RC vs LR Circuit: Time Constants Compared

The inversion, stated once

Put a resistor with a capacitor and you get a circuit that takes time to settle. Put a resistor with an inductor and you get the same thing. Both are described by a single characteristic time, both approach their final state exponentially, and both are the second half of a Physics C circuits problem. The similarity is real, and it is why the two get filed together.

The difference is in one symbol's position:

Resistance multiplies in one and divides in the other. Everything on this page follows from that.

Raise the resistance in an RC circuit and it gets slower. Double $R$ and the capacitor takes twice as long to charge.

Raise the resistance in an LR circuit and it gets faster. Double $R$ and the current reaches its final value in half the time.

That is not a curiosity, it is the thing students get wrong under time pressure, because "more resistance means more sluggish" is a good intuition that happens to be exactly backwards for half the cases. Worked example one puts numbers on it, and the section after next explains why the physics genuinely runs the other way rather than the algebra playing a trick.

The RC circuit and LR circuit entries define each circuit; the time constant entry defines $\tau$. This page is about the pair.

Side by side

The fourth row is the one people skip. Doubling the storage element slows both circuits down, and that part of the intuition is fine: a bigger capacitor needs more charge, a bigger inductor fights harder against a change in current. It is only resistance that behaves in opposite ways, and that is precisely because $R$ is on top in one expression and underneath in the other.

The last row means the comparison itself is a Physics C question. AP Physics 2 has RC circuits and no inductors at all, so a Physics 2 student never meets $\tau = L/R$ and has no reason to confuse the two.

Why resistance works in opposite directions

The algebra is one line, but the physical reasons are different in each case and it is worth having both.

In an RC circuit, the resistor is the throttle on the charge supply. Charging a capacitor means delivering coulombs to its plates, and the current that delivers them is limited by the resistor. A bigger resistor means a smaller current at every stage, which means the same final charge $Q = C\mathcal{E}$ takes longer to arrive. More resistance, slower charging. Nothing surprising.

In an LR circuit, the resistor sets the target the current is trying to reach. The final current is $I_{\infty} = \mathcal{E}/R$, so a bigger resistor means a smaller target. Meanwhile the inductor's opposition depends only on how fast the current is changing, through $\mathcal{E}_{\mathrm{ind}} = -L\frac{dI}{dt}$, and it does not know or care what the target is. So a bigger resistor asks the circuit to travel a shorter distance against the same opposition, and the trip is quicker. More resistance, faster settling, and a smaller final current.

Those two sentences are the page. Read them side by side and the inversion stops being a formula to remember:

• The resistor stands in the way of what an RC circuit is trying to do, so more of it takes longer.
• The resistor defines what an LR circuit is trying to do, so more of it means less to do.

One consequence catches people out. In an LR circuit, raising $R$ makes the current settle sooner and settle lower. Those are not in tension. "Faster" here means the exponential shape is compressed in time, not that more current arrives.

A second consequence is worth carrying into lab questions. To get a time constant of about a second from an RC circuit you need a large resistance, thousands of ohms with a typical capacitor. To get a time constant of about a second from an LR circuit you need a small resistance, a few ohms with a large inductor. Worked example two builds both, and the resistances differ by a factor of a thousand for the same $\tau$.

What the equation sheet prints, and what it leaves to you

Both time constants are printed on the C: E&M equation sheet, in different blocks of the same page:

• $\tau = R_{\mathrm{eq}}C_{\mathrm{eq}}$ sits with the series and parallel combination rules, at the end of the circuits group.
• $\tau = \dfrac{L}{R_{\mathrm{eq}}}$ sits near the bottom of the magnetism group, between $\mathcal{E} = -L\frac{dI}{dt}$ and $\omega_{LC} = \frac{1}{\sqrt{LC}}$.

Both carry the subscript "eq", which is the sheet telling you to reduce the network first: the $R$ in either expression is the equivalent resistance the storage element actually sees, not whichever resistor is drawn nearest to it.

