📘 Chapter 19: Electric Potential Due to a Point Charge (Class XII)

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🔷 1. Introduction

In electrostatics, one of the most fundamental results is the electric potential due to a point charge. It helps us understand how energy is distributed in an electric field without directly dealing with forces.

This concept forms the basis for studying more complex systems like multiple charges, continuous charge distributions, and electric circuits.

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🔷 2. Definition of Electric Potential

Electric potential at a point is defined as the work done per unit positive charge in bringing a test charge from infinity to that point.

V = W / q

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🔷 3. Derivation of Potential Due to Point Charge

Consider a point charge q. To find potential at a distance r, we calculate work done in bringing a unit charge from infinity.

Electric field due to point charge:

E = (1 / 4πε₀) (q / r²)

Work done:

W = ∫ E · dr

Integrating from ∞ to r:

V = (1 / 4πε₀) (q / r)

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📦 4. Important Results (Must Remember)

• Electric potential: V = (1/4πε₀)(q/r)
• Scalar quantity
• Unit: Volt (V)
• Depends on distance r
• Reference: V = 0 at infinity

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🔷 5. Nature of Potential

• For positive charge: V is positive
• For negative charge: V is negative

Potential decreases as distance increases.

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🔷 6. Variation with Distance

Electric potential varies inversely with distance:

V ∝ 1/r

• At r → ∞, V → 0
• At r → 0, V → ∞

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🔷 7. Equipotential Surfaces

For a point charge:

• Equipotential surfaces are concentric spheres
• Electric field is perpendicular to these surfaces

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🔷 8. Relation with Electric Field

Electric field is related to potential as:

E = − dV/dr

This shows field is negative gradient of potential.

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🧠 9. Solved Numerical Problems

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🔹 Q1

Find potential at 0.2 m from a charge of 2×10⁻⁶ C.

Solution:

V = kq/r

= 9×10⁹ × 2×10⁻⁶ / 0.2

V = 9 × 10⁴ V

Answer: 9 × 10⁴ Volt

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🔹 Q2

What happens to potential if distance doubles?

Solution:

V ∝ 1/r → becomes half

Answer: Potential reduces to half

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🔹 Q3

What is potential at infinity?

Answer:

Zero

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🔹 Q4

Can potential be negative?

Answer:

Yes, for negative charges.

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🔹 Q5

Why is potential scalar?

Answer:

Because work is scalar quantity and independent of direction.

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🔷 10. Advanced Conceptual Insight

Electric potential due to a point charge simplifies many problems in electrostatics. It allows use of energy methods instead of force-based methods, making analysis easier for complex systems.

This concept is also fundamental in atomic physics, quantum mechanics, and field theory.

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🔷 11. Applications

• Electrostatic energy calculations
• Capacitor analysis
• Charge distribution problems
• Electric field mapping

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🔷 12. Summary

The electric potential due to a point charge varies inversely with distance and is a scalar quantity. It plays a central role in electrostatics and helps in understanding energy distribution in electric fields.

✨ End of Chapter 19: Electric Potential Due to Point Charge ✨