Two ideal batteries of emf `V_1 and V_2` and three resistances `R_1, R_2 and R_3` are connected as shown in the figure. The current in resistance `R_2` would be zero if

Options

A. `V_1 = V_2 and R_1 = R_2 = R_3`
B. `V_1 = V_2 and R_1 = 2R_2 = R_3`
C. `V_1 = 2V_2 and 2R_1= 2R_2 = R_3`
D. `2V_1 = V_2 and 3R_1 = R_2 = R_3`

Answer

A, B, D

Explanation

(a,b,d) Applying KVL in MNOPQAM
`V_1 -iR_1 + V_2 -iR_3 = 0`
`:. i = (V_1+V_2)/(R_1+R_3)` ……..(i)
Applying KVL in NOPQN
`V_1 - iR_1 = 0 :. I = V/R_1` ……(ii)
From (i) & (ii) `V_1/R_1 = (V_1+ V_2)/(R_1 + R_3)`
`:. V_1R_1 + V_1R_3 = V_1R_1 + V_2R_1`
`rArr V_1R_3 = V_2R_1`
If `V_1 = V_2 then R_1 = R_3 = R_2 :. ` (a) is correct option.
If `V_1 = V_2 then R_1 = R_3 = 2R_2 :. ` (b) is correct option.
`(R_2 can have any value as there is no current flowing
through d)
If `V_1 = 2V_2 the 2R_3 = R_1 :.` (c) is incorrect option.
If `2V_1 = V_2 then R_3 = 2R_1 = R_2 :.` (d) is correct option.