The reading of an ammeter in the circuit

(i) `I` when key `K_(1)` closed key `K_(2)` is open
(ii) `I//2` when both keys `K_(1) and K_(2)` are closed Find the expression for the resistance of `X` in terms of the resistances of R and S

Answer

`(RS)/(R - S)`

Explanation

When key `K_(1)` is closed and `K_(2)` is open resistance of the circuit `= R + X`. So current
`I_(1) = I (epsilon)/( R+ X)` …..(i)
When key `K_(1)` is closed and `K_(2)` both are closed `S and X` will be in parallel then their effective resistance is
`R' = (SX)/(S + X)`
Total resistance of circuit `= R + R' = R + (SX)/(S + X)`
Total current `I_(2) = (epsilon)/(R + (SX)/(S + X))`
Current through resistance `X` will be
`I'_(2) = I_(2) xx (S)/(S + X) = (epsilon)/(R + (SX)/(S + X)) xx (S)/((R + X))`
`= (epsilon S)/((S + X) R + SX)`.......(ii)
As `I'_(2) = (I)/(2)`
so `(I)/(2) = (epsilonS)/((S + R) R +SX) ` [from (i) and (ii)]
or `(epsilon)/((R + X)2) = (epsilonS)/((S + X) R + SX)`
On similarlly `X = (RS)/(R - S)`