RL Time Constant (τ = L/R)

Close a switch on a coil and the current does not jump to its final value — it climbs. The inductor generates a back-EMF proportional to how fast the current is changing, and that back-EMF eats into the voltage available to drive the current, so the rise is self-limiting in exactly the way an RC charge is. The current covers 63.2% of its remaining gap in each interval $\tau = L/R$, and is within 1% of its final $V/R$ after 5τ. The units check out because a henry is a volt-second per ampere and an ohm is a volt per ampere: divide and the volts and amperes cancel, leaving seconds.

A 100 mH relay coil wound to 50 Ω has $\tau = 0.1/50 = 2\ \text{ms}$, so the current is essentially established after about 10 ms and the relay clicks in on that timescale — which is why mechanical relays are measured in milliseconds while transistors are measured in nanoseconds. Reduce the resistance and the coil gets slower, which is the counterintuitive part: τ is $L/R$, so halving $R$ doubles the time constant even though it doubles the final current. A superconducting loop has $R = 0$ and a time constant that is formally infinite, which is precisely why persistent-mode magnets hold their current for years.

The same constant governs the decay when the supply is removed, and that case is where the practical importance lies. The current will not stop, so it will find a path; if the only path is the opening switch contact, the inductor drives the voltage up until the air breaks down and the current continues as an arc. A flyback diode across the coil gives it a legal route instead. Note what the diode does to the timing: the decay's τ is set by the resistance in whatever loop the current ends up in, so a plain diode across the coil leaves only the winding resistance and the current takes a long time to die — which delays relay dropout — while adding a resistor or a Zener in that path shortens τ at the cost of a higher clamp voltage. Designers trade those two off deliberately.

Get the resistance right and the order of the division right. The $R$ that matters is every ohm in the current's loop — the winding's own DC resistance, the source's internal resistance, the sense resistor, the switch — and using only the external resistor when the coil itself has more resistance than it will give a τ that is badly wrong. Second, the arrangement is $L/R$, not $RL$; the RC constant is a product and the RL constant is a quotient, and swapping them is the most common slip on this page. A quick sanity check: more inductance should mean slower, more resistance should mean faster. Third, inductance is often given in millihenries or microhenries and must be converted before dividing. And as with any inductor equation, an iron or ferrite core makes $L$ a function of current, so a coil approaching saturation has a shrinking time constant and its current rises faster than the calculation predicts.