4.3 Capacitance & Magnetism

Parallel Plate Capacitors

A capacitor is a device that stores electric charge and energy by separating positive and negative charge across an insulating gap. Capacitance (C) measures how much charge a capacitor stores per volt of potential difference across it:

C = Q/V

Capacitance is measured in farads (F), where 1 F = 1 C/V. Because a farad is an enormous unit, most real capacitors are measured in microfarads (μF, 10⁻⁶ F) or picofarads (pF, 10⁻¹² F). Cell membranes have tiny absolute capacitances, but their capacitance per unit area is large enough to dominate the timing of voltage changes during action potentials.

The simplest capacitor geometry is the parallel plate capacitor: two flat conducting plates of area A, separated by a small distance d, with a uniform electric field between them. Its capacitance is:

C = ε₀A/d

where ε₀ is the permittivity of free space (ε₀ ≈ 8.85 × 10⁻¹² C²/N·m²). Capacitance increases with larger plate area, since there is more room to store charge, and decreases with greater plate separation, since the same charge produces a weaker field (E = V/d for a uniform field). The membrane-as-capacitor model uses the same dependence: intracellular and extracellular fluids are the "plates," and the lipid bilayer is the insulating gap of thickness d.

Energy of a Charged Capacitor

Charging a capacitor requires work, because each new increment of charge must be pushed onto a plate against the repulsion of charge already there. The energy stored in a charged capacitor is:

U = ½CV² = ½QV = Q²/(2C)

The three forms are algebraically equivalent (using Q = CV to convert between them); pick whichever form uses the two quantities you already know. The factor of one-half appears because voltage rises linearly as charge accumulates during charging from zero — average voltage during the process is V/2.

Worked example: A 4 μF capacitor is charged to 100 V. How much energy does it store?

U = ½CV² = ½(4 × 10⁻⁶ F)(100 V)² = ½(4 × 10⁻⁶)(10,000) = ½(0.04) = 0.02 J = 20 mJ

Square the voltage first, then multiply by capacitance, then halve the result — doing the steps in this order keeps the arithmetic manageable without a calculator. Defibrillators exploit the same energy storage idea on a much larger scale: a charged capacitor dumps stored energy through the chest to depolarize a critical mass of myocardium.

Capacitors in Series and Parallel

Capacitors combine by rules that are the mirror image of the resistor rules — this reversal is one of the most heavily tested facts in this content category.

• Parallel capacitors each experience the same voltage, and their charges add, so equivalent capacitance adds directly: Ceq = C₁ + C₂ + ...
• Series capacitors each store the same charge, and their voltages add, so equivalent capacitance adds as reciprocals: 1/Ceq = 1/C₁ + 1/C₂ + ...

Worked example: Two identical 6 μF capacitors are combined first in series, then in parallel.

Series: 1/Ceq = 1/6 + 1/6 = 2/6 = 1/3, so Ceq = 3 μF (less than either individual capacitor)

Parallel: Ceq = 6 + 6 = 12 μF (more than either individual capacitor)

A quick sanity check: series capacitance is always smaller than the smallest individual capacitor, while parallel capacitance is always larger than the largest individual capacitor — the opposite of how series and parallel resistances behave.

Dielectrics

A dielectric is an insulating material inserted between a capacitor's plates. Because the dielectric's molecules polarize in response to the field, they partially cancel the field between the plates, allowing more charge to be stored at the same voltage. Capacitance increases by the material's dielectric constant (κ, always ≥ 1 for a real material):

C = κC₀

where C₀ is the capacitance with only vacuum (or air, κ ≈ 1) between the plates. Beyond boosting capacitance, a dielectric also physically keeps the plates from touching, preventing a short circuit, and raises the maximum voltage the capacitor can hold before the insulator breaks down and charge arcs across the gap. Water and many biological materials have high κ, which is one reason membrane capacitance is large for such a thin structure.

