Prove that the distribution of charge between two conductors in the ratio of their capacitance

# Prove that the distribution of charge between two conductors in the ratio of their capacitance .

Let two conductor A and B of capacities C1 and C2. on charging them by charge Q1 and Q2 respectively then their potential become V1 and V2 here V1>V2 .

Thus before joining:
charge on conductor A, Q1 = C1.V1
Charge on conductor B, Q2 = C2.V2

After joining :
Two charged conductor joining together by the conducting wire L then charge flows from high potential to low potential till both the conductor getting common potential V.

Thus, total charge,.  Q = Q1 + Q2
                            Q = C1.V1 + C2.V2

Total capacity, C = C1 + C2

By the definition of capacitance
C = Q/V
Hence, V = Q/C

Putting the values

[V= (C1.V1+C2.V2) / C1+C2 ]

This is common potential

After joining, the charge on conductor A is
Q1'= C1.V

And charge on conductor B is
Q2' = C2.V

Hence ,  Q1'/Q2' = C1.V/C2.V

[Q1'/Q2' = C1/C2 ] Hence proved