Calculate the value of the ratio i_1/i_2 for the circuits shown in...

For the first circuit, the two cells are connected in series and aid each other.

The equivalent EMF is $E_1 = 6 + 6 = 12 \text{ V}$.

The equivalent internal resistance is $r_1 = 1 + 1 = 2 \text{ } \Omega$.

The current $i_1$ in the circuit is given by:

$i_1 = \dfrac{E_1}{R + r_1} = \dfrac{12}{10 + 2} = 1 \text{ A}$

For the second circuit, the two cells are connected in parallel.

The equivalent EMF is $E_2 = \dfrac{\dfrac{6}{1} + \dfrac{6}{1}}{\dfrac{1}{1} + \dfrac{1}{1}} = 6 \text{ V}$.

The equivalent internal resistance is $r_2 = \dfrac{1 \times 1}{1 + 1} = 0.5 \text{ } \Omega$.

The current $i_2$ in the circuit is given by:

$i_2 = \dfrac{E_2}{R + r_2} = \dfrac{6}{10 + 0.5} = \dfrac{6}{10.5} = \dfrac{4}{7} \text{ A}$

The ratio of the currents is:

$\dfrac{i_1}{i_2} = \dfrac{1}{\dfrac{4}{7}} = \dfrac{7}{4} = 1.75$