Question

Derive an expression for the loss of energy when two conductors at different potentials are brought into electrical contact. Account for this energy.

Answer 1

When two conductors of different potentials are brought close to each other, charge is shared between them. During this process, some amount of energy is lost which is disappeared in the form of heat.
In order to calculate the total loss of energy during this process, consider two capacitors having capacitance C₁ and C₂ having distinct potentials V₁ and V₂ respectively. Now we know that, the charge always flow from a capacitor of high potential to a capacitor of low potential.
Hence,
Their common potential is given as:
V = (C₁V₁ + C₂ V₂)  / C₁ + C₂  .......... (1)
Total energy of the system will be:
E₁ = 1/2C₁V₁² + 1/2 C₂ V₂²   ........... (2)
E₂ = 1/2 (C₁ + C₂) V²
E₂ = [ 1/2 (C₁ + C₂) ] . [ (C₁V₁ + C₂ V₂)  / (C₁ + C₂) ]²  (Using equation (1))
E₂ = [ 1/2 (C₁ + C₂) ] . [ (C₁V₁ + C₂ V₂)² / (C₁ + C₂)² ]
E₂ = (C₁V₁ + C₂ V₂)² / 2 (C₁ + C₂)   .......... (3)
Subtracting equation (2) and equation (3), we get:
E₁ - E₂ = [1/2 C₁V₁² + 1/2 C₂ V₂² ] - [ (C₁V₁ + C₂ V₂)² / 2 (C₁ + C₂) ]
E₁ - E₂ = [ C₁V₁² (C₁ + C₂) + C₂ V₂² (C₁ + C₂) - (C₁V₁ + C₂ V₂)² ] / 2 (C₁ + C₂)
E₁ - E₂ =C₁²V₁²+C₁C₂V₁²+C₁C₂V₂²+C₂²V₂²-(C₁²V₁²+C₂²V₂² +2C₁C₂V₁V₂)/2(C₁+C₂)
E₁ - E₂ =C₁²V₁²+C₁C₂V₁²+C₁C₂V₂²+C₂²V₂²- C₁²V₁²-C₂²V₂² - 2C₁C₂V₁V₂ /2(C₁ + C₂)
E₁ - E₂ = C₁C₂ (V₁ - V₂)² / 2(C₁ + C₂)
The above equation shows that E₁ - E₂ is greater then zero.
Or E₁ is greater then E₂
Or E₂ is less then E₁
Hope it helps. Thanks.

Answer 2

Answer:

Derive an expression for the loss of energy when two conductors at different potentials are brought into electrical contact. Account for this energy.