Electrostatics and Capacitors (AAMC 4C)

Electrostatics deals with charges at rest and the forces, fields, potentials, and energy associated with them. The AAMC MCAT content outline is dense in this chapter, from Coulomb's law and field calculations for points/sheets, to capacitors and RC charging/discharging. This is a high-yield topic on the MCAT.

MCAT Chem/Phys focus. Electrostatics sits in content category 4C (electrostatics, circuits, and magnetism). High-yield: Coulomb's law, the electric field of point charges and charged sheets, field lines, electric potential and potential energy, capacitance (C = Q/V), dielectrics, energy stored (½CV²), and RC charging/discharging.

Coulomb's Law

Two point charges q₁ and q₂ separated by distance r exert on each other a force directed along the line joining them:

F = k q₁q₂ / r²

where k = 1 / (4πε₀) ≈ 9 × 10⁹ N m² C⁻² in vacuum, and ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻².

Like charges repel; unlike charges attract.
The force obeys Newton's third law (equal and opposite).
In a medium of relative permittivity ε_r, divide the vacuum force by ε_r.
The principle of superposition: net force = vector sum of individual Coulomb forces.

Electric Field

An electric field at a point is the force per unit positive test charge placed there:

E = F / q

SI unit: N C⁻¹ = V m⁻¹. E is a vector quantity in the direction of force on a positive test charge.

Electric Field Intensity (Point Charge)

The field at distance r from a point charge Q in vacuum is

E = kQ / r²

Direction: radially outward for +Q, inward for −Q.

Standard electric-field formulas you should remember
Source	Formula for E	Direction / notes
Point charge Q (vacuum)	E = kQ/r² = Q/(4πε₀r²)	Radial, ∝ 1/r²
Long charged rod (linear density λ)	E = λ/(2πε₀r)	Radial, ∝ 1/r
Infinite charged sheet (σ)	E = σ/(2ε₀)	Perpendicular to sheet, independent of distance
Parallel-plate capacitor (between)	E = σ/ε₀ = V/d	Uniform between plates; zero outside
Inside a conductor (electrostatic)	E = 0	Charge resides on outer surface (Faraday cage)
At centre of a uniformly charged sphere	E = 0	Symmetry → net field cancels
Outside uniformly charged sphere	E = kQ/r²	Behaves like point charge at centre

Field inside a conductor

Under electrostatic equilibrium, the electric field inside a conductor is zero. Any excess charge resides on the outer surface; the conductor itself is at one constant potential (an equipotential volume). This is the principle behind electrostatic shielding.

Electric Field (Infinite Sheet)

A uniformly charged infinite plane of surface charge density σ (C m⁻²) produces a uniform field perpendicular to the sheet on either side:

E = σ / (2ε₀)

The field is independent of distance from the sheet (within the infinite-sheet idealisation).

Two parallel sheets: the parallel-plate capacitor

Two infinite sheets carrying opposite charge densities ±σ produce a uniform field between the plates and zero field outside:

E_between = σ / ε₀

This is twice the single-sheet result, the contributions from the two plates add up between, cancel outside.

Electric Field Lines

Field lines (Faraday's lines of force) provide a visual map of an electric field. Properties to remember:

They begin on positive charges and end on negative charges (or run to infinity).
The tangent at any point gives the direction of E there.
Density of lines (lines per unit perpendicular area) is proportional to the magnitude of E.
Two field lines never cross, the field has a unique direction at every point.
Field lines are perpendicular to a conductor's surface in equilibrium.

Common trap. Electric field lines are not paths of moving charges. They are an instantaneous direction map. A charge starts moving along the line at that instant, but it accelerates and may follow a curved trajectory.

Electric Potential Energy and Potential

The work done by an external agent in bringing a positive test charge q from infinity to a point against the field equals the electric potential energy U at that point.

Electric potential

V = U / q = W / q

Electric potential is potential energy per unit charge. SI unit: volt (V) = J C⁻¹. It is a scalar, no direction, only sign.

Potential due to a point charge

V = kQ / r

(Reference: V = 0 at infinity.) Note V ∝ 1/r, while E ∝ 1/r².

Potential energy of a charge pair

U = k q₁q₂ / r = q₂ V₁

Positive U → like charges (repulsive system). Negative U → unlike charges (bound system).

