The current enters at A and leaves at F. The values of some resistance are shown. What should be the value of resistance AB so that no current will flow through CB?

Options

A. `((5)/(7))R`
B. `(R )/(3)`
C. `((5)/(3))R`
D. `((5)/(9))R`

Answer

C

Explanation

Let `V_(A) -V_(F) =V`
if `V_(A) -V_(C ) = V_(A) -V_(B)`
Equivalent resistance along ADCEF is
`R((2)/(7) +(3)/(7) +(1)/(7) +(1)/(7)) =R` (Remove CB)
Along ADCEF, `V_(A) -V_(F) = V =I_(1)R`
`V_(1) = (V)/(R )`
`V_(A) -V_(C ) = I_(1)(5)/(7)R =(5)/(7)R = (5V)/(7) =V_(A) -V_(B)`
[From Eq. (i)]
Along path ABF, equivalent resistance is
`R_(AB) +(2)/(3)R`
`V_(A) -V_(F) =V = I_(2)(R_(AB)+(2)/(3)R)`
or `I_(2)=(V)/(R_(AB)+(2)/(3)R) or V_(A) -V_(B) =I_(2) R_(AB) -V_(A) -V_(C )`
`[(V)/(R_(AB)+(2)/(3)R)R_(AB)]=(5)/(7)R` [from Eq.(ii)]
or ` R_(AB)=(5)/(3)R`