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Combination of Resistances

In the circui...

Question

In the circuit shown in figure belwo, $E_{1} = 17 V, E_{2} = 21 V, R_{1} = 2 Ω, R_{2} 3 Ω$ and $R_{3} = 5 Ω$ . Using Kirchhoff's laws, find the currents flowing through the resistors $R_{1}, R_{2}$ and $R_{3}$ . (Internal resistance of each of the batteries is negligible)

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Solution

Let us suppose that $I_{1}$ current flows through $E_{1}$
and $I_{2}$ current flows through $E_{2}$

Using Kirchhoff's Law:
$I_{1} R_{1} + (I_{1} + I_{2}) R_{3} = E_{1}$
$I_{1} \times 2 + (I_{1} + I_{2}) 5 = 17$
$2 I_{1} + 5 I_{1} + 5 I_{2} = 17$
$7 I_{1} + 5 I_{2} = 17$

$I_{1} R_{2} + (I_{1} + I_{2}) R_{3} = E_{2}$
$I_{2} \times 3 + (I_{1} + I_{2}) 5 = 21$

$5 I_{1} + 8 I_{2} = 21$
$35 I_{1} + 25 I_{2} = 85$
$3 - 5 I_{1} + 5 - 6 I_{2} = 1 - 47$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 31 I_{2} = - 62$
$I_{2} = 2 a m p$

$7 I_{1} + 5 I_{2} = 17$
$7 I_{1} + 10 = 17$
$7 I_{1} = 7$
So, $I_{1} = 1 a m p$

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