Topic 13.6: Circuits with Capacitors and Inductors (LC Circuits)

What Topic 13.6 requires

Topic 13.6 closes Unit 13 and the whole AP Physics C: Electricity and Magnetism course. One learning objective, three essential-knowledge statements, and no boundary statement.

13.6.A, describe the physical and electrical properties of a circuit containing a combination of capacitors and a single inductor.

• 13.6.A.1 in circuits containing only a charged capacitor and an inductor (LC circuits), the maximum current in the inductor can be determined using conservation of energy within the circuit.
• 13.6.A.2 in LC circuits, the time dependence of the charge stored in the capacitor can be modeled as simple harmonic motion. The derived equation is

• 13.6.A.3 the angular frequency of an oscillating LC circuit can be derived from the differential equation that describes an LC circuit. The derived equation is

The suggested skills are 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.B, calculate or estimate an unknown quantity with units from known quantities by selecting and following a logical computational pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Three statements is the smallest required-content block in Unit 13, and 13.6.A.2 does almost all of the work by naming the model: simple harmonic motion. That single phrase imports an entire chapter of AP Physics C: Mechanics into this topic, which is what makes it the least new material and the most leverage of anything in the unit.

Note the objective's wording, exactly as in Topic 13.5: a combination of capacitors and a single inductor. Plural capacitors, one inductor. That has consequences, and they come later on this page.

How far the CED takes the differential equation

This is the question worth settling first, because it decides how much of a university treatment belongs here.

Topic 13.6 prints no boundary statement. Neither does 13.1, 13.3, 13.4 or 13.5. The whole of Unit 13 carries exactly one, and it sits under Topic 13.2, about the speed of light. So there is no sentence anywhere fencing off the LC mathematics, and the level is set by the essential-knowledge statements themselves and by the labels on their equations.

Both equations in this topic carry the Derived Equation label. The CED's Required Equations page defines it: not all equations in the framework appear on the sheet, and many "are provided for reference and guidance, or to demonstrate the final results of derivations expected of students on the exam. These equations are denoted as 'Derived Equations.'"

Now check them against the sheet, one at a time, because the label and the sheet do not track each other in this course.

So the result of the derivation is handed to you and the derivation is not. Read together with 13.6.A.3, which says the angular frequency "can be derived from the differential equation that describes an LC circuit", the expectation is clear enough: produce the differential equation, then get $\omega$ from it, even though you could have read $\omega$ off the sheet.

What the CED does not ask for, and it is worth being precise about the absences:

• No solved function of time is printed anywhere in Unit 13. Not for the LR current and not for the LC charge. Statement 13.6.A.2 says the time dependence "can be modeled as simple harmonic motion", which points you at $x = x_{\max}\cos(\omega t + \phi)$ in the reprinted mechanics table rather than printing an electrical version of it.
• The words RLC, damping and resonance appear nowhere in the AP Physics C: Electricity and Magnetism course description. Statement 13.6.A.1's own wording is "circuits containing only a charged capacitor and an inductor". A resistor in the loop is Topic 13.5's business, and a circuit with all three is outside this course.
• No driven or forced oscillation. The word "oscillating" appears exactly once in the whole course description, in 13.6.A.3.

Two supporting signals about register. The unit's Building the Science Practices page says students may benefit from "practice crafting clear, concise arguments, derivations, and calculations that follow a logical pathway", and names deriving a mathematical equation that shows the current in a solenoid as a function of time as a valuable skill. And the CED's own sample free-response Question 4, on an RC circuit, instructs students to "Derive, but do not solve, a differential equation" and then asks them to justify a prediction by referring to the equation they wrote. That is the shape to prepare for: the setup and the reasoning carry the points.

The oscillator you already own

Statement 13.6.A.2 says the charge can be modeled as simple harmonic motion. It is worth seeing exactly how literally that is meant, because the correspondence is not an analogy that mostly works. It is the same differential equation with different letters.

The AP Physics C: Mechanics course description, at essential knowledge 7.3.A.2, says the position as a function of time for an object exhibiting SHM is a solution of the second-order differential equation derived from the application of Newton's second law, and prints as its derived equation

Set that beside 13.6.A.2's $\dfrac{d^2q}{dt^2} = -\dfrac{1}{LC}q$ and the identification is immediate: $q$ plays the part of $x$, and $\omega^2 = \dfrac{1}{LC}$. Statement 13.6.A.3 is that one line of algebra.

