Three uncharged capacitors of capacities `C_(1),C_(2)` and `C_(3)` are connected as shown in the figure to one another and the potentials `V_(1),V_(2)` and `V_(3)` respectively. Then the potential at `O` will be
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Options

A. `(V_(A) + V_(B) + V_(D))/(C_(1) + C_(2) + C_(3))`
B. `(V_(A) C_(1) + V_(B) C_(2) + V_(D) C_(3))/(C_(1) + C_(2) + C_(3))`
C. `(V_(A) V)(B) + V_(B) V_(D) + (V_(D) V_(A))/(C_(1) + C_(2) + C_(3))`
D. `(V_(A) V_(B) V_(D))/(C_(1) C_(2) + C_(2) C_(3) + C_(3) C_(1))`

Answer

B

Explanation

Let `q_(1), q_(2), q_(3)` be the charges on `C_(1), C_(2), C_(3)` resectively From Fig
`V_(A) - V_(0) = (q_(1))/(C_(1))` or `q_(1) = (V_(A) - V_(0)) C_(1)`
`V_(B) - V_(0) = (q_(2))/(C_(2))` or `q_(2) = (V_(B) - V_(0)) C_(2)`
`V_(D) - V_(0) = (q_(3))/(C_(3))` or `q_(3) = (V_(D) - V_(0)) C_(3)`
As `q_(1) = q_(2) + q_(3)`
`:. (V_(A) - V_(0)) C_(1) = (V_(B) - V_(0)) C_(2) + (V_(D) - V_(0)) C_(3)`
which gives, `V_(0) = (V_(A) C_(1) + V_(B) C_(2) + V_(D) C_(3))/(C_(1) + C_(2) + C_(3))`