Question

A potentiometer wire \(AB\) having length \(L\) and resistance \(12 r\) is joined to a cell \(D\) of emf \(\epsilon\) and internal resistance \(r\). A cell \(C\) having emf \(\dfrac{\epsilon}{2}\) and internal resistance \(3r\) is connected. The length \(AJ\) at which the galvanometer as shown in fig. shows no deflection is


&

• A. $\frac{5}{12}\text{L}$
• B. $\frac{11}{12}\text{L}$
• C. $\frac{13}{24}\text{L}$
• D. $\frac{11}{24}\text{L}$

Answer 1

The correct option is C $\frac{13}{24}\text{L}$

Given:
Wire length $AB=L$
Resistance in $\left(AB\right),R_{AB}=12r$
emf of cell at $D$, $V_D=\in$
Resistance of cell at $D$, $R_D=r$
emf of cell at $C$, $V_C=\in /2$
Resistance of cell at $C$, $R_C=3r$

Current in circuit,
$\therefore i=\frac{ϵ}{12r+r}=\frac{ϵ}{13r}$

Let $x$ be the length $AJ$ at which galvanometer shows null deflection,

$\Rightarrow i\left(\frac{x}{L}12r\right)=\frac{ϵ}{2}$

$\Rightarrow \frac{ϵ}{13r}\left[\frac{x}{L}.12r\right]=\frac{ϵ}{2}$

$\Rightarrow x=\frac{13L}{24}$

Hence, option $\left(C\right)$ is correct.