Chapter 6: Circuits

A circuit is just a loop — wires connecting a battery to some stuff (a bulb, a motor, a resistor) and back to the battery. Plug a phone charger into the wall and you’ve completed a circuit. Flip a light switch and you’ve completed (or broken) one.

Before you can analyze any circuit on the MCAT, you need to know two basic things: what’s actually moving through the wires, and what’s pushing it. Those are current (the flow of charge) and EMF (the energy source pushing it). Get these two ideas right and the rest of circuit analysis is just bookkeeping.

What Is Electric Current?

Picture a packed school hallway between classes. No single student is sprinting — most are shuffling. But the crowd as a whole moves steadily toward the next class. Electric current works the same way. Individual electrons drift slowly through a wire — often less than a millimeter per second — but the collective movement of trillions of charges per second produces the measurable flow we call current.

Current is a scalar (just magnitude), but we still assign it a direction for circuit analysis — and that’s where the next quirk comes in.

Conventional Current vs. Electron Flow

A weird piece of physics history that the MCAT expects you to know:

When Benjamin Franklin first described electricity in the 1700s, he guessed that positive charges flowed from the + terminal to the − terminal. He guessed wrong. We now know that electrons (negatively charged) actually flow from − to +. But Franklin’s convention got baked into all the math and circuit symbols, and we never bothered to flip it.

So:

• Conventional current flows + → − through the external circuit. (This is the convention used in every formula and diagram.)
• Electron flow is the opposite: − → +.

On the MCAT, always use conventional current unless the question explicitly asks about electron flow.

The Water-Park Analogy

A complete circuit is a lot like a water park.

• Battery = the pump that lifts water to the top of the park (provides energy to the charges).
• Wires = wide pipes that carry water with very little resistance.
• Resistor = a narrow, twisting slide where water loses energy as it tumbles down (charges lose energy as they pass through).

Crucial rule: water has to return to the pump. If you break the loop (open circuit), nothing flows.

Electromotive Force (EMF)

Despite the misleading name, EMF is not actually a force. It’s a voltage — specifically, the voltage that a battery (or other energy source) provides to push charge around a circuit.

An ideal battery maintains constant EMF no matter how much current flows. Real batteries have internal resistance (covered in §6.5), which eats up some of the energy and reduces the voltage available to the rest of the circuit.

Requirements for Current to Flow

For steady current to actually exist, you need both:

1. A complete (closed) loop. Any break anywhere stops all current — not just at the break. This is exactly why a single burned-out bulb in a series string of holiday lights kills the whole strand.

2. A source of EMF. Something has to do work on the charges to keep them moving. Without a battery (or generator, or solar cell), any current would die out within microseconds as charges lose energy to resistance.

Worked Example

A wire carries a current of 0.5 A. How many coulombs of charge pass a given cross-section in 1 minute? How many electrons is that?

• Charge: $Q = It = 0.5 \times 60 = 30$ C.
• Each electron carries $1.6 \times 10^{-19}$ C, so number of electrons $= 30/(1.6 \times 10^{-19}) \approx 1.9 \times 10^{20}$ electrons.

That’s almost $10^{20}$ electrons per minute through a wire carrying just half an amp. Even a small current involves an unimaginable number of electrons.

If current is the flow of charge, resistance is what opposes that flow. Every material resists current to some degree. Copper barely fights it (great conductor — that’s why your wires are copper). Rubber fights it almost completely (great insulator — that’s why the wires are coated with rubber).

Understanding resistance and its relationship to voltage and current — captured by Ohm’s law — is the single most important skill for MCAT circuit problems. Ohm’s law shows up in 90% of circuit questions, often in disguise.

Resistance: The Opposition to Flow

Think of resistance as a narrow section of pipe in our water-park analogy. A narrow, long, twisty pipe is hard to push water through. A short, wide, straight pipe is easy. Same idea for charges in a wire.

When charges fight their way through a resistor, they lose energy — that energy becomes heat. This is exactly why phone chargers, laptops, light bulbs, and toaster elements all warm up: the resistance of the circuit is dissipating electrical energy as thermal energy.

