4.2 Circuit Elements: Current, Voltage & Resistance

Electric Current

Current (I) is the rate at which charge flows past a point in a circuit:

I = ΔQ/Δt

Current is measured in amperes (A), where 1 A = 1 C/s. By convention, conventional current is defined as the direction positive charge would flow — from higher to lower potential, i.e., from the positive terminal of a battery, through the external circuit, and back to the negative terminal. In a metal wire, the actual charge carriers are negatively charged electrons, which physically drift in the opposite direction, from low to high potential. The MCAT expects you to work with conventional current direction even while knowing that electrons are the physical carriers moving the other way.

In ionic solutions (blood, interstitial fluid, cytoplasm), both cations and anions carry current: positive ions migrate toward lower potential and negative ions toward higher potential, so conventional current still points in the net positive-charge-flow direction. That dual-carrier picture is the bridge from metal-wire circuits to nerve membranes and ECG volume conduction.

Sign conventions matter for tracking current through circuit elements and cell membranes alike: at any junction the total current in must equal the total current out — charge cannot accumulate at a point in a steady-state circuit. That statement is Kirchhoff's junction (node) rule, a direct expression of charge conservation.

Electromotive Force (EMF) and Voltage

A battery or other power source does not supply current directly — it supplies electromotive force (EMF), the energy per unit charge the source makes available to push current through a circuit, measured in volts (V), where 1 V = 1 J/C. Despite the name, EMF is not itself a force; it is best understood as an ideal, no-load voltage source.

Real batteries have internal resistance (r), so the voltage actually available at the battery's terminals — the terminal voltage — is slightly less than the EMF whenever current flows:

V = EMF − Ir

The internal resistance dissipates some energy as heat inside the battery itself, which is why a battery's usable voltage sags as it supplies more current, and why a nearly-dead battery can still read close to its rated EMF on an idle voltmeter but collapse under load. Kirchhoff's loop rule formalizes the same energy bookkeeping: around any closed loop, the algebraic sum of EMFs and IR voltage drops is zero — you cannot gain net energy by walking a charge around a closed path in a steady circuit.

Worked terminal-voltage check: EMF = 12 V, r = 0.5 Ω, I = 4 A → V = 12 − (4)(0.5) = 10 V at the terminals. The "missing" 2 V appears as heating inside the battery (P_internal = I²r = 8 W).

Ohm's Law, Resistance & Power

Resistance (R) quantifies how strongly a circuit element opposes current flow, measured in ohms (Ω), where 1 Ω = 1 V/A. For an ohmic resistor, current and voltage are related by Ohm's Law:

I = V/R (equivalently, V = IR)

Not every element is ohmic — diodes and many biological membranes have nonlinear I–V curves — but MCAT circuit problems usually assume ohmic resistors unless a passage states otherwise.

A resistor dissipates electrical energy as heat at a rate given by power, P = IV, which combines with Ohm's Law to give two equivalent forms: P = I²R and P = V²/R. These three power equations let you solve for dissipated power from whichever two variables (I, V, R) a passage provides. High current through a small resistance can still dump large power (I²R); high voltage across a large resistance can also dump large power (V²/R). Choose the form that matches the knowns.

Worked power example: A 100 Ω resistor has 20 V across it. I = V/R = 0.2 A, so P = IV = 4 W, or directly P = V²/R = 400/100 = 4 W. Same answer either way — a good self-check on exam day.

Resistors in Series and Parallel

• Series resistors share the same current, and their voltage drops add, so equivalent resistance adds directly: Req = R₁ + R₂ + ... Adding a resistor in series always increases total resistance.
• Parallel resistors share the same voltage, and their currents add, so equivalent resistance adds as reciprocals: 1/Req = 1/R₁ + 1/R₂ + ... Adding a resistor in parallel always decreases total resistance, since there is an additional path for current to take. For two resistors only, the shortcut Req = (R₁R₂)/(R₁ + R₂) is fast.

Worked example: A 12 V battery with negligible internal resistance is connected to a 4 Ω resistor (R₃) in series with a parallel combination of two 4 Ω resistors (R₁ and R₂). Find the total current and the current through each resistor.

