Find the magnitudes of currents \( i_1 \), \( i_2 \), and \( i \) with the help of the given circuit, when (i) just at the moment switch S is pressed and (ii) S is pressed for a long time.
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In an RL circuit, the inductor initially resists current flow and allows full current flow after a long time.
(i) Just at the moment switch \( S \) is pressed:
At \( t = 0 \), the inductor opposes any change in current, so the initial current through the inductor is zero.
\[
i_2 = 0, \quad i_1 = \frac{10}{5} = 2 \text{ A}
\]
\[
\boxed{i = i_1 = 2 \text{ A}}
\]
(ii) After a long time:
The inductor acts as a short circuit, and the total resistance becomes:
\[
R_{\text{eq}} = 5 + 10 = 15 \, \Omega
\]
\[
i = \frac{10}{15} = \frac{2}{3} \text{ A}
\]
Current division:
\[
i_1 = \frac{2}{3} \times \frac{10}{15} = \frac{2}{3} \text{ A}, \quad i_2 = \frac{2}{3} \times \frac{5}{15} = \frac{1}{3} \text{ A}
\]
\[
\boxed{i_1 = \frac{2}{3} \text{ A}, \quad i_2 = \frac{1}{3} \text{ A}}
\]