What the sheet does not print is the exponential itself. There is no $q(t) = Q(1 - e^{-t/\tau})$ and no $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$ anywhere on it. That absence is deliberate and it tells you what the exam wants. The CED gives each circuit as a differential equation to be set up from Kirchhoff's loop rule and then solved:

• Essential knowledge 11.8.B.1 states that the charge on a capacitor or the current in a resistor in an RC circuit can be described by a fundamental differential equation derived from Kirchhoff's loop rule, with the derived equation $\mathcal{E} = \frac{dq}{dt}R + \frac{q}{C}$.
• Essential knowledge 13.5.A.2 states that Kirchhoff's loop rule can be applied to a series LR circuit with a battery of emf $\mathcal{E}$, resulting in a differential equation that describes the current in the loop, with the derived equation $\mathcal{E} = IR + L\frac{dI}{dt}$.

Those two are the same equation with different labels: one unknown, one resistive term, one storage term, and the same exponential solution shape. Seeing that is worth more than memorising either.

On the AP Physics 2 sheet, $\tau = R_{\mathrm{eq}}C_{\mathrm{eq}}$ is printed and $\tau = L/R_{\mathrm{eq}}$ is not, along with no $L$ at all. The symbol $\tau$ appears in that sheet's symbol list defined as the time constant, and the only time constant that course has is the RC one.

The 63 and 37 percent statements, mapped correctly

The CED gives both circuits the same pair of milestones, and it attaches them to different quantities in each case. Getting the mapping right is the difference between a correct sentence and a nearly correct one.

For an RC circuit, essential knowledge 11.8.B.2.ii and 11.8.B.2.iii read: for a charging capacitor, the time constant represents the time required for the capacitor's charge to increase from zero to approximately 63 percent of its final asymptotic value; and for a discharging capacitor, the time constant represents the time required for the capacitor's charge to decrease from fully charged to approximately 37 percent of its initial value. Both statements are about charge.

For an LR circuit, essential knowledge 13.5.A.3.iii and 13.5.A.3.iv read: for an inductor that has zero initial current, the time constant represents the time required for the current in the inductor to reach approximately 63 percent of its final asymptotic value; and for an inductor with an initial current, the time constant represents the time required for the current in the inductor to reach approximately 37 percent of its initial value. Both statements are about current.

So the growing quantity is charge in one circuit and current in the other, which is the same swap that runs through the whole inductor and capacitor comparison: the capacitor's continuous variable is voltage, and charge tracks it exactly; the inductor's continuous variable is current.

One more CED line worth having, because it tells you what else follows the same curve. For the RC case, 11.8.B.3.iii says the potential difference across a capacitor, the current in its branch, and the electric potential energy stored in it all change with respect to time and asymptotically approach steady state conditions. For the LR case, 13.5.A.4.ii says the potential difference across an inductor, the current in the inductor, and the energy stored in the inductor are exponential with respect to time and have asymptotes determined by the initial conditions of the circuit. In both circuits, all three quantities share one time constant.

The case that separates them: same resistor, then change it

Build two circuits with the same resistor and tune the storage element so that both have exactly the same time constant. Then change the resistor and watch them go opposite ways.

Take $R = 25 \ \Omega$ in both. Choose $C = 800 \ \mu\mathrm{F}$ and $L = 0.50 \ \mathrm{H}$:

• $\tau_{RC} = (25)(800 \times 10^{-6}) = 0.020 \ \mathrm{s}$
• $\tau_{LR} = 0.50/25 = 0.020 \ \mathrm{s}$

Identical. Watch them on an oscilloscope and you could not tell the two traces apart by their timing.

Now raise the resistor to $100 \ \Omega$, a factor of four, and change nothing else:

• $\tau_{RC} = (100)(800 \times 10^{-6}) = 0.080 \ \mathrm{s}$, four times slower
• $\tau_{LR} = 0.50/100 = 0.0050 \ \mathrm{s}$, four times faster

The two time constants started equal and now differ by a factor of sixteen. That is the sharpest way to see the inversion: not that the formulas look different, but that a single change made to both circuits at once sends them in opposite directions and separates them by the square of the factor you applied.

Worked example three runs this through and checks the arithmetic. It is also the shape of a common multiple-choice question, phrased as "the resistance in each circuit is doubled; which of the following is true", where three of the four options assume both circuits do the same thing.

When it costs a mark

Writing $\tau = LR$. It produces a number with the wrong units and the wrong dependence, and it looks like the RC formula, which is why the hand writes it anyway. Run the units check: henries times ohms is not seconds. $\mathrm{H}/\Omega$ is.

