The distance between the positions of two null points obtained in a metre brige wire of length 100 cm , by interchaging a known resistance of `2.5Omega` and an unknown resistance in the two gaps , is 28.6 cm . Find the value of the unknown resistance .

Explanation

In first case , if the unknown resistance r is kept at the left gap and resistance of `2.5Omega` at the right gap , the position of null point is `l_(1)`.
`therefore" " (r)/(2.5)=(l_(1))/(100-l_(1)) " " ...(1) `
In second case , after interchanging the resistance , the position of full point is `l_(2)`
`therefore" " (r)/(2.5)=(l_(1))/(100-l_(2)) " " ...(2)`
According to the question ,
`l_(1)-l_(2)=28.6 " " ...(3)`
`By (1)xx(2)` we get ,
`1=(l_(1)l_(2))/((100-l_(1))(100-l_(2)))`
`"or, "10^(4)-100l_(1)-100l_(2)+l_(1)l_(2)=l_(1)l_(2)`
`"or, "l_(1)+l_(2)=100" "... (4)`
Solving equation (3) and (4) ,
`therefore" " l_(1)=64.3`
and`l_(2)=100-l_(1)=35.7`
`therefore` The value of the unknown resistance
`r=2.5xx(l_(1))/(100-l_(1))=2.5xx(64.3)/(100-64.3)=4.5Omega`