Relation to field

For a uniform field, V = E d (where d is distance along the field). In general, E = −dV/dx, the field points "downhill" in potential.

Charging and Discharging Capacitor

A capacitor stores charge. For any capacitor: C = Q / V (units: farad, F). For a parallel-plate capacitor in vacuum:

C = ε₀ A / d

Inserting a dielectric of permittivity ε_r multiplies C by ε_r.

Combinations

Series: 1/C_s = 1/C₁ + 1/C₂ + ... ; same charge on each, voltages add.
Parallel: C_p = C₁ + C₂ + ... ; same voltage across each, charges add.
Energy stored: U = ½ CV² = ½ QV = Q²/(2C).

Biological connection, the cell membrane as a capacitor

A neuron's plasma membrane is a thin insulator (the lipid bilayer) separating two conducting salt solutions, so it behaves like a parallel-plate capacitor. Separated Na⁺/K⁺ charge across it sets up the resting membrane potential (≈ −70 mV), and the membrane stores charge exactly as C = Q/V predicts. This is why electrostatics recurs in MCAT physiology passages on nerve signaling.

Charging through a resistor (RC circuit)

When a capacitor is charged through a resistor R from a battery of EMF V₀:

V(t) = V₀ (1 − e^−t/τ)

The charge on the capacitor builds up exponentially. The time constant

τ = RC

is the time taken for the capacitor to reach 63% of its final voltage. After 5τ it is essentially fully charged.

Discharging

When a charged capacitor discharges through a resistor:

V(t) = V₀ e^−t/τ

Voltage drops to 37% of its initial value after one time constant.

Time-constant memory aid. "63 up, 37 down", after one τ, a charging capacitor reaches 63% and a discharging one falls to 37%. After ~5τ the transient is over and the circuit is in steady state.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter. Read the explanation even when you get the answer right, it's where the deeper concept lives.

Q1. The Coulomb force between two point charges, when the distance between them is doubled, becomes:

Double the original
Half the original
One-third the original
One-quarter the original

F ∝ 1/r². Doubling r quarters the force. The same inverse-square dependence governs gravitation and the electric field of a point charge.

Q2. The electric field inside a conductor in electrostatic equilibrium is:

Zero
Equal to the surface field
Maximum at the centre
Equal to σ/ε₀

If E were non-zero inside a conductor, free electrons would accelerate, not "equilibrium." Excess charge sits on the outer surface, leaving zero field inside. This is why a Faraday cage shields the interior from external fields.

Q3. A 10 µF capacitor is charged through a 200 kΩ resistor. The time constant of the circuit is:

0.2 s
2 s
20 s
200 s

τ = RC = (2 × 10⁵) × (10⁻⁵) = 2 s. After 2 s the capacitor reaches 63% of the supply voltage; after ~10 s (5τ) it is essentially fully charged.

Q4. The electric field due to an infinite plane sheet of charge with surface density σ is:

σ/ε₀
σ/(2ε₀)
σ/(4πε₀)
2σ/ε₀

Single infinite sheet: E = σ/(2ε₀), independent of distance. Between two parallel oppositely charged sheets the fields add and E = σ/ε₀ (and zero outside), that's the parallel-plate capacitor.

Q5. The energy stored in a capacitor of capacitance C charged to potential V is:

CV
CV²
½ CV²
2 CV²

Energy U = ½ CV² = ½ QV = Q²/(2C). The factor of ½ arises because the voltage rises from 0 to V as charge accumulates, the work integral gives half of QV.

Quick Recap

Coulomb: F = kq₁q₂/r²; k = 9 × 10⁹ N m² C⁻².
Field of a point charge: E = kQ/r²; potential: V = kQ/r.
Inside a conductor in equilibrium: E = 0.
Infinite sheet: E = σ/(2ε₀); parallel-plate capacitor inside: E = σ/ε₀.
Field lines: + to −, never cross, density = magnitude.
Potential energy: U = kq₁q₂/r; V = U/q.
Capacitor: C = Q/V; parallel plate C = ε₀A/d; energy U = ½CV².
Combinations: series 1/C_s = Σ1/C; parallel C_p = ΣC.
RC circuit: τ = RC. Charging V = V₀(1 − e^−t/τ); discharging V = V₀e^−t/τ.

Test yourself. Take a timed Electrostatics quiz or browse all Physics MCQs to lock these concepts in.