This formula gets tested heavily — the MCAT loves to change one variable and ask what happens:

• Double the length → resistance doubles ($R \propto L$).
• Double the cross-sectional area → resistance halves ($R \propto 1/A$).
• Switch to a higher-resistivity material → resistance goes up ($R \propto \rho$).

Ohm’s Law

This is the single most-used equation in circuit physics.

Ohm’s law tells you three different things depending on which form you use:

• $V = IR$ — The voltage drop across a resistor equals the current times the resistance.
• $I = V/R$ — More voltage drives more current; more resistance reduces it.
• $R = V/I$ — Resistance is the ratio of voltage drop to current.

Temperature Dependence of Resistance

For metals (conductors), resistance increases as temperature increases. Higher temperature → metal atoms vibrate more → more collisions with the drifting electrons. It’s like trying to walk through a crowd where everyone is dancing chaotically — the more they move, the harder it is to get through.

For semiconductors (silicon, germanium), the opposite is true: resistance decreases as temperature increases. The thermal energy frees up extra charge carriers that more than make up for the increased atomic vibration.

Ohmic vs. Non-Ohmic Materials

An ohmic material obeys Ohm’s law: its resistance stays constant regardless of the voltage applied. On a $V$ vs. $I$ graph, an ohmic material produces a straight line through the origin — and the slope of that line is $R$.

A non-ohmic material has resistance that changes with voltage or current. Examples:

• Diodes — only conduct in one direction; almost no current in the other direction.
• Light bulb filaments — heat up sharply when current flows, raising R as they warm.
• Batteries under heavy load — internal resistance changes as the battery drains.

The MCAT defaults to assuming resistors are ohmic unless the passage explicitly says otherwise.

Worked Example

A copper wire has resistance 5 Ω. You replace it with a piece of the same copper but twice as long and half the cross-sectional area. What’s the new resistance?

• $R \propto L/A$.
• Length doubles → factor of 2.
• Area halves → factor of 2 (since R is inversely proportional to A).
• Combined: $R_{new} = 5 \times 2 \times 2 = 20$ Ω.

When resistors are lined up end-to-end so that all the current must pass through every one of them, they’re in series. This is the simplest circuit configuration and the first one to master — once you nail series, parallel and combinations become much easier to reason about.

The defining feature of series: there’s only one path for current to take. No branching, no shortcuts, no alternate routes.

The Series Rules

Picture three narrow water-park slides connected one after another. All the water entering slide 1 must flow through slide 2 and then slide 3. None of it can take a shortcut. That’s the defining property of a series circuit:

The current is the same through every resistor in series.

Since the same current flows through every resistor, Ohm’s law tells us each resistor drops a voltage proportional to its resistance:

• $V_1 = IR_1$, $V_2 = IR_2$, $V_3 = IR_3$.

Those voltage drops have to add up to the total voltage supplied by the battery:

$V_{total} = V_1 + V_2 + V_3 + \ldots$

This is conservation of energy: every coulomb of charge gets some energy from the battery, then “spends” it dropping through each resistor before returning to the battery.

The Voltage Divider

When the MCAT gives you a series circuit and asks for the voltage across one specific resistor, use the voltage divider formula:

Example. A 12 V battery is connected to a 2 Ω and a 4 Ω resistor in series. $R_{total} = 6$ Ω.

• Voltage across 4 Ω: $12 \times (4/6) = 8$ V.
• Voltage across 2 Ω: $12 \times (2/6) = 4$ V.
• Check: $8 + 4 = 12$ V. ✓

The bigger resistor “hogs” more of the voltage drop, in direct proportion to its share of the total resistance.

Interactive Circuit Explorer

Use this simulation to build series and parallel circuits. Adjust resistor values and battery voltage, then watch current flow and voltage distributions update in real time. Toggle between series and parallel to see how the same resistors behave completely differently.

Series Circuit Summary

| Property | Series rule |
|----------|-------------------|
| Current | Same through all resistors |
| Voltage | Divides among resistors ($V = IR$ for each) |
| Total resistance | $R_{total} = R_1 + R_2 + \ldots$ (sums up) |
| If one resistor breaks (open) | Entire circuit stops — no current anywhere |

When resistors are connected so that each one provides a separate path for current, they’re in parallel. Parallel circuits behave very differently from series circuits, and the MCAT tests these differences relentlessly.

