Time constant

Also called RC time constant, tau

The time constant is the characteristic time that sets how quickly a circuit settles toward its final state. For a resistor and capacitor it is resistance times capacitance, and one time constant takes a charging capacitor to about 63 percent of its final charge.

The time constant τ\tau answers how long a circuit takes to get most of the way to its final state. For a resistor and a capacitor, both the AP Physics 2 and the AP Physics C: E&M sheets print

τ=ReqCeq\tau = R_{\text{eq}} C_{\text{eq}}

and the C: E&M sheet adds the inductive version, τ=L/Req\tau = L / R_{\text{eq}}, for a resistor and an inductor. AP Physics 2 has no inductors, so only the first form applies there. Check the units once and you will trust both: an ohm times a farad is a second, and a henry divided by an ohm is a second.

What one time constant buys. For a charging capacitor, τ\tau is the time for the charge to rise from zero to roughly 63 percent of its final value. For a discharging one, it is the time for the charge to fall to roughly 37 percent of its initial value. Those figures are stated about charge, and since Q=CΔVQ = C \Delta V with CC unchanged, the potential difference across the capacitor tracks them. The current does not: while a capacitor charges, the current starts large and decays as the charge climbs.

The dependence is a plain product, so doubling either RR or CC doubles τ\tau and doubling both quadruples it. For times much greater than τ\tau the branch may be modeled with steady-state conditions, which is the license behind every question that says "after a long time". See RC circuit for the two states themselves.

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