Question

An unknown resistance $X$ is to determined using resistances $R_1,R_2\text{or}{R}_3$. Their corresponding null points are $\text{A},\text{B}$ and $\text{C}$. Which of the above will give the most accurate reading ?
$($Length of $\text{OD}$ is $1\text{m}$. Consider jockey to move gradually on rheostat for changing values)

• A. $R=R_1$
• B. $R=R_2$
• C. $R=R_3$
• D. Can't be determined

Answer 1

The correct option is B $R=R_2$

All the null points will correspond to the balanced condition of formed Wheatstone bridge.

At null point,

$\frac{X}{R}=\frac{l_1}{l_2}$

$X=R\left(\frac{l_1}{l_2}\right)$

Here, $R$ is a constant and $l_1,l_2$ are variables.

So, the maximum fractional error is

$\frac{\Delta X}{X}=\frac{\Delta l_1}{l_1}+\frac{\Delta l_2}{l_2}$

Here, for small variables let $\Delta l_1=\Delta l_2=y$,

$\frac{\Delta X}{X}=y\left[\frac{l_2+l_1}{l_1{l}_2}\right]$

Now for $\frac{\Delta X}{X}$ to be minimum (most accurate readings)

$l_1{l}_2$ should be maximum, because $l_1+l_2=1\text{m}=\text{constant}$

Now, let $z=l_1{l}_2$

$z=l_1(1-l_1)=l_1-l_1^2$

differentiate both sides w.r.t $l_1$ ans equally to zero.

$\frac{dz}{d{l}_1}=1-2l_1=0$

$l_1=1/2\text{m}$

Similarly, we get

$l_2=1-\frac{1}{2}=\frac{1}{2}m$

$\therefore l_1=l_2=\frac{1}{2}\text{m}$

It corresponds to null point $\text{B}$ i.e $R=R_2$ where distance of null point is half the length.

$\therefore R=R_2$ gives the most accurate reading.