Two ideal batteries of `emf V_(1) and V_(2)` and three resistance `R_(1)R_(2) and R_(3)` are connected 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 2R_(1) = R_(2) = R_(3)`

Answer

a,b,d

Explanation

`V_(1) = IR_(1) + 0 xx R_(2) or I = (V_(1))/(R_(1))` ….(i)
In closed circuit `ABCDEFA` we get
`V_(1) + V_(2) = IR_(1) + IR_(2) or I = (V_(1) + V_(2))/(R_(1) + R_(3))`….(ii)
From (i) and (ii), `(V_(1))/(R_(1)) = (V_(1) + V_(2))/(R_(1) + R_(2))`....(ii)
or `V_(1)R_(1)+ V_(1)R_(3) = V_(1)R_(1) + V_(2) R_(1)`
or `V_(1) R_(3) = V_(2)R_(1)`.....(iii)
Since current in `R_(2)` is zerto `R_(2)` can have any value from (ii) we note that if `V_(1) = V_(2)` Then
`R_(1) = R_(3) = R_(2) or R_(1) = 2R_(1) + R_(3)`
If `V_(1) = 2V_(2) ,R_(1) = 2R_(3)`
If `2V_(1) = V_(2),R_(3) = 3R_(1) = R_(2)`
Thus option (a,b and d) are true .