A parallel combination of an `8 Omega` resistor and an unknown resistor R is connected in series with a `16 Omega` resistor and a battery. This circuit is then disassembled, and the three resistors are then connected in series with each other oand the same battery. in both arrangements, the current through the `8 Omega` resistor is the same. What is the unknown resistance R?

Explanation

Let current through `8Omega` be I in both cases.
,
In case `(I) V_1 = V_2 or ixx8 = R xxi_2 ori_2 ori_2 = (8i)/(R )`
`I= i+i_2 = i+(8i)/(R )`
`V =I (R_("Total"))`
`=( i +(8i)/(R ) (8R)/(R +8 +16)) = i ((8+R)/(R)) ((8R)/(R+8)+ 16)` (i)
in case (ii) , V = i(24 +R) (ii)
Now, from Eqs. (i) and (ii) we, get
`(24R+ 128)/(R ) = 24 +R or R = sqrt128 Omega`