The reading of an ideal ammeter, in the circuit shown here, equals (i) I when key `K_1` is closed but 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 resistance R and S.

Explanation

We know that resistance of an ideal ammeter is zero.
(i) When only key `K_1` is closed, the current flowing through the ammeter is given as : (ii) When key `K_2` is also closed, the circuit current becomes
`I. = (epsi)/( R + (SX)/(S + X))`
and the current flowing through the ammeter is given as :
`I/2 = (S)/(S + X).I. = ( (S)/(S+X)) .(epsi)/(R +(SX)/(S + X)) = (S epsi)/((S + X)R+SX)` ...(ii)
Multiplying (ii) by 2 and then equating with (i), we get
`(epsi)/(R+X) = (2S epsi)/((S + X)R+ SX)`
` rArr (S+X) R + SX = 2S (R+X) " or " SR + XR + SX = 2SR + 2SX`
` rArr XR - SX = SR` or `X = (RS)/((R - S))`