The defining feature of parallel: there are multiple paths current can take. Each resistor has its own branch.

The Parallel Rules

Picture the water park installing three slides side-by-side, all starting at the same upper pool and ending at the same lower pool. Each slide is a separate path. Water splits up among the three slides — but every slide starts and ends at the same elevation. That’s a parallel circuit.

The voltage is the same across every resistor in parallel.

This makes intuitive sense: adding another slide (even a narrow one) gives water more paths, so total flow goes up and total opposition goes down.

Current Division

Since every resistor has the same voltage across it, Ohm’s law tells you that more current flows through smaller resistors:

• $I_1 = V/R_1$, $I_2 = V/R_2$, $I_3 = V/R_3$.
• Total current: $I_{total} = I_1 + I_2 + I_3 + \ldots$.

The Product-Over-Sum Shortcut

For exactly two resistors in parallel, there’s a much faster formula than the reciprocal one:

This only works for two resistors. For three or more, you need the reciprocal formula (or pair them up two at a time).

N Equal Resistors in Parallel

When $N$ identical resistors (each $R$) are in parallel, the formula simplifies beautifully:

Parallel Circuit Summary

Series vs. Parallel: The One Table to Remember

If you internalize this contrast, series-and-parallel problems become almost mechanical.

There’s no such thing as an ideal battery. Every real battery has some internal resistance — a small but real opposition to current inside the battery itself. That resistance steals a portion of the EMF before any current reaches the external circuit, which is why your phone charger feels warm and why old batteries can no longer power what they once could.

The MCAT requires you to understand this concept and the formula that goes with it. Without internal resistance, every battery would deliver its full rated voltage forever — but then we couldn’t explain why an old AA can’t run a toy any longer despite having “almost full” voltage when measured.

Terminal Voltage

Think of a real battery as a tiny circuit inside itself: an ideal EMF source in series with a small internal resistance $r$. When current flows, some voltage gets “lost” inside the battery — converted to heat by the internal resistance. The voltage you actually measure at the battery’s terminals is less than the EMF.

When no current flows (open circuit), the terminal voltage equals the EMF exactly — because the $Ir$ drop is zero. This is why a voltmeter reading across an unused battery shows the full EMF (a voltmeter draws negligible current).

Why Batteries “Die”

As a battery ages, its internal resistance increases due to chemical degradation inside the cell. The $Ir$ drop gets bigger and bigger, leaving less voltage for the external circuit. The battery’s EMF may still be close to its rated value — but so much voltage gets lost internally that the device can’t function any more.

This is why an “old” AA battery often shows nearly full voltage on a multimeter (where almost no current is drawn) but can’t power a flashlight (which needs to draw real current). Multimeter sees ~1.5 V open-circuit; flashlight sees, say, 0.6 V because $Ir$ has eaten the rest.

The Complete Circuit Equation

For a simple circuit with a battery (EMF $\varepsilon$, internal resistance $r$) and an external resistance $R$:

So the terminal voltage across the external resistor is:

$V_{terminal} = IR = \dfrac{\varepsilon R}{R + r}$

When $R \gg r$ (external resistance much larger than internal): nearly all the EMF appears across the external resistor (negligible internal loss). When $R \ll r$ (small external resistor): most of the EMF gets dropped inside the battery.

Short Circuits

A short circuit is when the external resistance approaches zero — a bare wire connects the battery terminals directly, with no load in between. In that case:

$I_{short} = \dfrac{\varepsilon}{r}$

This is the maximum possible current the battery can deliver. All the EMF drops across the internal resistance, dumping massive power into the battery as heat. In real life, short circuits are dangerous — batteries can rupture or catch fire, lithium-ion cells in particular can vent flames. (Hence safety circuits in modern devices.)

Series and parallel rules handle simple circuits beautifully. But what about messier ones — circuits with multiple batteries, branching paths, or components that don’t fit neatly into “series” or “parallel” boxes? That’s where Kirchhoff’s two laws come in.

These aren’t new physics. They’re just conservation of charge and conservation of energy applied specifically to circuits — dressed up in slightly intimidating names. Understand the analogies and the laws become almost obvious.

