Question

Five identical cells each of internal resistance $1\Omega$ and emf $5\text{V}$ are connected in series and in parallel with an external resistance '$R$'. For what value of '$R$', current in series and parallel combination will remain the same?

• A. $10\Omega$
• B. $1\Omega$
• C. $5\Omega$
• D. $25\Omega$

Answer 1

The correct option is B $1\Omega$

When cells are connected in the series combination :
The equivalent emf is :
${ϵ}_s={ϵ}_1+{ϵ}_2+{ϵ}_3+{ϵ}_4+{ϵ}_5$
${ϵ}_s=25\text{V}$
The equivalent internal resistance is :
$r_s=r_1+r_2+r_3+r_4+r_5=5\Omega$
Therefore, the current in series combination is :
$I_s=\frac{{ϵ}_s}{R+r_s}=\frac{25}{R+5}$

When cells are connected in parallel combination :
The equivalent internal resistance is :
$\frac{1}{r_p}=\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}+\frac{1}{r_5}$
$r_p=\frac{1}{5}\Omega$
The equivalent emf is :
$\frac{{ϵ}_p}{r_p}=\frac{{ϵ}_1}{r_1}+\frac{{ϵ}_2}{r_2}+\frac{{ϵ}_3}{r_3}+\frac{{ϵ}_4}{r_4}+\frac{{ϵ}_5}{r_5}$
${ϵ}_p=5\text{V}$
Therefore, the current in parallel combination is :
$I_p=\frac{{ϵ}_p}{R+r_p}=\frac{5}{R+1/5}$

According to the question :
$I_p=I_s$
$\frac{25}{R+5}=\frac{5}{R+1/5}$
$\Rightarrow R=1\Omega$