Question

In the circuit shown below, the current in the $1\Omega$ resistor is

• A. $1.3\text{A}$, from $\text{P}$ to $\text{Q}$
• B. $0\text{A}$
• C. $0.13\text{A}$, from $\text{Q}$ to $\text{P}$
• D. $0.13\text{A}$, from $\text{P}$ to $\text{Q}$

Answer 1

The correct option is C $0.13\text{A}$, from $\text{Q}$ to $\text{P}$

The circuit can be visualized as a parallel combination of two non-ideal cells shown below.


The equivalent emf of above arrangement is given by

$E_{eq}=\frac{\frac{E_1}{r_1}-\frac{E_2}{r_2}}{\frac{1}{r_1}+\frac{1}{r_2}}$

$\Rightarrow E_{eq}=\frac{\frac{9}{5}-\frac{6}{3}}{\frac{1}{5}+\frac{1}{3}}=\frac{\left(\frac{27-30}{15}\right)}{\left(\frac{3+5}{15}\right)}$

$\therefore E_{eq}=-\frac{3}{8}\text{V}$

And effective internal resistance of parallel combination is given by

$\frac{1}{r_{eq}}=\frac{1}{3}+\frac{1}{5}\Rightarrow \frac{1}{r_{eq}}=\frac{8}{15}$

$\therefore r_{eq}=\frac{15}{8}\Omega$

Redrawing circuit,


Thus, for simplified circuit, effective resistance will be

$R_{eq}=\frac{15}{8}+1=\frac{23}{8}\Omega$

Current in circuit,

$i=\frac{E_{eq}}{R_{eq}}$

$\Rightarrow i=\frac{\left(3/8\right)}{\left(23/8\right)}=\frac{3}{23}=0.13\text{A}$

Thus, direction of current in $1\Omega$ resistor will be form $\text{Q}$ and $\text{P}$ and its value is $0.13\text{A}$.
$$\begin{array}{c}\text{Why this question?}\\ \text{Tip:If the equivalent battery is having}\\ \text{negative emf, that means that polarity}\\ \text{of equivalent cell will be opposite to}\\ \text{the reference positive polarity battery.}\end{array}$$