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     When  an Inductor L and a resistor R are connected in series with a DC power supply V, the inductor does not allow immediately the full current V/R to flow through the circuit.  The initial few milliseconds duration is called the transient response duration of the circuit.

     As the current tries flow across the inductor, voltage drops across the inductor. from DC voltage V to 0 in the steady state.   In the steady state the change in current in the circuit is 0.  Hence, the voltage across the inductor is 0.

The voltages across elements in the LR circuit is:
           V_dc =  R i(t) +  L di/dt

The solution for this differential equation is :
             i(t) = (V/R) [ 1 - e^{-\frac{R}{L}t} ]

So at t = 0, i(0) = 0.    at  t = infinity ,  i(∞) = V/R.    Thus current steadily increases.

   An inductor opposes change in the current that passes through it.  Inductor stores energy inside the magnetic field developed in the space or magnetic core in the cylindrical space among the coil turns.  These are according to Faraday's law of induction.

   When the current tries to increase through the inductor, an opposing emf is developed in the direction opposite to the flow of current.  The voltage drop across the coil is L di/dt.  Initially  di/dt is very high.  It slowly reduces.  The rate at which energy is stored inside the coil decreases.  Energy stored increases till the steady state.  Finally the current increases slowly, the voltage drop across the inductor reduces, induced emf is reduced.  Finally in steady state an inductor becomes a conducting wire, with no voltage difference across terminals.

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why will the current try to increase through the inductor? DC supply is a source of constant is not AC SOURCE
initially there is 0 current. then there is a voltage applied across L and R. An inductor is nothing but a coil of conducting wire. Hence current passes through it. The difference between R and L is that, due to the turns and coil structure, there is a magnetic field developed inside the coil. The magnetic flux opposes the current. Still as the inductor is a conductor wire, the current passes through it. The energy to overcome the opposing flux is given by the battery.
Most of the energy is absorbed initially by the Inductor and only a small current passes through the resistance. At t = 0, I = 0. Due to potential difference, there is current passing through it. Hence, the inductor becomes a short circuit after the transient period. It is the capacitor which is an open circuit for DC current.
for DC source, inductor is a short circuit with 0 resistance and capacitor is an open circuit with infinite resistance... X_L = w L and X_C = 1 / (w C)
thank you very much