Question

A $6\text{V}$ battery with negligible internal resistance is connected across a uniform wire of length $100\text{cm}$. The positive terminal of another battery of emf $4\text{V}$ and internal resistance $1\Omega$ is connected at $A$ as shown in the figure. Take the potential at $B$ to be zero and find out which of the following is correct?

• A. The potential at $A$ is $6\text{V}$ and at $C$ is $2\text{V}$.
• B. The potential is $4\text{V}$ on $AB$ at a distance of $66.67\text{cm}$ from $A$.
• C. If $C$ is connected at $66.67\text{cm}$ from $A,$ then there will be no current in the $4\text{V}$ battery.
• D. If the $6\text{V}$ battery is replaced by a $12\text{V}$ battery, then the potential at $A$ would be $8\text{V}$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A The potential at $A$ is $6\text{V}$ and at $C$ is $2\text{V}$.
B The potential is $4\text{V}$ on $AB$ at a distance of $66.67\text{cm}$ from $A$.
C If $C$ is connected at $66.67\text{cm}$ from $A,$ then there will be no current in the $4\text{V}$ battery.

Given,
$V_B=0$
As $AB$ is connected to battery of $6\text{V}$
$V_A-V_B=6\text{V}$
$V_A=6\text{V}$
As C is open,
$V_A-V_C=4$
$V_C=4\text{V}$

Potential gradient
$K=\frac{V}{l}=\frac{6}{100}=0.06\text{V/cm}$

When $C$ is connected to $AB$ such that $AC=x\text{cm}$

As no current flows,

$V_{AC}=4\text{V}=xK$
$x=\frac{4}{0.06}=66.67\text{cm}$

If the $6\text{V}$ battery is replaced by a $12\text{V}$ battery, then $V_A-V_B=12\text{V}\Rightarrow V_A=12\text{V}$