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Redistribution of Charges: Common Potential

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Estimated time: 9 minutes
CISCE: Class 12

Introduction

When two charged conductors with different potentials are connected by a thin conducting wire, charge flows from the conductor at higher potential to the one at lower potential — not until charges are equal, but until potentials become equal. This equalized value is called the common potential.

Think of two water tanks at different water levels connected by a pipe. Water flows from the tank with the higher level to the one with the lower level until both reach the same level — similarly, charge flows until potentials equalize, not until the water (charge) quantities are equal.

CISCE: Class 12

Derivation

Assumptions

  • The two conductors are far apart, so they do not electrostatically influence each other.
  • The connecting wire has negligible capacitance.
  • No charge is lost to the surroundings during transfer.

Derivation

Step 1: Initial Setup

  • Two conductors with capacitances C1​ and C2​ carry initial charges q1​ and q2​, at potentials V1​ and V2​ respectively, where V1 = \[\frac {q_1}{C_1}\]​​ and V2 = \[\frac {q_2}{C_2}\]​​.

Step 2: Connection

  • When connected by a wire, charge flows until both conductors reach a common potential V.

Step 3: Charge Conservation

  • Total charge before = Total charge after:
    q1 + q2 = q1′ + q2′

Step 4: Common Potential Formula

  • V = \[\frac{q_1+q_2}{C_1+C_2}=\frac{C_1V_1+C_2V_2}{C_1+C_2}\]

Step 5: Final Charges

  • q1′ = C1V, q2′ = C2V

Step 6: Charge Ratio

  • \[\frac{q_1^{\prime}}{q_2^{\prime}}=\frac{C_1}{C_2}\]

Final charges distribute in the ratio of capacitances.

CISCE: Class 12

Before vs After Comparison

Quantity Conductor 1 (Before) Conductor 2 (Before) Conductor 1 (After) Conductor 2 (After)
Potential V1 V2 V V
Charge q1 q2 C1V C2V
Capacitance C1 C2 C1​ (unchanged) C2 (unchanged)
CISCE: Class 12

Energy Loss on Connection

Charge Transferred

If V1 > V2​, charge flows from conductor 1 to conductor 2. Magnitude transferred: \[q_1-q_1^{\prime}=\frac{C_1C_2(V_1-V_2)}{C_1+C_2}\]

Initial energy: \[U=\frac{1}{2}\left(C_1V_1^2+C_2V_2^2\right)\]

Final energy: \[U^{\prime}=\frac{1}{2}(C_1+C_2)V^2\]

Energy Lost: \[U-U^{\prime}=\frac{C_1C_2}{2(C_1+C_2)}(V_1-V_2)^2\]

CISCE: Class 12

Example

Given: Two spheres, C1 = 3.0 μF, C2 = 5.0 μF, initially at V1 = 300 V, V2 = 500 V.

Step Calculation Result
Common potential \[\frac {3(300)+5(500)}{3+5}\] 425 V
Charge on 3 µF sphere 3 × 10−6 × 425 1.275 × 10-3 C
Charge on 5 µF sphere 5 × 10−6 × 425 2.125 × 10-3 C
Energy lost \[\frac {3×5}{2(8)}\](200)2 × 10−6 0.0375 J
CISCE: Class 12

Real-Life Application

This energy loss appears as heat in the connecting wire, and as light/sound if sparking occurs — the same principle behind static discharge sparks and safety earthing in electrical systems.

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