$1.($a$)$ State three advantages of the Kelvin Double Bridge $($KDB$)$ for the satisfactory measurement of $0\mathrm{.}1$

resistance value.

$[1\mathrm{.}5$marks$]$

$($b$)($i$)$ Show that the balance equation for the KDB circuit in Figure $1($b$)$ is:

$x=\frac{Q}{M}S+\frac{mr}{m+q+r}(\frac{Q}{M}-\frac{q}{m}),$ where $Q,q,M,$ and $m$ are four known non$-$inductive resistances. $x$

is the unknown resistance, $r$ is a low resistance link, and $S$ is a standard resistance. $[4$marks$]$

Figure $1($b$)$

$($ii$)$ Explain why in practice the expression in b$($i$)$ is usually simplified to $x=\frac{Q}{M}S.[$Imark$]$

$($c$)$ In operating the bridge of Figure $1($b$),$ balance was obtained with $Q-q=1025,S=0\mathrm{.}010Q,r=$

$0\mathrm{.}001,M-1000Q$ and, by accident, $m=100.$ If the operator used the simplified expression lor