Electrostatics

Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force in an electromagnetic field. Charge comes in two varieties, positive and negative, and is quantized: all observable charge is an integer multiple of the elementary charge e = 1.60 x 10^-19 C. The proton carries +e, the electron carries -e. The SI unit of charge is the coulomb (C). The principle of conservation of charge states that the net charge of an isolated system remains constant: charge can be transferred (as when electrons move from one object to another during rubbing) but never created or destroyed. Objects become charged by gaining or losing electrons, never by transferring protons. The three methods of charging are: friction (rubbing transfers electrons), conduction (contact with a charged object redistributes charge), and induction (a charged object brought near, but not touching, a conductor causes charge separation, and grounding then traps net charge). Like charges repel and unlike charges attract. The magnitude of charge is typically expressed in microcoulombs (uC = 10^-6 C), nanocoulombs (nC = 10^-9 C), or elementary charges. On the MCAT, charge quantization rarely matters for macroscopic problems but is tested in the context of Millikan's oil-drop experiment, which demonstrated that charge is quantized.

Coulomb's Law

Coulomb's law quantifies the electrostatic force between two point charges. The magnitude of the force is F = k|q1q2|/r^2, where k = 1/(4piepsilon0) = 8.99 x 10^9 Nm^2/C^2, q1 and q2 are the charges, and r is the distance between their centers. The direction of the force is along the line joining the charges: repulsive if the charges have the same sign, attractive if they have opposite signs. Coulomb's law is an inverse-square law, meaning doubling the separation reduces the force to one-quarter. This is directly analogous to Newton's law of gravitation (F = Gm1m2/r^2), but with one critical difference: gravitational force is always attractive, while electrostatic force can be attractive or repulsive. For problems with more than two charges, the net force on any charge is the vector sum of the individual Coulomb forces (superposition principle). On the MCAT, common calculations include: finding the net force on a charge placed between two other charges, determining where a third charge would experience zero net force (equilibrium point), and comparing the relative magnitudes of electrostatic and gravitational forces between subatomic particles. A worked example: two point charges of +2 uC and -3 uC are separated by 10 cm. The force magnitude between them is F = (8.99 x 10^9)(2 x 10^-6)(3 x 10^-6)/(0.10)^2 = 5.39 N, attractive. The permittivity of free space epsilon0 = 8.85 x 10^-12 C^2/(Nm^2) appears when Coulomb's law is written as F = (1/(4piepsilon0))|q1q2|/r^2.

Electric Field

An electric field E is a vector field that represents the force per unit charge that a test charge would experience at any point in space: E = F/q0. The SI unit is N/C (or equivalently V/m). For a point charge Q, the electric field at a distance r is E = kQ/r^2, directed radially outward if Q is positive and radially inward if Q is negative. The superposition principle applies: the net electric field at a point is the vector sum of the fields due to each source charge. Electric field lines are a visual tool: they originate on positive charges, terminate on negative charges, never cross, and are denser where the field is stronger. The number of lines is proportional to the magnitude of the charge. For a uniform electric field, the most common MCAT scenario, the field lines are parallel and equally spaced, as between two oppositely charged parallel plates. In a uniform field, the force on a charge q is constant: F = qE, and the acceleration of a charged particle of mass m is a = qE/m. This allows kinematic analysis: a charged particle entering a uniform field behaves like a projectile under constant acceleration, leading to parabolic trajectories. The MCAT frequently tests the motion of electrons and protons in uniform fields: an electron (negative charge) accelerates opposite the field direction; a proton (positive charge) accelerates along the field direction.

Electric Potential and Voltage

Electric potential V (also called voltage) is the electric potential energy per unit charge: V = U/q. It is a scalar quantity measured in volts (V), where 1 V = 1 J/C. For a point charge Q, the electric potential at a distance r is V = kQ/r, with the zero of potential conventionally taken at infinity. Unlike the electric field (a vector), potential is a scalar, making it much easier to sum contributions from multiple charges: V_net = k * sum(q_i/r_i). The potential difference deltaV between two points A and B is the negative of the work done by the electric field per unit charge in moving a test charge from A to B: deltaV = V_B - V_A = -W_AB/q0. Equivalently, the work done by an external agent to move a charge q through a potential difference deltaV is W = qdeltaV. For a uniform electric field (e.g., between parallel plates separated by distance d), the relationship simplifies to deltaV = -Ed, or in magnitude: E = deltaV/d. This means an electric field can be expressed in V/m instead of N/C. The electronvolt (eV), the energy gained by one electron moving through 1 V, is 1 eV = 1.60 x 10^-19 J and is the standard unit for atomic-scale energies.

