Current Decay and the L over R Time Constant

An inductor resists any change in the current through it. While the battery is connected, current builds up and energy gets stored in the inductor's magnetic field, given by (1/2)LI^2. Cut the battery out of the circuit, and the inductor refuses to let that current stop instantly. It acts like a source of its own, pushing charge around the loop in the same direction as before, with the resistor now the only thing draining the energy away. The current doesn't vanish; it decays smoothly as the stored magnetic energy gets converted to heat in the resistor.

The decay follows I(t) = I0 e^−tR/L, where I0 is the current at the moment the battery is removed. The exponent tR/L is dimensionless, so L/R has units of time; call it tau, the time constant. A bigger inductance means the coil fights harder against change, stretching out the decay. A bigger resistance burns off the stored energy faster, shortening it. After one time constant the current has fallen to about 37 percent of I0; after a few tau it is essentially gone.

This mirrors the charging and discharging of a capacitor through a resistor, with L/R playing the role RC played there, and current standing in for charge. The same exponential math reappears whenever energy drains from a reservoir at a rate proportional to what remains, which is why this topic sets up the comparison to RC circuits and, later, the oscillations that show up when an inductor trades energy with a capacitor instead of a resistor.