JEE Main & Advanced Physics Electro Magnetic Induction Growth and Decay of Current In LR- Circuit

If a circuit containing a pure inductor L and a resistor R in series with a battery and a key then on closing the circuit current through the circuit rises exponentially and reaches up to a certain maximum value (steady state). If circuit is opened from it's steady state condition then current through the circuit decreases exponentially.

(1) The value of current at any instant of time t after closing the circuit (i.e. during the rising of current) is given by \[i={{i}_{\mathbf{0}}}\left[ \mathbf{1}-{{e}^{-\,\frac{R}{L}t}} \right]\]; where \[{{i}_{0}}={{i}_{\max }}=\frac{E}{R}\]= steady state current.

(2) The value of current at any instant of time t after opening from the steady state condition (i.e. during the decaying of current) is given by  \[i={{i}_{\mathbf{0}}}{{e}^{-\,\frac{R}{L}t}}\]

(3) Time constant \[(\tau )\]: It is given as \[\tau =\frac{L}{R}\]; It's unit is second. In other words the time interval, during which the current in an inductive circuit rises to 63% of its maximum value at make, is defined as time constant or it is the time interval, during which the current after opening an inductive circuit falls to 37% of its maximum value.