Kirchhoff’s Current Law (KCL) — The Junction Rule

This is just conservation of charge. Charges can’t pile up at a junction or disappear — whatever flows in has to flow out.

Think of a river fork: if 10 m³/s flows into the fork and 6 m³/s goes left, then 4 m³/s must go right. The water didn’t vanish, and it didn’t accumulate at the split.

Example. If 5 A flows into a junction and splits into three branches with 2 A in the first branch and 1 A in the second, the third branch carries $5 - 2 - 1 = 2$ A out.

Kirchhoff’s Voltage Law (KVL) — The Loop Rule

This is conservation of energy. A charge that travels around a complete loop and returns to its starting point ends up with the same potential energy it started with — so the energy gained from batteries must exactly equal the energy lost across resistors. Net change: zero.

Sign Conventions for KVL

When you “walk” around a loop in a chosen direction, here’s how to tally up voltage changes:

The choice of loop direction doesn’t matter — pick one, stay consistent, and the math works out.

Applying Kirchhoff’s Laws — A Strategy

1. Label all currents with assumed directions. If you guess wrong, the math returns a negative value — that just means the current actually flows opposite your guess. No need to redo the diagram.
2. Apply KCL at each junction to relate the branch currents.
3. Apply KVL around enough independent loops to solve for all unknowns.
4. Solve the system of equations.

For the MCAT, you rarely need more than one junction equation and one or two loop equations. The exam favors conceptual understanding over algebraic complexity.

Quick Example

A simple loop has a 10 V battery and two series resistors: $R_1 = 3$ Ω and $R_2 = 2$ Ω. Apply KVL clockwise starting at the battery:

$+10 \text{ V} - I(3) - I(2) = 0$

$10 = 5I \Rightarrow I = 2$ A. Then $V_{R_1} = 6$ V, $V_{R_2} = 4$ V. They add to 10 V — the battery’s EMF. KVL satisfied. ✓

This tiny example shows the recipe: walk around the loop, write down each voltage change with the correct sign, set the sum to zero, solve.

Power in a circuit tells you how fast electrical energy is being converted into other forms — usually heat (toaster, hair dryer), light (bulb), or motion (motor). When you pay your electricity bill, you’re literally paying for power × time = energy.

The MCAT tests circuit power with three equivalent formulas. The trick isn’t memorizing any one of them — it’s knowing which form to reach for depending on what the question gives you.

The Three Power Formulas

When to use each:

• $P = IV$ — when you know current and voltage directly.
• $P = I^2R$ — when you know current and resistance. Common in series circuits, where current is the same through every resistor.
• $P = V^2/R$ — when you know voltage and resistance. Common in parallel circuits, where voltage is the same across every resistor.

Brightness of Bulbs

The MCAT loves asking which bulb in a circuit is brightest. The rule is simple:

Brightness ∝ power dissipated by the bulb.

But which power formula to use depends on the configuration:

• In a series circuit, all bulbs carry the same current. So $P = I^2R$ — the bulb with the highest resistance is brightest.
• In a parallel circuit, all bulbs see the same voltage. So $P = V^2/R$ — the bulb with the lowest resistance is brightest.

Energy Dissipated

Power is energy per unit time. To find total energy dissipated by a resistor over a time $t$:

This is exactly how electricity bills work — you pay for energy in kilowatt-hours (kWh), not power. A 100 W bulb running 10 hours = 1 kWh ≈ 15¢.

RMS Values (AC Circuits)

The MCAT occasionally touches on alternating current (AC). In AC circuits, voltage and current oscillate sinusoidally — they’re constantly changing direction. To get a meaningful “average” power, we use root-mean-square (RMS) values.

The average power dissipated in an AC circuit uses RMS values: $P_{avg} = I_{rms} V_{rms} = I_{rms}^2 R$.

Resistors dissipate energy (turn it into heat). Capacitors store energy — and release it later, on demand.

A camera flash is the textbook example. The battery slowly trickles charge into a capacitor over a couple of seconds. Then, when you press the shutter, the capacitor dumps all that stored energy in less than a millisecond — producing a brilliant burst of light far brighter than the battery alone could deliver. Same total energy as the battery provided, just released at a much higher rate.

