Current Growth And Decay In RL Circuits

When the circuit closes, the increasing current produces an induced emf across the inductor that opposes the applied emf, following Lenz's law. This opposing emf counteracts the current increase, forcing the current to start at zero and rise gradually toward its steady-state value ε/R rather than jumping instantly.

The inductive time constant is the ratio of inductance to resistance (L/R). It measures how quickly current builds toward its final value. At time equal to one time constant, current grows to 63% of its final value during growth, or decays to 37% of its original value during decay.

As current increases from zero toward its steady-state value ε/R, the energy stored in the inductor increases asymptotically from zero to a maximum value. This stored energy in the magnetic field grows proportionally with the increasing current until the circuit reaches steady state.

When the emf source is disconnected and the circuit forms a single loop with only the resistor and inductor, the initial current ε/R decreases exponentially with time. The energy stored in the inductor is gradually depleted through the resistor until the current reaches zero.

Kirchhoff's loop rule states that the sum of emfs around a closed loop equals zero. In an RL circuit with applied emf, the applied emf equals the sum of the emf across the resistor and the induced emf across the inductor. Rearranging and integrating this equation yields the current growth equation.

During decay, the inductor's induced emf is proportional to the rate of current change. As current decreases, the induced emf also decreases, resulting in a slower decay rate at each moment. This proportional relationship produces exponential decay rather than a constant linear decrease.

Both RL and RC circuits exhibit exponential responses to switching, but with different time constants. RL circuits have time constant L/R, while RC circuits have time constant RC. Both show asymptotic approach to steady state, though the physical mechanisms differ—inductors oppose current change while capacitors oppose voltage change.