An unknown resistance X is to be determined using resistances `R_1, R_2 or R_3`. Their corresponding null points are A, B and C. Find which of the above will give the most accurate reading and why?
. `R= R_1 or R_2 or R_3`.

Explanation

At all null points, the wheatstone bridge will be
balanced. Therefore,
`(X)/(r_(1)) = (R)/(r_(2))` or `X = R (r_(1))/(r_(2))`
where `R` is aconstant and `R_(1)` and `r_(2)` are varibles. The maximum faction error is
`(Delta X)/(X) = (Deltar_(1))/(r_(1)) + (Deltar_(2))(r_(2))`
Here `Deltar_(1) = Deltar_(2) = y` (say),
then
`(Delta X)/(X) = y[(r_(2) + r_(1))/(r_(1)r_(2))]`
For `Delta X//X` to be minimum,
`r_(1) xx r_(2)` should be maximum
`["since" r_(1) + r_(2) = c ("constant")]`.
Let `E = r_(1) xx r_(2) = r_(1) xx (c - r_(1))`
`= r_(1) c - r_(1)^(2)`
`:. (dE)/(dr_(1)) = c - 2r_(1) = 0` or `r_(1) = (c)/(2)`
or `r_(2) = (c)/(2)` or `r_(1) = r_(2)`
Thus, `R_(2)` gives the most accurate value.