Electrostatics – Stage 2 (Page 5)

Electric Potential & Electric Field Relationship

                                           

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1️⃣ What is Electric Potential?

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

V = W / q

Unit: Volt (V)

👉 Potential is a scalar quantity, unlike electric field.

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2️⃣ Electric Potential due to a Point Charge

For a point charge Q at distance r:

V = 1/(4πε₀) · Q / r

• If Q > 0 → V positive
• If Q < 0 → V negative

⚠️ Compare with E ∝ 1/r², but V ∝ 1/r

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3️⃣ Electric Potential due to Charged Spherical Shell

(a) Outside the shell (r ≥ R)

V = 1/(4πε₀) · Q / r

Same as point charge.

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(b) Inside the shell (r < R)

V = 1/(4πε₀) · Q / R

👉 Potential is constant inside shell, but field is zero.

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4️⃣ Electric Potential due to Solid Charged Sphere

(a) Outside (r ≥ R)

V = 1/(4πε₀) · Q / r

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(b) Inside (r < R)

V = 1/(4πε₀) · (Q / 2R) · (3 − r²/R²)

👉 Potential varies smoothly inside solid sphere.

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5️⃣ Relation between Electric Field and Potential

Electric field is the negative gradient of potential.

E = − dV / dr

• Field points from high V to low V
• Steeper V → stronger E

⚠️ Extremely important for JEE Advanced problems.

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6️⃣ Potential Difference

Potential difference between two points A and B:

V_B − V_A = − ∫ E · dr

Path independent → conservative field.

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7️⃣ Equipotential Surfaces

• Same potential everywhere
• No work done along surface
• E ⟂ equipotential surface

👉 Equipotential surfaces never intersect.

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8️⃣ Graph-Based Insights (Very Important)

• V vs r for shell → flat inside, 1/r outside
• E vs r → zero inside shell, sudden jump at surface

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9️⃣ IIT Concept Check

Q: Can potential be zero but field non-zero?

A: Yes (example: midpoint between equal charges).

Q: Can field be zero but potential non-zero?

A: Yes (inside conducting shell).

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What’s Next?

👉 Stage 2 – Page 6: Electric Potential Energy & System of Charges

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