4.1 Electrostatics: Charge, Coulomb's Law & Electric Fields

Charge, Conductors & Insulators

Content Category 4C on the MCAT Chemical and Physical Foundations section opens with electrostatics because every later circuit, cell, and membrane-potential idea is built on charge, force, field, and potential. Electric charge is a fundamental property of matter that comes in two types, positive and negative, and is measured in coulombs (C). The smallest unit of charge that exists freely is the elementary charge, e ≈ 1.6 × 10⁻¹⁹ C — the magnitude of the charge on a single proton (+e) or electron (−e). Because charge is quantized, any observable charge is an integer multiple of e. One mole of elementary charges is one faraday of charge (≈ 96,500 C), a quantity that reappears when electrochemical cells deposit metal or liberate gas.

Materials differ in how easily charge moves through them. Conductors (metals like copper and silver, and, in biology, ionic solutions such as cytoplasm and extracellular fluid) have free charge carriers that redistribute quickly. Insulators (rubber, glass, most plastics, and the hydrophobic core of a phospholipid bilayer) hold charges tightly, so charge placed on one region tends to stay put rather than spreading.

Charge conservation is a strict law: the net charge of an isolated system never changes. Charge can be transferred between objects — by direct contact (conduction) or by bringing a charged object close without touching (induction, which polarizes but does not by itself transfer net charge) — but it is never created or destroyed. When a charged rod touches a neutral conductor, charge flows until the two objects share the same potential; the total charge among them afterward still equals the total charge before contact.

MCAT trap: induction charges an object by polarization and grounding, without any charge ever crossing between the two original objects in the first step — the final charge on the induced object is opposite in sign to the inducing object, unlike conduction, which leaves both objects with the same sign of charge.

Charge-transfer methods compared:

• Conduction: charged object touches neutral object; charge of the same sign redistributes so both share the same sign
• Induction (with grounding): nearby charge polarizes object; ground allows opposite charge to remain after separation
• Friction (triboelectric): electrons transfer by rubbing; one object becomes positive, the other negative
• Conservation always holds: total charge before = total charge after, even when signs redistribute
• MCAT tip: opposite charges attract; like charges repel; the magnitude of the Coulomb force is identical on both charges (Newton's third law)

Coulomb's Law

Coulomb's Law gives the electrostatic force between two point charges:

F = kq₁q₂ / r²

where k is Coulomb's constant (k ≈ 8.99 × 10⁹ N·m²/C², rounded to 9.0 × 10⁹ N·m²/C² for MCAT mental math), q₁ and q₂ are the charge magnitudes, and r is the distance between their centers. Like Newton's law of gravitation, Coulomb's Law is an inverse-square law — doubling the distance cuts the force to one-fourth, and tripling the distance cuts it to one-ninth. The product q₁q₂ is positive for like charges (repulsion) and negative for unlike charges (attraction) if you track signed charges carefully; many MCAT problems give magnitudes and ask you to state the direction separately.

The force is a vector: like charges repel, unlike charges attract, and the force acts along the line connecting the two charges. When more than two charges are present, the net force on any one charge is the vector sum of the individual pairwise Coulomb forces (superposition). Break multi-charge problems into pairwise forces, resolve into components if needed, then add — never try to invent a single multi-body formula.

Worked example: Two point charges, q₁ = +2 μC and q₂ = +3 μC, sit 3 m apart. What is the force between them?

F = kq₁q₂ / r² = (9 × 10⁹)(2 × 10⁻⁶)(3 × 10⁻⁶) / (3)²

Multiply the charges first: (2 × 10⁻⁶)(3 × 10⁻⁶) = 6 × 10⁻¹². Multiply by k: (9 × 10⁹)(6 × 10⁻¹²) = 54 × 10⁻³ = 0.054. Divide by r² = 9:

F = 0.054 / 9 = 0.006 N = 6 mN (repulsive, since both charges are positive)

Keeping powers of 10 separate from the leading digits, as shown above, is the fastest no-calculator route to a Coulomb's Law answer. A second quick check: if the same charges sat 1 m apart, F would be nine times larger (0.054 N), which is consistent with the inverse-square law.