Using the nearest resistor rather than the equivalent resistance. Both printed formulas carry the subscript "eq". In a network, work out what resistance the capacitor or inductor actually sees, which usually means shorting the ideal battery and reducing the network from the component's terminals.

Assuming more resistance always slows things down. True for RC, false for LR. If a question changes $R$ and asks about timing, name which circuit you are in before answering.

Reading 63 percent as a definition rather than a milestone. One time constant is not the time to finish; it is the time to get about 63 percent of the way. Full settling is asymptotic and never technically complete, which is why the CED phrases the steady state as something the circuit approaches. Three time constants gets you to about 95 percent, five to about 99 percent.

Attaching the 63 percent to the wrong quantity. In an RC circuit it is the charge (and therefore the voltage) that rises to 63 percent; the current is falling through the same interval and is at about 37 percent of its initial value at $t = \tau$. In an LR circuit it is the current that rises to 63 percent while the inductor voltage falls to about 37 percent of the applied value.

Quoting an exponential formula that is not printed and getting a sign wrong in it. Since neither $q(t)$ nor $I(t)$ is on the sheet, you either derive it or recall it. The safer habit on free response is to write the loop rule, show the differential equation the CED names, and let the exponential follow, which is also what the graded steps are looking for.

Treating an LC circuit as a slow version of these. An LC circuit has no resistor and does not settle at all: it oscillates, at $\omega_{LC} = \frac{1}{\sqrt{LC}}$, which is a frequency and not a time constant. Different topic, 13.6, and different behaviour.

What the two circuits share

The inversion is the headline, but the structural similarity is what makes it worth learning once instead of twice.

• Same differential equation shape. Both come out of Kirchhoff's loop rule as a first-order linear equation with a constant driving emf, and both have the same exponential solution. Solve one and you can pattern-match the other.
• One time constant governs everything. Voltage, current and stored energy in either circuit all evolve on the same $\tau$, which the CED states for both cases.
• The resistor gets the energy in the end. On discharge, whatever the capacitor or the inductor was holding is dissipated in the resistance. Essential knowledge 13.5.A.1 says so for the inductor case directly: a resistor will dissipate energy that was stored in an inductor as the current changes.
• Both settle into a resistor-network problem. After a long time the capacitor is a break and the inductor is a wire, so both circuits become ordinary DC networks with no calculus in them. That steady-state part is usually where the easy marks are.
• Both are asked about at three moments. The instant the switch closes, one time constant later, and a long time afterwards. Getting the first and third right is worth more than any formula, and those are set entirely by which component you have.

So the practical routine is identical for both: work out the $t = 0$ state from the continuity rule, work out the $t \to \infty$ state by replacing the component with a wire or a break, compute $\tau$, and connect the two ends with an exponential. Only the third step and the direction of travel differ.

Where these sit in the courses

RC circuits are Topic 11.8, the closing topic of Unit 11, Electric Circuits, which the CED weights at 15 to 25 percent of the AP Physics C: Electricity and Magnetism multiple-choice section over roughly 12 to 24 class periods. Learning objective 11.8.B is to describe the behaviour of a circuit containing combinations of resistors and capacitors, and the topic also carries the equivalent-capacitance rules under 11.8.A.

LR circuits are Topic 13.5 in Unit 13, Electromagnetic Induction, weighted at 10 to 20 percent over roughly 10 to 20 class periods. Learning objective 13.5.A is worded for a circuit containing a combination of resistors and a single inductor, so unlike the capacitor case you are never asked to combine two of the storage elements.

The two units are separated by Unit 12 on magnetic fields, which is why they arrive as separate ideas rather than as a matched pair. Reading Topic 13.5 immediately after Topic 11.8 is the cheapest way to make the second one feel like revision, because the derivation, the milestones and the graph shapes are the same and only the roles of the variables move.

In AP Physics 2, Topic 11.8 covers RC circuits qualitatively and its equation sheet prints $\tau = R_{\mathrm{eq}}C_{\mathrm{eq}}$. That course has no inductance topic, no inductors, no $L$ in its symbol list and no LR circuits, so "RC vs LR" is a question only a Physics C student has. If you are revising for AP Physics 2, the RC half of this page applies and the rest is beyond the course.

What comes after both is Topic 13.6, where a capacitor and an inductor share a loop with no resistor and trade energy indefinitely rather than settling.