Two identical batteries of emf $\mathit{\epsilon }\mathbf{=}\mathbf{12}\mathbf{.0}\mathbf{\text{V}}$and internal resistance $\mathit{r}\mathbf{=}\mathbf{0}\mathbf{.200}\mathit{\Omega }$are to be connected to an external resistance$\mathit{R}$ , either in parallel (Figure a) or in series (Figure b). (a) If ,$\mathit{R}\mathbf{=}\mathbf{2}\mathbf{.00}\mathit{r}$ whatis the current in the external resistance in the parallel arrangement? (b) If $\mathit{R}\mathbf{=}\mathbf{2}\mathbf{.00}\mathit{r}$,what is the current $\mathit{i}$in the external resistance in the series arrangements? (c) For which arrangement is$\mathit{i}$greater? (d) If$\mathbf{}\mathit{R}\mathbf{=}\mathit{r}\mathbf{/}\mathbf{2}\mathbf{.00}$ , what is in the external resistance in the parallel? (e) If $\mathbf{}\mathit{R}\mathbf{=}\mathit{r}\mathbf{/}\mathbf{2}\mathbf{.00}$, what is $i$ in the external resistance in the series arrangements? (f) For which arrangement is $i$ greater now?

The emf voltage,$\epsilon =12.0\text{V}$

The internal resistance,$r=0.200\Omega$

From the given information, if you apply the loop rule,you can find the current in the circuit. Using this, you can find the current across the external resistance in various arrangements.

Formula:

Current in block is define by,

$i=\frac{V}{R}$

For parallel combination,according to the junction rule, the current in $R$ is$2i$ .

Apply the loop rule for the given circuit.

$\begin{array}{l}\epsilon -ir-2iR=0\\ i=\frac{\epsilon }{r+2R}\end{array}$

And for the series combination,

$\begin{array}{l}\epsilon -2ir-iR=0\\ i=\frac{\epsilon }{2r+R}\end{array}$

The current across external resistance is.$i_r=2i$

Value of current $i$in parallel arrangement for$R=2.00r$.

$\begin{array}{c}{i}_r=2\left(\frac{\epsilon }{r+2R}\right)\\ =\frac{2\left(12.0\right)}{[0.200+2\left(0.400\right)]}\\ =\frac{24.0}{1.000}\\ =24.0\text{A}\end{array}$

Hence, the value of current in parallel arrangement for$R=2.00r$ is$24.0\text{A}$ .

The value of current in series arrangement for$R=2.00r$is,

$\begin{array}{c}{i}_r=2\left(\frac{\epsilon }{2r+R}\right)\\ =\frac{2\left(12.0\right)}{[2\left(0.200\right)+0.400]}\\ =\frac{24.0}{0.800}\\ =30.0\text{A}\end{array}$

Hence, the value of current $i$in series arrangement for$R=2.00r$is$30.0\text{A}$.

From the above results, it is clear that the current is greater in series combination.

The value of current $i$in series arrangement for$R=\frac{r}{2.0}$.

$\begin{array}{c}{i}_r=2\left(\frac{\epsilon }{r+2R}\right)\\ =\frac{2\left(12.0\right)}{[0.200+2\left(0.100\right)]}\\ =\frac{24.0}{0.400}\\ =60\text{A}\end{array}$

Hence, the value of current $i$ in parallel arrangement for$R=\frac{r}{2.0}$ is $60\text{A}$.

Value of current $i$in series arrangement for$R=\frac{r}{2.0}$.

$\begin{array}{c}{i}_r=2\left(\frac{\epsilon }{2r+R}\right)\\ =\frac{2\left(12.0\right)}{[2\left(0.200\right)+0.100]}\\ =\frac{24.0}{0.500}\\ =48.0\text{A}\end{array}$

Hence, the value of current $i$in series arrangement for$R=\frac{r}{2.0}$is$48.0\text{A}$.

From the above results, it is clear that the current is greater in parallel combination.