removed at B, then electric force pushes +q away → stored potential energy converts to kinetic energy.
• At every point in the field, charge q has some potential energy.
  • Work done in moving q from A → B = increase in its potential energy.
  • Depends only on initial and final positions (path independent).

Definition:

• Electrostatic potential energy difference (between A and B): Minimum work done by external force in moving test charge q (without acceleration) from A to B.
• Absolute potential energy at a point is not meaningful → only difference matters.
• Convention: Potential energy = 0 at infinity.

Therefore:

• Electrostatic potential energy of charge q at point B = Work done by external force in bringing q from infinity → B (without acceleration).

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Electrostatic Potential

• Definition (Potential Difference):
  • The electrostatic potential difference between two points A and B = work done in moving a unit positive test charge (without acceleration) from A to B against the electric field.

• Key Features:
  • Work depends only on initial (A) and final (B) points, not on the path.
  • If potentials are VA and VB:
    U_B–U_A/q = W_AB/q = ΔV

• Dimensional Formula:
  [ML²T⁻³A⁻¹]
• SI Unit:
  • Volt (V)
  • 1 volt = 1 joule / coulomb = 1 J C⁻¹ = 1 N m C⁻¹
• Definition (1 volt):
  • Potential difference between two points is 1 volt if 1 joule of work is done in moving a 1 coulomb positive charge without acceleration from one point to another.
• Important Fact:
  • Only potential difference is physically significant, not the absolute potential at a point.
  • Convention: Potential at infinity = 0.
• Electrostatic Potential (at a point):
  • The minimum work done in bringing a unit positive charge (without acceleration) from infinity to that point.
  • It is a scalar quantity.
• Definition (1 volt at a point):
  • A point has potential of 1 volt if 1 joule of work is done in bringing 1 coulomb positive charge from infinity to that point (without acceleration).

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Electrostatic Forces are Conservative

• Statement: Electrostatic forces are conservative → work done in moving a unit positive test charge around a closed path = zero.
• Proof Idea:
  • Work done in moving unit positive charge from A → B along path L = W_AB.
  • Work done from B → A along another path = W_BA.
  • Total work around closed path A → B → A:
    W_AB+W_BA=(V_B–V_A)+(V_A–V_B)=0
• Conclusion:
  • No work is done in moving a unit charge over a closed path.
  • Hence, the electrostatic field is conservative and electrostatic forces are conservative in nature.
• Mathematical Form:
  (Line integral of electric field over a closed path = zero)

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Electrostatic Potential Due to a Point Charge

Electrostatic Potential due to a Point Charge

• Situation: A point charge q is placed at the origin O. We want to calculate the potential at point P where OP = r.
• Definition: Electrostatic potential at point P = work done in bringing a unit positive test charge from infinity (∞) to P without acceleration.
• Electric Field:

  E = (1 / 4πɛ₀) · (q / x²)

  For small displacement dx: dW = -E dx

• Total Work (∞ → r):

  W = ∫_∞^r -(1 / 4πɛ₀) · (q / x²) dx

  Potential: V = q / (4πɛ₀ r)

• Key Points:
  • If q > 0 ⇒ V is positive
  • If q < 0 ⇒ V is negative
  • At r → ∞, V = 0
  • At equal distance r, potential is same → spherical symmetry
• Variation:

  V ∝ 1/r and E ∝ 1/r²

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Potential at a point due to group of electric charges (Updating soon)

Must Read :

1. Electric Charges And Fields Complete Notes