The same physics powers defibrillators (slow charge from a battery, sudden discharge through a patient’s chest), camera flashes, audio amplifiers (capacitors smooth out the power supply), and the timing circuits in nearly every electronic device.

What Is a Capacitor?

At its simplest, a capacitor is just two conducting plates separated by a small gap. When connected to a battery, positive charge piles up on one plate and negative charge on the other. The plates don’t touch, so charge can’t flow between them — it just builds up, creating an electric field across the gap.

Capacitance

Capacitance measures how much charge a capacitor stores per volt applied.

Parallel Plate Capacitor

The most common capacitor geometry on the MCAT is the parallel plate capacitor: two flat plates of area $A$ separated by distance $d$.

The MCAT loves “what happens when you change one variable” questions:

• Double the plate area → capacitance doubles ($C \propto A$).
• Double the plate separation → capacitance halves ($C \propto 1/d$).
• Insert a dielectric → capacitance increases by a factor $\kappa$ (covered in §6.10).

The Electric Field Between the Plates

The electric field between the plates of a parallel plate capacitor is uniform (constant everywhere) and given by:

$E = \dfrac{V}{d}$

So field strength grows when voltage grows or plate separation shrinks. This uniform field is exactly why parallel plate capacitors are the go-to setup for MCAT problems about charged particles flying through electric fields — the math is much simpler with a uniform field than with the messy fields around point charges.

Energy Stored in a Capacitor

A charged capacitor stores electrical potential energy in the electric field between its plates.

That $\tfrac{1}{2}$ factor isn’t arbitrary. The voltage builds gradually as the capacitor charges. The first bit of charge is easy to add (low voltage opposing it), but the last bit has to push against the high voltage that’s already built up. Average voltage during the charging process is half the final voltage — hence the $\tfrac{1}{2}$.

Here’s one of the most reliable traps the MCAT sets: capacitors combine in the exact opposite way from resistors.

If you memorize the resistor combination rules and then blindly apply them to capacitors, you’ll get every capacitor problem wrong. This section exists to make sure that never happens.

The rules to lock in:

• Resistors in series add directly. Capacitors in series use reciprocal addition.
• Resistors in parallel use reciprocal addition. Capacitors in parallel add directly.

The pattern is exactly reversed. Once you internalize “capacitors are contrary,” every capacitor combination problem becomes mechanical.

Capacitors in Parallel

When capacitors are connected in parallel, they all share the same voltage (just like parallel resistors). But for the capacitances, you add directly — not reciprocally.

Why does this make sense? Each parallel capacitor stores its own charge independently. Total charge stored: $Q_{total} = Q_1 + Q_2 + Q_3$. They all share the same voltage $V$, so:

$C_{total} = \dfrac{Q_{total}}{V} = \dfrac{Q_1 + Q_2 + Q_3}{V} = C_1 + C_2 + C_3$

Capacitors in Series

When capacitors are connected in series, the same charge $Q$ ends up on each capacitor (the inner connected plates have nowhere else for charge to go — it gets stuck between the plates of adjacent capacitors). The voltages then add up.

For exactly two capacitors in series, you can use the same product-over-sum shortcut you learned for parallel resistors:

$C_{total} = \dfrac{C_1 \cdot C_2}{C_1 + C_2}$

The Opposite-of-Resistors Rule

The full lookup table, all in one place:

Why the Reversal Makes Physical Sense

Think about parallel-plate capacitors:

• Adding capacitors in parallel is like increasing the total plate area. More area = more charge storage = more capacitance. So adding directly makes sense.
• Adding capacitors in series is like increasing the separation between the outermost plates. Bigger separation = less capacitance ($C \propto 1/d$). So putting them in series reduces total capacitance — and the reciprocal addition reflects that.

This physical intuition matches the math exactly.

Worked Example

A 6 μF and a 3 μF capacitor are connected in parallel; that combination is then put in series with a 4 μF capacitor. What is the total capacitance?

• Parallel pair first: $C_{12} = 6 + 3 = 9$ μF.
• Then in series with 4 μF: $C_{total} = (9 \times 4)/(9 + 4) = 36/13 \approx 2.77$ μF.

