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In the given circuit, the current through the 5mH inductor in steady state is

$\left( {\text{A}} \right){\text{ }}\dfrac{2}{3}A$
$\left( {\text{B}} \right){\text{ }}\dfrac{8}{3}A$
$\left( {\text{C}} \right){\text{ }}\dfrac{1}{3}A$
$\left( {\text{D}} \right){\text{ }}\dfrac{5}{3}A$

Last updated date: 01st Mar 2024
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Hint: The inductors are connected in parallel; think which quantities will be the same for this combination and which quantities would change as the current flows in the circuit.

Formulae used:
$V = L\dfrac{{di}}{{dt}}$ , $I = \dfrac{V}{R}$ here, \[I\] is current, \[R\] is resistance and \[V\] is potential difference.

Complete step by step solution:
The steady current that flows in the circuit is given by-
$I = \dfrac{V}{R}$
From the figure, $V = 20V$ and $R = 5\Omega $
$\therefore I = \dfrac{{20}}{5} = 4A$
Now, when the current passes through the inductors, current \[I\] splits into \[{i_1}\] (across \[5{\text{ }}mH\] inductor) and i2 (across \[10mH\] inductor)
$\therefore I = {i_1} + {i_2}$ ---------(1)
We need to find current across 5mH inductor i.e. \[\;{i_2}\]
Now since the inductors are connected in parallel then, the voltage across them will be same
$\therefore {V_1} = {V_2}$ (${V_1}$ for \[5mH\] and ${V_2}$ for\[10mH\])
$ \Rightarrow {L_1}\dfrac{{d{i_1}}}{{dt}} = {L_2}\dfrac{{d{i_2}}}{{dt}}$
After integration,
${L_1}{i_1} = {L_2}{i_2}$ --------(2)
From equation 1,
${i_2} = I - {i_1}$
Substituting in equation 2, we get
${L_1}{i_1} = {L_2}(I - {i_1})$
$ \Rightarrow ({L_1} + {L_2}){i_1} = {L_2}I$
$\therefore {i_1} = I\left( {\dfrac{{{L_2}}}{{{L_1} + {L_2}}}} \right)$
Substituting the values of \[{L_1}\] and \[{L_2}\] we get,
${i_1} = \left( {\dfrac{{10}}{{10 + 5}}} \right) = 4\left( {\dfrac{{10}}{{15}}} \right) = \dfrac{8}{3}A$

Additional information:
Four applications of inductors you should know:
1. Filters: It is used with capacitors and a resistor to form LCR circuits to create filters for analog circuits and in signal processing.
2. Sensors: Inductors detect magnetic field or the presence of magnetically permeable material from a distance therefore they are used in sensors. Combining inductors that have shared magnetic path forms a transformer.
3. Transformers are the main components used in power transmission.
4. It is used to store energy for switch mode power supplies like in PCs etc.

Note: For any devices that are connected in parallel, the potential difference for both the devices remains same and the current changer, for series connection, the current across the devices remains same but the potential difference across them changes. This holds good for all electrical devices such as resistors, inductors, capacitors, bulbs, etc.