Step 1 — combine the parallel branch: since R₁ = R₂ = 4 Ω, Rp = (R₁ × R₂)/(R₁ + R₂) = (4 × 4)/(4 + 4) = 16/8 = 2 Ω

Step 2 — combine with the series resistor: Req = R₃ + Rp = 4 Ω + 2 Ω = 6 Ω

Step 3 — apply Ohm's Law for total current: Itotal = V/Req = 12 V / 6 Ω = 2 A

Step 4 — find the voltage split: V₃ = Itotal × R₃ = (2 A)(4 Ω) = 8 V, so the parallel branch carries the remaining Vp = 12 V − 8 V = 4 V (check: Itotal × Rp = (2 A)(2 Ω) = 4 V, which matches)

Step 5 — split the current across the parallel branch: since R₁ and R₂ are equal, they split the 2 A total current evenly: I₁ = I₂ = Vp/R = 4 V/4 Ω = 1 A each (1 A + 1 A = 2 A, matching the total)

Step 6 — power check (optional): total power from the battery is P = IV = (2 A)(12 V) = 24 W. Sum of I²R on each resistor: (1)²(4) + (1)²(4) + (2)²(4) = 4 + 4 + 16 = 24 W. Energy conservation holds.

Resistivity and Biological Conductors

While resistance describes a specific object, resistivity (ρ) is an intrinsic property of the material itself, independent of the object's shape or size. The two are related by:

ρ = RA/L (equivalently, R = ρL/A)

where A is the wire's cross-sectional area and L is its length, and ρ is measured in ohm-meters (Ω·m). For a given material, a longer wire has more resistance, and a thicker wire has less resistance — exactly mirroring how a longer, narrower pipe resists fluid flow more than a short, wide one. Conductivity σ is the reciprocal of resistivity (σ = 1/ρ).

Biological context: this length-and-area dependence previews how a neuron's axon behaves as an electrical cable — a longer or thinner axon has greater internal (axoplasmic) resistance, which is one reason myelination and larger axon diameter both speed up nerve signal propagation, a relationship developed later in this chapter. Tissue conductivity also matters for surface recordings: extracellular fluid is a relatively good ionic conductor, while bone and fat are poorer conductors, so currents from the heart distribute unevenly through the body volume.

The Body as a Circuit: ECG/EKG as an Analog

An electrocardiogram (ECG or EKG) does not measure membrane voltage of a single myocyte directly. Instead, the heart's coordinated depolarization and repolarization produce time-varying current sources in the chest, and those currents flow through the resistive, ionic volume of the torso. Surface electrodes sample potential differences between skin sites — exactly the job of a voltmeter wired in parallel with a portion of a distributed resistive network.

Useful circuit analogies for MCAT passages:

• Cardiac muscle cells act like many tiny batteries/current sources that turn on and off in a coordinated wave
• Blood and interstitial fluid act like a three-dimensional resistive mesh (a volume conductor)
• Lead wires and amplifiers act like high-impedance voltmeters reading ΔV between points
• Lead placement changes which projection of the heart's electrical vector you capture (different "loop paths")
• Asystole or a flatline is the absence of coordinated sources, not "zero body resistance"

You do not need clinical ECG interpretation for Chem/Phys, but you do need to recognize that surface potentials arise from real currents in a conductive body and that Kirchhoff-style conservation still applies to those distributed currents.

Common MCAT Traps

• Reversing electron flow and conventional current. Work problems with conventional current unless a passage specifically asks about electron drift direction.
• Using EMF as terminal voltage under load. Terminal voltage is always EMF − Ir when current is flowing.
• Adding parallel resistances as if they were series. Parallel always lowers Req below the smallest branch resistance.
• Forgetting power has three equivalent forms. If you know V and R, use V²/R; if you know I and R, use I²R.
• Treating resistivity as if it changed when you cut a wire shorter. Resistivity is material-intrinsic; resistance changes with geometry.
• Thinking an ECG measures absolute potential at one electrode. It measures potential difference between leads, just as a voltmeter does.