A cell is a source of electric current in the electrical circuit. The Potential Difference between terminals of a cell in an open circuit (when no current is drawn) is called electromotive force (emf) of the cell. When electrical circuit is closed and current is drawn from the terminal Potential Difference between two terminals is called terminal potential difference (v) of the cell. The cells can be connected in series as well as in parallel combinations. Like resistor cell also offers opposition to the flow of current. This opposition offered by cell is called internal resistance of the cell.

Two identical cells each of emf ε and internal resistance r when connected in series or in parallel across external resistance R give the same value of current. Then the relation between r and R is:

r = R

The correct answer is Option (1) → $r = R$

Case 1: Cells in Series

for two identical cell connected in series:-

Total emf: $ε_{series}=ε+ε=2ε$

Total internal resistance: $r_{series}=r+r=2r$

∴ By Ohm's law: $V=IR$

Total current in the circuit,

$I_{series}=\frac{ε_{total}}{r_{total}}=\frac{2ε}{R+2r}$

Case 2: Cells in parallel

for two identical cell connected in series:-

Total emf: $ε_{parallel}=ε$

[since emf in same for identical in parallel]

ans, $r_{parallel}=\frac{r}{2}$

For resistance connected in parallel

$\frac{1}{r_{parallel}}=\frac{1}{r_1}+\frac{1}{r_2}+....$

∴ $I_{parallel}=\frac{ε_{total}}{r_{total}}$ [By Ohm's law]

$I_{parallel}=\frac{ε}{R+\frac{r}{2}}$

Now,

$I_{series}=I_{parallel}$ [Given]

$\frac{2ε}{R+2r}=\frac{ε}{R+\frac{r}{2}}$

$\frac{2}{R+2r}=\frac{1}{R+\frac{r}{2}}$

$2\left(R+\frac{r}{2}\right)=R+2r$

$R=r$