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

Explanation

When key `K_(1)` is closed, `K_(2)` is open resistance of circuit = R+X
So, `I=(E)/(R+X)`
When `K_(1)andK_(2)` both closed, S and X will be in parallel
So, `(1)/(R')=(1)/(S)+(1)/(X)impliesR'=(SX)/(S+X)`
Total resistance `=R+(SX)/(S+X)`
Total current `=(E)/(R+(SX)/(S+X))`
Current through resistance X will be
`(S)/(S+X)xx(E)/(R+(SX)/(S+X))=(ES)/((S+X)R+SX)`
`implies(I_(2))/(2)=(ES)/((S+X)R+SX)" "....(ii)`
From (i) and (ii)
`(E)/((R+X)2.)=(EX)/((S+X)R+SX)`
On simplifying `x=(RS)/(R-S)`.