Here is the full dictionary. Everything in the right-hand column that is printed on the C: E&M sheet is marked, and because that sheet reprints the entire C: Mechanics table, everything in the middle column is on the same page as everything in the right.

Three things to take from the table rather than from the individual rows.

The inductor is the mass. That is the row students find least obvious and it is the most useful one. Inductance opposes a change in current exactly as mass opposes a change in velocity, and both store energy in proportion to the square of the rate, $\frac{1}{2}LI^2$ against $\frac{1}{2}mv^2$. The identification is already made on the Topic 13.4 page, where inductance is called electrical inertia.

Capacitance is the reciprocal of stiffness. A large capacitor is a weak spring: it takes a lot of charge to build up a given potential difference, just as a floppy spring takes a lot of stretch to build up a given force. That reciprocal is the reason $C$ appears under the square root in the numerator's denominator rather than where $k$ sits, and it is why a bigger capacitor gives a slower oscillation, which catches people who pattern-match from $\omega = \sqrt{k/m}$ without checking.

The period follows from two printed equations. $T = 2\pi\sqrt{LC}$ is not printed anywhere, but $\omega_{LC} = 1/\sqrt{LC}$ is on the sheet and $T = 2\pi/\omega = 1/f$ is on the same sheet in the reprinted mechanics table. One substitution.

If you want the mechanical side refreshed, the simple harmonic motion guide covers the routine, and the AP Physics C: Mechanics Unit 7 hub covers what that course requires.

Deriving the differential equation, twice

Statement 13.6.A.2 gives the equation and not the route. There are two routes, and both are worth having, because a question can hand you either starting point.

Route one: Kirchhoff's loop rule. This is the same move that produced the LR equation at 13.5.A.2.

Take a capacitor of capacitance $C$ carrying charge $q$ on one plate, in a closed loop with an inductor of inductance $L$ and nothing else. Declare the convention first and keep it: let $q$ be the charge on the plate the current flows into, so that $I = dq/dt$, which is the definition the sheet prints.

Going once around the loop, the two potential changes must sum to zero:

1. Across the capacitor, from $C = Q/\Delta V$ on the sheet, the magnitude of the potential difference is $q/C$.
2. Across the inductor, from 13.4.A.3 and the sheet, the self-induced emf is $-L\,dI/dt$.

Now substitute $I = dq/dt$, so $dI/dt = d^2q/dt^2$:

which is exactly the derived equation at 13.6.A.2.

Route two: conservation of energy. This one starts from 13.6.A.1, and it is the more physical of the two.

There is no resistor in the loop, so nothing dissipates energy. Statement 13.4.A.2.ii says the transfer of energy generated in an inductor to other forms obeys conservation laws, and 13.4.A.2.i names one of those forms explicitly: the stored energy can be "used to charge a capacitor". So the total is constant:

Differentiate both sides with respect to time. A constant differentiates to zero, and the chain rule handles both terms:

Substitute $I = dq/dt$ throughout and factor out that common $I$:

The current is not zero at all times, so the bracket must vanish, and the same differential equation drops out.

Why the minus sign is the whole point. The equation says the second derivative of the charge is proportional to the charge and points the other way. That is the definition of a restoring relationship, and AP Physics C: Mechanics statement 7.1.A.2 gives the general form: SHM results when the magnitude of the restoring force exerted on an object is proportional to that object's displacement from its equilibrium position. Here "displacement from equilibrium" is the charge on the capacitor, since $q = 0$ is the state with no stored electric energy, and the further the charge is from zero the harder the circuit pushes it back. A plus sign would give runaway exponential growth rather than oscillation, so the sign is not a bookkeeping detail.

From the equation to the frequency

Statement 13.6.A.3 says the angular frequency "can be derived from the differential equation that describes an LC circuit". Two ways to run that derivation, and the second is worth knowing because it works even if you have forgotten the mechanical result.

By comparison. The C: Mechanics derived equation is $d^2x/dt^2 = -\omega^2 x$. Matching coefficients with $d^2q/dt^2 = -(1/LC)q$ gives $\omega^2 = 1/LC$, so

By substitution. Propose the solution $q = q_{\max}\cos(\omega t + \phi)$, which is the electrical version of the $x = x_{\max}\cos(\omega t + \phi)$ printed in the reprinted mechanics table. Differentiate twice, using the derivative rules from the booklet's CALCULUS box, which prints $\frac{d}{dx}\left[\cos(ax)\right] = -a\sin(ax)$ and $\frac{d}{dx}\left[\sin(ax)\right] = a\cos(ax)$:

Substituting into the differential equation gives $-\omega^2 q = -(1/LC)q$, which holds at all times only if $\omega^2 = 1/LC$. The substitution route also hands you the current for free, since $I = dq/dt$:

Check the units, because this is a place where a slip is easy to miss. $L$ is in henries and $C$ in farads. A henry is $\mathrm{V \cdot s/A}$ and a farad is $\mathrm{C/V} = \mathrm{A \cdot s / V}$, so a henry-farad is $\mathrm{s^2}$, its square root is a second, and its reciprocal is $\mathrm{s^{-1}}$. Angular frequency in radians per second, as it should be.