Notice the answer (~2.77 μF) is smaller than the smallest capacitor (3 μF) — that’s the series step doing its work. If we’d done the operations in the wrong order, the answer would be very different.

A dielectric is an insulating material placed between the plates of a capacitor. Glass, plastic, ceramic, paper, even air — all are dielectrics. They’re called “insulators” in everyday speech (because they don’t conduct), but in capacitor land they have a special name and a specific job.

Inserting a dielectric always increases capacitance. But the downstream effects — what happens to voltage, charge, and energy — depend on whether the battery is still connected. This distinction is one of the highest-yield topics in MCAT circuit physics. The exam loves writing both versions of the question and watching students get one right and the other wrong.

How Dielectrics Increase Capacitance

Why Does This Happen?

The dielectric molecules can polarize — their positive and negative ends shift slightly in response to the electric field between the plates. The polarized molecules create their own small internal electric field that points opposite to the original field. The two fields partially cancel, leaving a weaker net field.

A weaker field means less voltage is needed to support a given amount of charge — equivalently, more charge can be stored at the same voltage. That’s exactly what “higher capacitance” means.

The Two Scenarios: Battery Connected vs. Disconnected

This is where the MCAT gets sneaky. You have to know what stays constant in each scenario.

Scenario 1: Battery Stays Connected

The battery enforces constant voltage across the capacitor.

Scenario 2: Battery Disconnected, Then Dielectric Inserted

The charge is trapped on the plates — no battery to add or remove any.

Dielectric Breakdown

Every dielectric has a maximum electric field strength it can withstand. Push above this limit (called the dielectric strength) and the material suddenly starts conducting — the capacitor discharges all at once, often destructively.

The most familiar example: lightning. The atmosphere is normally a great insulator (air’s dielectric strength is ~3 MV/m). But during a storm, charge separation builds up such enormous voltages between cloud and ground that the field exceeds air’s breakdown strength. Air abruptly becomes conducting along a path → boom, lightning bolt.

The same principle limits how small modern electronics can be: as components shrink, fields get bigger ($E = V/d$), and engineers have to choose dielectrics with very high breakdown strengths to avoid catastrophic failure.

This section covers two AAMC-listed topics that often share the same MCAT passage: how materials conduct electricity, and how we measure current and voltage without messing up the very circuit we’re trying to study.

The conductivity material connects physics to general chemistry (electrolytes), and the meter material connects to lab-skills questions. Both are bite-sized topics with high test-yield-per-page-of-prep.

Conductivity

Conductivity is the inverse of resistivity. Where resistivity tells you how much a material opposes current, conductivity tells you how easily current flows.

Two Types of Conduction

The MCAT distinguishes between two fundamentally different ways charge can move through a material.

Metallic Conduction

In metals like copper or silver, the charge carriers are free electrons — a “sea” of delocalized electrons that aren’t tied to specific atoms. When a voltage is applied, those electrons drift toward the positive terminal.

Key features:

• Charge carriers: electrons.
• Conductivity decreases as temperature rises (more atomic vibrations means more collisions with the drifting electrons).
• No chemical change in the conductor — it’s a physical process.

Electrolytic Conduction

In ionic solutions (NaCl in water, sulfuric acid, biological fluids), the charge carriers are ions — both positive cations moving one way and negative anions moving the other.

Key features:

• Charge carriers: dissolved ions.
• Conductivity increases as temperature rises (ions move faster, may dissociate more).
• Conductivity increases with ion concentration.
• Chemical changes can occur at electrodes (this is electrolysis — the basis of car batteries, electroplating, electrolytic cells in chemistry).

Ammeters and Voltmeters

Ammeters: Measuring Current

An ammeter measures current and must be placed in series with the component you’re measuring. All the current you’re measuring has to flow through the ammeter.

For an ammeter to avoid disturbing the circuit, it must have very low internal resistance (ideally zero). If the ammeter had high resistance, it would behave like an extra resistor in series, reducing the very current you were trying to measure.

Voltmeters: Measuring Voltage

A voltmeter measures the potential difference between two points and must be placed in parallel with the component you’re measuring.

