Question

A cell having an emf $\epsilon$ and internal resistance $r$ is connected across variable resistance $R$. As the resistance $R$ is increased, the plot of potential difference $V$ across the resistance $R$ $vs$ its value is given by

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

The battery with emf $\epsilon$, internal resistance $r$ and variable resistance $R$ will make a closed circuit.


The variable current $i$ in the circuit will be,

$i=\frac{\epsilon }{R+r}$

The potential difference across resistance $R$ is given as,

$V=iR$

$\Rightarrow V=\frac{\epsilon R}{R+r}$

Dividing by $R$ the numerator and denominator, we get,

$V=\frac{\epsilon }{1+\frac{r}{R}}......\left(1\right)$

From eq.(1) we infer that

When $R=0$, then $\left(\frac{r}{R}\right)\to \infty$, thus the denominator becomes very large.

$\Rightarrow V=\frac{\epsilon }{1+\infty }=0$

$\therefore V\to 0$

Again if $R$ is very large, $R\to \infty$
In that case $\left(\frac{V}{R}\right)\to 0$.

$\Rightarrow V=\frac{\epsilon }{1+\left(\frac{r}{\infty }\right)}$

$\therefore V=\epsilon$ when $R\to \infty$

However, infinity is not a definite value hence $V=\epsilon$ will form the asymptote for plot of $VvsR$.

Therefore, the graph will look like option $\left(c\right)$.


$$\begin{array}{c}\text{Why this question?}\\ \text{Tip:When the resistance}R\text{infinitely}\\ \text{large}\left(\infty \right)\text{, then no current will pass}\\ \text{through circuit due to high resistance.Thus}\\ \text{circuit can be visualized as a open circuit,}\\ \text{hence potential difference across resistance}\\ R\text{will be equal to Emf}\left(E\right)\text{of battery.}\end{array}$$