Reading the dependence, which is skill 2.C. Everything sits under a square root, so factors of change are gentle.

The last row is the one worth stating out loud. The frequency of an LC circuit does not depend on how much charge you started with. That is the electrical version of the fact that a mass on a spring has the same period whatever its amplitude, and it comes from the same place: the differential equation contains $L$ and $C$ and nothing about the initial conditions. Change the starting charge and every current and voltage scales, while the timing does not move at all.

The energy exchange, quarter period by quarter period

Statement 13.6.A.1 says the maximum current in the inductor can be determined using conservation of energy within the circuit, so this is the topic's headline calculation. It takes one line:

All of the energy that was in the capacitor's electric field when the charge was greatest is in the inductor's magnetic field when the current is greatest, because there is nowhere else for it to go. That the answer equals $\omega Q$ is a useful cross-check, and it also confirms that the energy route and the SHM route are the same physics.

Follow one full cycle, starting from a fully charged capacitor at rest.

Four facts to carry out of that table:

1. The two maxima are a quarter period apart. Charge is greatest exactly when current is zero, and current is greatest exactly when charge is zero. Nothing in an LC circuit peaks at the same moment as anything else except by that quarter-cycle rule.
2. The capacitor's charge reverses. After half a period the plate that started positive is negative. This is not decay, it is oscillation, and it is the row that distinguishes an LC circuit from the RC and LR circuits in the rest of the course, which approach an asymptote and stay there.
3. Energy oscillates at twice the frequency of charge. $U_C \propto q^2$, and squaring a cosine doubles the frequency. Over one period of the charge, the energy makes two complete round trips between the two stores. This is the single most common mistake in a sketch of $U_C$ against time.
4. Nothing is lost. With no resistor there is no dissipation, so the oscillation in this model continues indefinitely. Statement 13.6.A.1's phrasing, "circuits containing only a charged capacitor and an inductor", is what licenses that, and the sheet's exam conventions add that wires and batteries are ideal.

At an arbitrary instant, with $q = Q\cos(\omega t)$ and $I = -\omega Q \sin(\omega t)$:

To see that the second of those is right, substitute $I_{\max} = \omega Q$ and $\omega^2 = 1/LC$ into $\frac{1}{2}LI^2$. Their sum is constant because $\sin^2\theta + \cos^2\theta = 1$, which is one of the four identities printed in the booklet's IDENTITIES box. That is a genuinely satisfying place for a printed trigonometric identity to earn its keep.

The energy splits evenly when $\cos^2 = \sin^2 = \frac{1}{2}$, which is at $\omega t = 45$ degrees, one eighth of a period after the start, and then again every quarter period after that.

A combination of capacitors and a single inductor

Read the learning objective again: a combination of capacitors and a single inductor. Plural capacitors. That is deliberate, it matches the wording of 13.5.A exactly (a combination of resistors and a single inductor), and it creates the same trap.

The $C$ in $\omega = 1/\sqrt{LC}$ is the equivalent capacitance the inductor sees, not the value printed on any one component. The sheet gives you the two combination rules:

and the direction of each is the reverse of the resistor case, which is where the errors come from. Capacitors in parallel add; capacitors in series add reciprocally.

There is a notation detail on the sheet worth flagging, because it is the sort of thing that makes a student doubt a correct method. The two time constants are printed with explicit equivalent subscripts, $\tau = R_{\text{eq}}C_{\text{eq}}$ and $\tau = L/R_{\text{eq}}$, while the LC angular frequency is printed as $\omega_{LC} = 1/\sqrt{LC}$ with no subscripts at all. Do not read that as permission to use a single capacitor's value when there are several. The learning objective says "a combination of capacitors", and the physics is unambiguous: what governs the oscillation is the total capacitance in the loop.

