Question 111: Current Electricity

Question

For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6 \Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is $\_\_\_\_$ $\Omega$.

Accepted Answer

Case 1 :
when cells are in series :
When two identical cells ( $\mathrm{E}, \mathrm{r}$ ) are connected in series with an external resistor $\mathrm{R}=6 \Omega$ :
Total $\mathrm{EMF} \mathrm{E}_{\mathrm{eq}}=\mathrm{E}+\mathrm{E}=2 \mathrm{E}$
Total internal resistance $\mathrm{r}_{	ext {eq }}=\mathrm{r}+\mathrm{r}=2 \mathrm{r}$
So, the current in the circuit is,
$$ i_s=\frac{E_{e q}}{R+r_{e q}}=\frac{2 E}{6+2 r} $$
Case 2 :
Cells in Parallel
When two identical cells ( $\mathrm{E}, \mathrm{r}$ ) are connected in parallel with an external resistor $\mathrm{R}=6 \Omega$ :
Equivalent EMF ( $\mathrm{E}_{\mathrm{eq}}$ ) in parallel is given as $\frac{\frac{\mathrm{E}_1}{r_1}+\frac{\mathrm{E}_2}{\mathrm{r}_2}+\cdots+\frac{\mathrm{E}_{\mathrm{n}}}{\mathrm{r}_{\mathrm{n}}}}{\frac{1}{\mathrm{r}_1}+\frac{1}{\mathrm{r}_2}+\cdots+\frac{1}{\mathrm{r}_{\mathrm{n}}}}$
Here, $\mathrm{E}_1=\mathrm{E}_2=\mathrm{E}$ and $\mathrm{r}_1=\mathrm{r}_2=\mathrm{r}$,
$$ E_{e q}=\frac{\frac{E}{r}+\frac{E}{r}}{\frac{1}{r}+\frac{1}{r}}=\frac{\frac{2 E}{r}}{\frac{2}{r}}=E $$
As the internal resistances are in parallel, equivalent internal resistance $\left(\mathrm{r}_{\mathrm{eq}}ight)$ :
$$ \frac{1}{\mathrm{r}_{\mathrm{eq}}}=\frac{1}{\mathrm{r}}+\frac{1}{\mathrm{r}}=\frac{2}{\mathrm{r}} \Rightarrow \mathrm{r}_{\mathrm{eq}}=\frac{\mathrm{r}}{2} $$
Hence, the current through external resistor is
$$ i_p=\frac{E_{e q}}{R+r_{e q}}=\frac{E}{6+\frac{r}{2}} $$
According to the question, $\mathrm{i}_{\mathrm{s}}=\mathrm{i}_{\mathrm{p}}$ :
$$ \begin{gathered} \frac{2 E}{6+2 r}=\frac{E}{6+\frac{r}{2}} \Rightarrow \frac{1}{6+2 r}=\frac{1}{12+r} \ \Rightarrow 12+r=6+2 r \Rightarrow 12-6=2 r-r \ \Rightarrow r=6 \Omega \end{gathered} $$
So, the value of internal resistance r is $6 \Omega$.
Hence, the correct option is (B).

Answer

3

Answer

6

Answer

4

Answer

9

Current Electricity

For the two cells having same EMF E and internal resistance r, the current passing through the external resistor 6 \Omega is same when both the cells are connected either in parallel or in series. The value of internal resistance r is \_\_\_\_ \Omega.

Choose the correct answer: