Question

Two capacitors C1 and C2 are charged to potential V1 and V2 resp. and then connected in parallel. Calculate common potential, charge on each capacitor and energy in the system after connection??

Answer 1

Hello Buddy,

◆ Answer -
¶ V = (C1V1+C2V2)/(C1+C2)
¶ Q1 = C1(C1V1+C2V2)/(C1+C2)
¶ Q2 = C2(C1V1+C2V2)/(C1+C2)
¶ E = (C1V1+C2V2)^2 /2(C1+C2)

◆ Explaination -
When capacitors are separated, total charge on capacitors
Qi = C1V1 + C2V2

When capacitors are connected in parallel with voltage V, total charge is
Qf = (C1+C2)V

By law of conservation of momentum,
Qi = Qf
C1V1 + C2V2 = (C1+C2)V
V = (C1V1+C2V2)/(C1+C2)

Charge on capacitor 1 is -
Q1 = C1V
Q1 = C1(C1V1+C2V2)/(C1+C2)

Charge on capacitor 2 is -
Q2 = C2V
Q2 = C2(C1V1+C2V2)/(C1+C2)

Energy in the system is calculated by-
E = 1/2 (C1+C2)V^2
E = (C1V1+C2V2)^2 /2(C1+C2)

Hope it helps...

Answer 2

Explanation:

Given That,

Two capacitors C₁ and C₂ are charged to potential V₁ and V₂ respectively.

The total charges before sharing = C₁V₁+C₂V₂......(I)

If V is the common potential on sharing charges then

The total charges after sharing=(C₁+C₂)V......(II)

We need to calculate the common potential

From equation (I) and (II)

$C_{1}V_{1}+C_{2}V_{2}=(C_{1}+C_{2})V$

The common potential is given by

$V=\dfrac{C_{1}V_{1}+C_{2}V_{2}}{(C_{1}+C_{2})}$

We need to calculate the charge on each capacitor

Formula of charge is given by

$Q= CV$

Where, C = capacitor

V = potential

Q = charge

The charge for first capacitor

$Q_{1}=C_{1}V_{1}$

The charge for second capacitor

$Q_{2}=C_{2}V_{2}$

We need to calculate the energy in the system after connection

The energy stored in capacitor is

$E =\dfrac{1}{2}CV^2$

Total energy before sharing charges

$E_{1}=\dfrac{1}{2}C_{1}V_{1}^2+\dfrac{1}{2}C_{2}V_{2}^2$

Total energy after sharing charges

$E_{2}=\dfrac{1}{2}(C_{1}+C_{2})V^2$.......(III)

Now, Put the value of common potential in equation (III)

$E_{2}=\dfrac{(C_{1}V_{1}+C_{2}V_{2})^2}{2(C_{1}+C_{2})}}$

The total energy is

$E'=E_{1}-E_{2}=\dfrac{C_{1}C_{2}(V_{1}-V_{2})^2}{2(C_{1}+C_{2})}$

Hence, This is the required solution.