Which way the combination pushes the frequency:

• Capacitors in parallel give a larger $C_{\text{eq}}$, so a smaller $\omega$ and a longer period. More total capacitance is a floppier spring.
• Capacitors in series give a $C_{\text{eq}}$ smaller than the smallest of them, so a larger $\omega$ and a shorter period.

The second worked example below runs the same two capacitors both ways and gets angular frequencies differing by a factor of $\sqrt{4.5}$, with neither equal to what either capacitor alone would give.

One inductor, though. The objective says "a single inductor" and the framework never combines them, so there is no equivalent-inductance rule anywhere in the course description or on the sheet. If a question shows two inductors, look again at the circuit before assuming a combination rule you have not been given.

Where the initial charge comes from. Statement 13.4.A.2.i says the energy stored in an inductor can be dissipated through a resistor or used to charge a capacitor, which is the handoff into this topic. In practice an LC problem starts one of two ways: a capacitor charged by a battery which is then switched out of the loop, or an inductor already carrying a current which is then switched into a loop with an uncharged capacitor. The second is the same oscillation started a quarter period later, so $q = q_{\max}\sin(\omega t)$ rather than a cosine, and the phase constant $\phi$ in $q = q_{\max}\cos(\omega t + \phi)$ is what encodes that choice.

Graphs, and how Topic 13.6 is tested

Skill 1.B, create quantitative graphs with appropriate scales and units including plotting data, is listed first for this topic. Note that it is 1.B and not 1.C: this is the quantitative-graph skill, so axes get labels, units and a numerical scale, not just a shape.

Five curves for one LC circuit started from a fully charged capacitor at $t = 0$:

Four things a scoring guideline can identify:

1. The two energy curves are never negative and both oscillate between zero and the same maximum. Drawing an energy curve that dips below the axis is an immediate error.
2. The two energy curves are at half the period of the charge, so two full humps per cycle of $q$.
3. They sum to a horizontal line at the total energy. If you draw both on one set of axes, that is a free self-check, and it is the same trick that works for the two voltages in an LR circuit.
4. Charge and current are a quarter period out of step, with the current at its extreme where the charge crosses zero and vice versa. Marking one such pair of points is often what earns the alignment mark.

Skill 3.A, create experimental procedures. Only two topics in Unit 13 list 3.A, this one and 13.2. The natural question here is how to measure an unknown inductance or capacitance from an oscillation. Build the LC loop with the known component, charge the capacitor, switch it into the loop, record the potential difference across the capacitor against time, measure the period from the trace, and invert $T = 2\pi\sqrt{LC}$ to get the unknown. Reading several cycles and dividing gives a better period than reading one. That mirrors the CED's own Unit 13 sample activity for Topic 13.5, which has students measure a solenoid's inductance from an LR time constant, and the unit's exam-preparation page says laboratory investigations about the behavior of circuits when a solenoid is in the circuit are valuable exercises that concretely demonstrate principles and ideas that are often abstract.

Skill 2.B is a straight calculation with units, and skill 2.C is a comparison between two circuits or two instants. Skill 3.B is a claim backed by the model, and the strongest one available is that an LC circuit oscillates because its governing equation is the simple harmonic motion equation.

None of the fifteen sample multiple-choice questions or four sample free-response questions in the CED aligns to 13.6.A. Unit 13 as a whole is weighted at 10 to 20 percent of the multiple-choice section over about 10 to 20 class periods.

Four traps.

• Thinking a bigger capacitor means a faster oscillation. It does not. $C$ is under a square root in the denominator, and capacitance behaves like the reciprocal of stiffness.
• Drawing the energy curves at the same period as the charge. They run at twice the frequency.
• Using one capacitor's value when the circuit holds several. Use the equivalent capacitance, and remember that capacitors combine the opposite way to resistors.
• Expecting the oscillation to decay. Not in this model. Statement 13.6.A.1 specifies a circuit containing only a charged capacitor and an inductor, and the words RLC, damping and resonance do not appear anywhere in the AP Physics C: Electricity and Magnetism course description.

How this topic sits against the rest of the unit. Topic 13.5 and the RC circuits of Unit 11 are both first-order: one derivative, an exponential approach to an asymptote, and a time constant. Topic 13.6 is the only second-order circuit in the course, and second order is what makes oscillation possible rather than mere approach. There is no AP Physics 2 counterpart, because that course contains no inductors at all: the words inductance and inductor appear nowhere in its course description. The nearest algebra-based relative is AP Physics 2 Topic 11.8 on RC circuits, which is for AP Physics 2 students and covers a first-order circuit; this page is for AP Physics C: Electricity and Magnetism students.