Electric Field

An electric field (E) is the force per unit positive test charge at a point in space: E = F/q. Electric field is a vector, with SI units of newtons per coulomb (N/C) or equivalently volts per meter (V/m), and it points in the direction a small positive test charge would be pushed if placed at that point.

For a single point charge Q, the field it creates at distance r is:

E = kQ / r²

which points radially away from a positive source charge and radially inward toward a negative source charge. The force on a charge q sitting in a field E is simply F = qE — positive charges are pushed in the direction of E; negative charges are pushed opposite E. That last point is crucial for ions: Na⁺ and K⁺ feel force along E, while Cl⁻ feels force opposite E for the same field.

Field lines are a visual tool for electric fields: they originate on positive charges and terminate on negative charges, never cross one another, and their density (how closely spaced they are) represents the field's magnitude — tightly packed lines mean a strong field, widely spaced lines mean a weak one. A uniform field, such as the field between the plates of a parallel-plate capacitor (or, approximately, across a thin membrane patch), is drawn as evenly spaced, parallel straight lines.

When a charge distribution creates a field, the total field at any point is the vector sum of the fields from every individual charge — the same superposition principle used for forces. The MCAT typically tests conceptual symmetry results (for example, the field is zero at the center of a uniformly charged ring, since opposite contributions cancel) rather than requiring an integral.

Biological scale note: a resting membrane potential of about −70 mV across a ~7 nm thick membrane corresponds to an enormous field magnitude on the order of 10⁷ V/m (E ≈ V/d for a roughly uniform field). Living cells sit in extreme local fields; that is why tiny voltage changes open voltage-gated channels so effectively.

Electrostatic Energy & Electric Potential

Moving charges against an electric field stores electric potential energy (U), the electrical analog of gravitational potential energy. For two point charges:

U = kq₁q₂ / r

U is positive for like charges (energy is required to push them together) and negative for unlike charges (the field does positive work pulling them together).

Electric potential (V), commonly called voltage, is the electric potential energy per unit charge at a point in space: V = U/q. For a point charge Q, the potential at distance r is:

V = kQ / r

Unlike the electric field, potential is a scalar (no direction), measured in volts (V), where 1 V = 1 J/C. Because potential is additive without regard to direction, the net potential from several charges is just the algebraic sum of the individual potentials — simpler than the vector sum required for the field. Surfaces of constant potential are equipotential surfaces; no work is done moving a charge along an equipotential, and field lines are always perpendicular to equipotentials.

The key relationship linking energy and potential is U = qV, and the work needed to move a charge q between two points equals the charge times the potential difference: W = qΔV. This is exactly the relationship that defines voltage across circuit elements in the next section, and it is also the basis of the resting membrane potential across a neuron's cell membrane — a charge separation that stores electrical potential energy the cell later releases to fire an action potential.

Potential versus potential energy (memorize this distinction):

• Potential V is a property of the source charges and the location alone (units: volts)
• Potential energy U = qV also depends on the test charge you place there (units: joules)
• Two different ions at the same membrane location share the same local potential but have different potential energies if their charges differ
• ΔV = Ed for a uniform field over distance d; this links field problems to voltage problems
• Membrane "voltage" on the MCAT almost always means transmembrane potential difference, not absolute potential relative to infinity

Common MCAT Traps

• Treating Coulomb force as proportional to 1/r instead of 1/r². Tripling distance divides force by 9, not by 3.
• Confusing field (vector) with potential (scalar). Fields cancel by opposing directions; potentials of opposite signs cancel by algebraic addition.
• Mixing potential V with potential energy U. U depends on the test charge; V does not.
• Forgetting that negative charges accelerate opposite the field direction. A positive test-charge definition of E misleads students about anions.
• Assuming induction always leaves the induced object with the same sign as the rod. Without grounding, a nearby rod only polarizes; net charge stays zero until electrons can enter or leave through a ground path.
• Ignoring that force magnitudes on a pair of charges are equal even when the charges themselves are unequal — action-reaction pairs have equal magnitude.