Battery connected versus disconnected (a classic MCAT trap pair):

• If the capacitor stays connected to a battery, V is fixed by the battery. Inserting a dielectric raises C, so Q = CV rises and stored energy U = ½CV² rises (the battery supplies extra charge and energy).
• If the capacitor is disconnected after charging, Q is fixed. Inserting a dielectric raises C, so V = Q/C falls and U = Q²/(2C) falls (the dielectric's polarization does work reducing field energy).

Always ask first: what is held constant, Q or V?

Conductivity and Meters

Conductivity describes how readily a material carries current, and the MCAT distinguishes two mechanisms:

• Metallic conductivity: mobile electrons carry the current through a fixed lattice of metal cations. Metallic conductivity typically decreases as temperature rises, since increased lattice vibration scatters the moving electrons more often.
• Electrolytic conductivity: mobile ions, both cations and anions, carry the current through a solution, molten salt, or biological fluid. Electrolytic conductivity generally increases with temperature, since ions move faster and more of a weak electrolyte dissociates, and it depends directly on ion concentration and mobility — the mechanism behind current flow across nerve cell membranes and through blood and interstitial fluid.

Circuits are measured with dedicated meters: an ammeter measures current and is wired in series in the branch of interest, built with very low internal resistance so it does not reduce the current it is measuring; a voltmeter measures the potential difference across a component and is wired in parallel with it, built with very high internal resistance so it draws negligible current from the circuit. Both are built around a galvanometer, a sensitive detector of small currents. Surface ECG leads behave like carefully placed, high-impedance voltage sensors on a distributed electrolytic conductor — the same design philosophy as a voltmeter.

Magnetism: Magnetic Field and the Lorentz Force

A magnetic field (B), measured in tesla (T), is produced by moving charge — electric current in a wire, or the intrinsic magnetic moments of atoms in a permanent magnet. Unlike the electric field, which exerts a force on any charge whether moving or stationary, the magnetic field exerts a force only on a moving charge:

F = qvB sin θ

where v is the charge's speed, B is the field magnitude, and θ is the angle between the velocity vector and the field vector. The force is maximal when velocity is perpendicular to the field (θ = 90°) and zero when the charge moves parallel to the field. Its direction is given by the right-hand rule and is always perpendicular to both v and B. For a current-carrying wire of length L, the magnitude form is F = ILB sin θ.

Because the magnetic force is always perpendicular to velocity, it does no work on the charge — it changes the direction of motion but never the speed. A charged particle moving perpendicular to a uniform magnetic field therefore travels in a circular path at constant speed, with the magnetic force acting as the centripetal force (mv²/r = qvB → r = mv/(qB)); if the particle also has a velocity component parallel to B, it traces a helix instead. When both an electric field and a magnetic field act on a charge simultaneously, the total electromagnetic force is the Lorentz force, F = q(E + v × B) — a combination MRI machines, mass spectrometers, and cyclotrons all exploit to steer charged particles along controlled paths.

MCAT biology note: ordinary household or geomagnetic fields are not what open ion channels; voltage-gated channels respond to electric field / membrane potential. Magnetic-force questions on the MCAT are almost always pure physics (path shape, work = 0, radius of curvature) even when embedded in a medical-device passage.

Common MCAT Traps

• Using resistor combination rules for capacitors. Series capacitors use reciprocal addition; parallel capacitors add directly — the reverse of resistors.
• Forgetting the ½ in capacitor energy. U = ½CV², not CV².
• Inserting a dielectric without stating whether Q or V is fixed. Outcomes for Q, V, E, and U differ in the two cases.
• Claiming magnetic force changes a particle's kinetic energy. Perpendicular force does no work; speed is constant in a pure magnetic field.
• Wiring an ammeter in parallel or a voltmeter in series. Wrong placement either shorts the circuit or inserts huge resistance in series.
• Confusing metallic and electrolytic conduction temperature trends in passages about nerves or blood.