Question

A potentiometer wire $\text{AB}$ is $100\text{cm}$ long and has a total resistance of $10\Omega$. if the galvanometer shows zero deflection at the position $\text{C}$, then find the value of unknown resistance $\text{R}$.

• A. $4\Omega$
• B. $6\Omega$
• C. $5\Omega$
• D. $2.5\Omega$

Answer 1

The correct option is A $4\Omega$

The potential difference across $\text{AB}$ is
$V_{AB}=10\text{volt}$

Potential gradient of wire $\text{AB}$,

$\frac{dV}{dl}=\frac{10}{100}=\frac{1}{10}\text{V/cm}$

$\therefore V_{AC}=L_{AC}\times \frac{dV}{dl}$

$V_{AC}=40\text{cm}\times \frac{1}{10}\text{V/cm}=4\text{V}$

Redrawing the circuit at balanced state,

$i=\frac{5}{1+R}$

Also equating the potential difference,

$V_{DE}=V_{AC}$

$V_D-5+1\left(\frac{5}{1+R}\right)=V_E$

$\Rightarrow V_D-V_E=5-\frac{5}{1+R}$

$V_A-V_C=5-\frac{5}{1+R}$

$4=5-\frac{5}{1+R}\Rightarrow 1=\frac{5}{1+R}$

$\Rightarrow 1+R=5$

$\therefore R=4\Omega$

Hence, option $\left(a\right)$ is the correct answer.