For a voltmeter to avoid disturbing the circuit, it must have very high internal resistance (ideally infinite). If the voltmeter had low resistance, it would provide an alternate current path through itself, changing the circuit’s behavior.

What Happens with Non-Ideal Meters?

• A real ammeter has some small resistance (not zero). It reduces the current in the circuit slightly. The measured current is lower than the true value would be without the ammeter present.
• A real voltmeter has finite (not infinite) resistance. It draws a tiny bit of current through itself, which slightly changes the voltage distribution in the circuit. The measured voltage is lower than the true value would be.

In practice, both effects are usually negligible because modern meters approach ideal behavior closely enough — but the MCAT may test the concept.

An RC circuit contains a resistor (R) and a capacitor (C). When you flip a switch, the capacitor doesn’t charge or discharge instantly — it does so gradually, following an exponential curve. The resistor sets the pace; the capacitor sets the size of the storage.

The MCAT doesn’t ask you to derive the exponential equations, but you must know the qualitative behavior and the time constant $\tau = RC$. RC circuits show up everywhere: camera flashes, defibrillators, audio filters, the timing circuits in just about every electronic device.

This section also wraps up the chapter with a general circuit-problem-solving strategy that you can apply to anything from simple series/parallel questions to multi-loop monsters.

Charging an RC Circuit

When a battery is connected to a resistor and an uncharged capacitor in series, charge flows onto the capacitor plates. At first the capacitor is empty, so nothing opposes the current — initial current is high. As charge builds up on the plates, the voltage across the capacitor grows, opposing further current flow. Current gradually decreases until the capacitor is fully charged and current stops entirely.

During charging:

• Charge $Q(t)$ starts at 0 and grows exponentially toward $Q_{max} = CV_{battery}$.
• Current $I(t)$ starts at $I_{max} = V_{battery}/R$ and decays exponentially toward 0.
• Capacitor voltage $V_C(t)$ starts at 0 and grows toward $V_{battery}$.

Discharging an RC Circuit

When a fully charged capacitor is disconnected from the battery and connected through a resistor, the stored charge flows back out through the resistor. Current starts high (large $V$ across the capacitor) and decays toward zero as the capacitor drains.

During discharging, all three quantities ($Q$, $I$, $V_C$) start at their initial values and decay exponentially toward zero — same time constant.

The Time Constant

The time constant is just a number with units of seconds — and the percentages of completion at each multiple of $\tau$ are the same for every RC circuit:

Why 63% and 37%?

These come from the exponential function. After one time constant, $e^{-1} \approx 0.37$ (37%). For charging, the capacitor reaches $1 - 0.37 = 0.63$ (63%) of its maximum. You don’t need to work with $e$ on the MCAT — just memorize the $\frac{63}{37}$ split at 1 τ, and the “essentially done at 5 τ” rule.

How R and C Affect the Time Constant

• Larger $R$ → larger τ → slower charging/discharging (current is more restricted).
• Larger $C$ → larger τ → slower charging/discharging (more charge to move on/off the plates).
• Smaller $R$ or $C$ → smaller τ → faster response.

Practical consequence: if you want a fast camera flash (rapid discharge), you use a small resistor and a moderate capacitor. If you want a slow circuit (a windshield-wiper delay), you use a big resistor.

Circuit Problem-Solving Strategy

Here’s a general game plan for any MCAT circuit problem:

1. Simplify. Identify resistors and capacitors that are in series or parallel. Combine them into equivalent values. Redraw the simplified circuit.
2. Apply the big rules. Ohm’s law ($V = IR$), Kirchhoff’s junction rule (current in = current out), Kirchhoff’s loop rule (voltage gains = voltage drops around any loop).
3. Identify what’s constant. In series: current is the same. In parallel: voltage is the same. For capacitors with battery connected: $V$ is constant. For isolated capacitors: $Q$ is constant.
4. Use the right power formula. $P = IV$, $P = I^2R$, or $P = V^2/R$ — pick the one that matches the variables you already know.
5. Check your answer. Does total resistance look reasonable (parallel < smallest)? Do voltage drops sum to the EMF (KVL)? Do currents balance at junctions (KCL)?