Potential Energy in Electric Fields

Electric potential energy U is the energy stored in a configuration of charges due to their relative positions. For two point charges q1 and q2 separated by distance r, U = kq1q2/r. The sign of U follows the sign of the product q1q2: positive for like charges (repulsive; energy must be added to bring them together), negative for unlike charges (attractive; energy is released when they come together). The electrostatic force is conservative, meaning the work done by the electric field in moving a charge between two points is independent of the path taken. This path independence is why a scalar potential V and potential energy U can be defined. The change in potential energy when a charge q moves through a potential difference deltaV is deltaU = qdeltaV. For a charge in a uniform electric field E, moving a distance d parallel to the field: deltaU = -qEd (the negative sign indicates that a positive charge loses potential energy moving along the field direction). Conservation of energy applies: KE_i + U_i = KE_f + U_f for a charged particle moving under only electric forces. A classic MCAT problem: an electron accelerates from rest through a potential difference of 100 V. Its final kinetic energy is KE = |qdeltaV| = (1.60 x 10^-19)(100) = 1.60 x 10^-17 J = 100 eV, and its final speed from (1/2)mv^2 = qdeltaV is v = sqrt(2qdeltaV/m). The analogy to gravitational potential energy is powerful: U_elec = kq1q2/r mirrors U_grav = -Gm1*m2/r (though the electric version uses signed charges while gravity uses positive masses with a negative sign convention).

Equipotential Lines

Equipotential lines (in 2D) or equipotential surfaces (in 3D) are loci of points at the same electric potential. No net work is required to move a charge along an equipotential surface because deltaV = 0 along that path. Key properties: (1) Equipotential surfaces are always perpendicular to electric field lines. If they were not, there would be a component of the field along the surface, and a charge would experience a force and gain or lose potential energy along the surface. (2) Electric field lines point from higher potential to lower potential. (3) Equipotential surfaces are closer together where the field is stronger (since E = -deltaV/delta_r, steep potential gradients mean strong fields). For a single point charge, equipotential surfaces are concentric spheres centered on the charge. For a uniform electric field (parallel plates), equipotential surfaces are planes perpendicular to the field. For an electric dipole, equipotential surfaces have a characteristic figure-eight pattern. The MCAT tests equipotential reasoning through questions like: Given equipotential lines labeled 10 V, 20 V, 30 V, where is the electric field strongest? Answer: where the lines are closest together (steepest potential gradient). Or: What is the work done moving a charge along an equipotential? Answer: zero.

Electric Dipole

An electric dipole consists of two equal and opposite charges (+q and -q) separated by a small distance d. The dipole moment is a vector p = qd, pointing from the negative charge to the positive charge, with units of Cm. The electric field of a dipole falls off as 1/r^3 at large distances (faster than the 1/r^2 of a monopole), because the fields of the two opposite charges partially cancel. In a uniform external electric field E, a dipole experiences zero net force but a net torque tau = p x E, with magnitude tau = pEsin(theta), where theta is the angle between p and E. This torque tends to align the dipole with the field (minimum potential energy at theta = 0 degrees, maximum at theta = 180 degrees). The potential energy of a dipole in an external field is U = -pE = -pEcos(theta). The MCAT tests dipoles primarily through: (1) torque on a dipole in a uniform field, (2) energy of alignment, and (3) the biological relevance. Many molecules (water, proteins) are permanent electric dipoles, and their interactions determine solubility, protein folding, and cell membrane structure. An electric dipole also creates its own non-uniform field, which is responsible for dipole-dipole intermolecular forces. While magnetic dipoles (PH-010) are a separate topic, the mathematical form of torque (tau = mu x B) is directly analogous to the electric dipole torque (tau = p x E).

Conductors and Insulators

Conductors are materials in which electric charges (usually electrons) move freely. Metals are the primary example: their conduction-band electrons are delocalized and can move through the material with minimal resistance. In electrostatic equilibrium, a conductor has three key properties: (1) The electric field inside the conductor is zero. If it were not, free charges would move until it became zero. (2) Any net charge resides on the surface of the conductor; like charges repel each other to maximize separation. (3) The electric field just outside a conductor is perpendicular to the surface; any tangential component would cause surface charges to move. The surface charge density is highest at points of sharpest curvature (lightning rods exploit this). Insulators (dielectrics) are materials in which charges are NOT free to move; electrons are tightly bound to their atoms. Examples include rubber, glass, plastic, and dry wood. When an insulator is placed in an external electric field, it polarizes: individual atoms or molecules develop induced dipole moments (the electron cloud shifts slightly opposite the field), but there is no net bulk movement of charge. The degree of polarization is characterized by the dielectric constant kappa, where kappa >= 1. A dielectric inserted between capacitor plates increases capacitance by a factor of kappa: C = kappa*C0. The MCAT also tests semiconductors, which have conductivity intermediate between conductors and insulators and whose conductivity increases with temperature (unlike metals).

Electrostatic Induction

Electrostatic induction is the process by which a charged object brought near, but not touching, a conductor causes a redistribution of charge within the conductor without any net charge transfer. When a negatively charged rod is brought near a neutral conducting sphere, electrons in the sphere are repelled to the far side, leaving the near side positively charged. The sphere remains neutral overall but develops a charge separation (it is polarized as a whole object). If the sphere is then grounded (connected to the Earth by a conducting path), electrons repelled by the rod flow to ground. When the ground connection is removed before the rod is withdrawn, the sphere is left with a net positive charge. This is charging by induction. The key sequence is: (1) bring charged object near, (2) ground the conductor, (3) remove ground, (4) remove charged object. Induction explains how a charged balloon sticks to a neutral wall: the balloon's charge induces opposite charge on the wall's surface, creating an attractive force. In the laboratory, an electroscope detects charge using induction: a charged object brought near the terminal causes the leaves to separate, even without touching. The MCAT also tests the distinction between induction in conductors (macroscopic charge separation) versus polarization in insulators (microscopic dipole alignment at the atomic/molecular level). Both phenomena produce attractive forces between charged and neutral objects, but through different mechanisms. The concept of electrostatic shielding, that a conducting shell blocks external electric fields from its interior, is a direct consequence of induction: charges in the conductor rearrange to cancel the external field inside.

Capacitors

A capacitor is a device that stores electric charge and energy in an electric field. The simplest capacitor consists of two parallel conducting plates separated by a distance d. When connected to a voltage source, equal and opposite charges (+Q and -Q) accumulate on the plates, creating a uniform electric field E = deltaV/d between them. Capacitance C is defined as the ratio of stored charge to potential difference: C = Q/deltaV, measured in farads (F), where 1 F = 1 C/V. For a parallel-plate capacitor: C = epsilon0A/d, where A is the plate area and epsilon0 = 8.85 x 10^-12 F/m is the permittivity of free space. Capacitance increases with plate area and decreases with plate separation. The energy stored in a capacitor is U = (1/2)QdeltaV = (1/2)CdeltaV^2 = Q^2/(2C). Inserting a dielectric material (insulator) between the plates increases capacitance by a factor of the dielectric constant kappa: C = kappaC0 = kappaepsilon0A/d. The dielectric weakens the electric field between plates (E = E0/kappa) because polarized molecules in the dielectric create an opposing field, allowing more charge to be stored at the same voltage. On the MCAT, capacitors are typically tested in the context of parallel-plate geometry, energy storage, and the effect of dielectrics. Capacitors also appear in RC circuits (PH-010) where their time-dependent charging/discharging behavior (tau = RC) is a high-yield topic.

How it works

MCAT electrostatics problems follow a decision tree. First, identify whether the question asks about forces/fields (vector analysis) or energy/potential (scalar analysis). For forces: use Coulomb's law and superposition for point charges, or F = qE if the field is given. For fields: use E = kQ/r^2 for point charges with vector addition, or E = deltaV/d for uniform fields between plates. For potential energy: U = kq1q2/r for pairs, deltaU = qdeltaV for moving a charge through a potential difference. For potential: V = kQ/r (scalar sum, no vectors). When a charged particle moves in a field, use energy conservation: (1/2)mv^2_f = (1/2)mv^2_i + qdeltaV. For dipoles: tau = pEsin(theta) (torque), U = -pEcos(theta) (energy of orientation). For conductors at equilibrium: E_inside = 0, charge on surface, E perpendicular to surface. For induction problems: follow the grounding sequence. The gravity analogy is your best friend: replace mass with charge and G with k, but remember the critical difference that electric forces can repel.

How it works

MCAT electrostatics problems follow a decision tree. First, identify whether the question asks about forces/fields (vector analysis) or energy/potential (scalar analysis). For forces: use Coulomb's law and superposition for point charges, or F = qE if the field is given. For fields: use E = kQ/r^2 for point charges with vector addition, or E = deltaV/d for uniform fields between plates. For potential energy: U = kq1q2/r for pairs, deltaU = qdeltaV for moving a charge through a potential difference. For potential: V = kQ/r (scalar sum, no vectors). When a charged particle moves in a field, use energy conservation: (1/2)mv^2_f = (1/2)mv^2_i + qdeltaV. For dipoles: tau = pEsin(theta) (torque), U = -pEcos(theta) (energy of orientation). For conductors at equilibrium: E_inside = 0, charge on surface, E perpendicular to surface. For induction problems: follow the grounding sequence. The gravity analogy is your best friend: replace mass with charge and G with k, but remember the critical difference that electric forces can repel.

