Question

All the batteries in the infinite resistance circuit shown in the figure have emf $E$ and internal resistance $r$. Then, choose the correct options.

• A. Effective internal resistance of battery system across 'AB' is $r(\sqrt{3}-1)$.
• B. Effective emf of battery system across 'AB' is $2E\sqrt{3}(\sqrt{3}-1)$.
• C. If a resistance of value $r$ is connected across 'AB', the terminal voltage across the battery system is $2E$.
• D. Short circuit current across 'AB' for the above system is $E/3$.

Answer 1

Answer: (A), (B)

The correct options are
A Effective internal resistance of battery system across 'AB' is $r(\sqrt{3}-1)$.
B Effective emf of battery system across 'AB' is $2E\sqrt{3}(\sqrt{3}-1)$.


Let the effective resistance be $x$ . The circuit can be redrawn as

$\Rightarrow x=\frac{r(2r+x)}{2r+x+r}\Rightarrow x^2+2rx-2r^2=0\Rightarrow x=|\sqrt{3}-1|r$
Let effective emf be $y$

=
They are in parallel
Hence, $y=\frac{\frac{y+2E}{(\sqrt{3}+1)r}+\frac{E}{r}}{\frac{1}{r}+\frac{1}{(\sqrt{3}+1)r}}\text{as}(E=\frac{\frac{E_1}{r_1}+\frac{E_2}{r_2}}{\frac{1}{r_1}+\frac{1}{r_2}})\Rightarrow y=2E\sqrt{3}(\sqrt{3}-1).$