Comparisons

• C/P (Forces): Coulomb's law is structurally identical to Newton's law of gravitation. MCAT passages often ask you to compare electric vs. gravitational forces between subatomic particles; the electric force is roughly 10^39 times stronger between two protons.
• C/P (Energy): Electric potential energy (U = kq1q2/r) is directly analogous to gravitational PE. Both are conservative, path-independent, and associated with inverse-square force laws (PH-004).
• C/P (Kinematics): A charged particle in a uniform electric field undergoes constant acceleration (a = qE/m), making all constant-acceleration kinematic equations applicable; projectile motion with E replacing g (PH-002).
• C/P (Circuits): Voltage, current, and resistance (PH-010) depend on the concepts of electric potential and field established here. The E field inside a wire drives current (E = rho*J).
• B/B (Membrane potential): The resting membrane potential (roughly -70 mV) and action potentials are diffusion potentials governed by ion concentration gradients; conceptually grounded in electrostatic potential differences across a selectively permeable membrane.
• B/B (Protein structure): Hydrogen bonds, ionic bonds, and van der Waals forces between amino acid side chains are electrostatic interactions; Coulomb's law governs their strength and distance dependence.
• B/B (Enzyme catalysis): Electrostatic stabilization of transition states and substrate binding involves charge-charge interactions and dipole alignment in the active site.
• C/P (Molecular polarity): Water's permanent dipole moment makes it an excellent solvent for ions; the energy of solvation (ion-dipole interactions) follows the dipole potential energy form U = -pE*cos(theta).

Common confusions

• Confusing electric field (vector) and electric potential (scalar). The field at a point can be zero while the potential is nonzero (e.g., midway between two equal positive charges, E = 0 but V != 0). Conversely, the potential can be zero where the field is nonzero (e.g., midway between equal opposite charges, V = 0 but E != 0).
• Using the wrong sign for potential energy. U = kq1q2/r: if q1 and q2 have opposite signs, U is negative (attractive). A negative U means the system is bound; energy must be added to separate the charges. Students often treat all potential energies as positive by default.
• Forgetting that conductors at equilibrium have zero internal field. If the MCAT says 'a conducting sphere in electrostatic equilibrium,' E = 0 inside regardless of the surrounding field, charges, or geometry. This also means the entire conductor is at a single potential.
• Misapplying E = deltaV/d. This formula is only valid for uniform fields (parallel plates). For point charges, you must use E = kQ/r^2. Students often incorrectly use E = deltaV/d for point charges by dividing the potential difference by some arbitrary distance.
• Ignoring the vector nature of electric fields. When summing fields from multiple charges, you must add them as vectors. A common error is to add magnitudes: E_net = kQ1/r1^2 + kQ2/r2^2 (scalar) instead of E_net = E1 + E2 (vector).
• Confusing the direction of force on a test charge. A positive test charge feels a force in the direction of E; a negative test charge feels a force opposite E. Students often forget the sign when computing F = qE.
• Treating equipotential lines as field lines. Equipotentials are perpendicular to field lines, not parallel. Moving along an equipotential requires zero work; moving along a field line changes the potential.

Quick review

• Charge is quantized (e = 1.60 x 10^-19 C), conserved, and exists in two types. Like charges repel; opposite charges attract.
• Coulomb's law: F = k|q1q2|/r^2, where k = 8.99 x 10^9 N*m^2/C^2. Inverse-square; double r, force drops to 1/4. Vector superposition for multiple charges.
• Electric field: E = F/q0 (N/C or V/m). Point charge: E = kQ/r^2, radial. Uniform field: E = deltaV/d, parallel lines between plates.
• Electric potential (voltage): V = U/q (scalar). Point charge: V = kQ/r. DeltaV = -W/q0. Electronvolt: 1 eV = 1.60 x 10^-19 J.
• Electric PE: U = kq1q2/r. DeltaU = q*deltaV. Conservative, path-independent. KE_i + U_i = KE_f + U_f.
• Equipotential surfaces: perpendicular to E; no work to move charge along them; closer spacing = stronger field. E points from high V to low V.
• Dipole: p = qd. Torque tau = pEsin(theta). Energy U = -pEcos(theta). Field decays as 1/r^3. Biological dipoles: water, proteins.
• Conductors at equilibrium: E_inside = 0, charge on surface, E perpendicular to surface. Insulators: polarize but no bulk charge flow. Dielectric constant kappa.
• Induction: bring charged object near conductor, charge separates, ground, remove ground, remove object, net charge trapped.