3.1 Coulomb's Law, Electric Fields & Electric Potential

3.1 Coulomb's Law, Electric Fields & Electric Potential

Electrostatics investigates the behavior of stationary electric charges, the forces they exert on one another, and the spatial energy fields they create. In biological systems, living cells exploit electrostatic forces and separated ionic charges across thin lipid membranes to store metabolic energy, propagate neural impulses, and regulate cardiovascular rhythmicity.

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1. Coulomb's Law and Fundamental Charge Quantization

All observable electrostatic phenomena originate from quantized electric charges. The fundamental unit of charge ($e$) represents the charge carried by a single proton ($+e$) or electron ($-e$):

$$\ne = 1.602 \times 10^{-19}\text{ C}$$

Coulomb's Law Formulation

Coulomb's law describes the electrostatic force ($F_e$) acting between two stationary point charges ($q_1$ and $q_2$) separated by distance $r$ in a vacuum:

$$F_e = \frac{k |q_1 q_2|}{r^2}$$

Where:

• $k$ is Coulomb's constant, defined in terms of the permittivity of free space ($\epsilon_0 = 8.854 \times 10^{-12}\text{ C}^2/(\text{N}\cdot\text{m}^2)$):

$$k = \frac{1}{4\pi \epsilon_0} \approx 8.99 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2 \approx 9.0 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$$

Key Characteristics of Electrostatic Force

1. Inverse-Square Dependency: The electrostatic force varies inversely with the square of the separation distance ($F \propto 1/r^2$). Doubling the distance between two charges reduces the force to one-fourth its original value; tripling the distance reduces the force to one-ninth.
2. Vector Nature and Newton's Third Law: The electrostatic force acts along the line joining the centers of the two charges. Like charges experience mutually repulsive forces directed away from each other, while opposite charges experience mutually attractive forces directed toward each other. The force exerted by $q_1$ on $q_2$ is equal in magnitude and opposite in direction to the force exerted by $q_2$ on $q_1$ ($\vec{F}_{12} = -\vec{F}_{21}$), satisfying Newton's third law regardless of unequal charge magnitudes.
3. Dielectric Attenuation: In polar solvents such as water (dielectric constant $\kappa \approx 80$), electrostatic forces between dissolved ions are attenuated by a factor of $\kappa$, reducing effective Coulombic attraction from $F_0$ to $F_0 / \kappa$.

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2. Electric Fields, Field Lines, and Conductors in Equilibrium

An electric field ($\vec{E}$) represents the spatial distortion generated by a source charge ($Q$), through which it exerts forces on other charges introduced into the region.

Electric Field Definition and Point Charge Calculation

The electric field at any point in space is defined as the electrostatic force experienced per unit positive test charge ($q_0$):

$$\vec{E} = \frac{\vec{F}_e}{q_0} = \frac{k Q}{r^2} \hat{r}$$

• Units: Newtons per Coulomb ($\text{N}/\text{C}$) or Volts per meter ($\text{V}/\text{m}$). Both units are dimensionally identical: $1\text{ N}/\text{C} = 1\text{ V}/\text{m}$.
• Direction: The electric field vector points in the direction that a hypothetical positive test charge would accelerate. Consequently, electric fields radiate radially outward from positive source charges and radially inward toward negative source charges.

Superposition Principle

When multiple source charges are present, the net electric field at any given coordinate is the vector sum of the individual electric fields generated by each charge:

$$\vec{E}_{\text{net}} = \sum_{i=1}^n \vec{E}_i = \vec{E}_1 + \vec{E}_2 + \dots + \vec{E}_n$$

Electric Field Lines

Electric field lines provide a geometric visualization of field geometry:

• Lines always originate on positive charges (or at infinity) and terminate on negative charges (or at infinity).
• The local tangent to a field line gives the direction of $\vec{E}$ at that spatial coordinate.
• The spatial density of field lines reflects field magnitude: closer lines indicate a stronger field.
• Field lines never cross; a crossing would imply two distinct field directions at a single physical point, violating vector uniqueness.

Conductors in Electrostatic Equilibrium

When an ideal electrical conductor reaches electrostatic equilibrium:

1. The internal electric field is strictly zero ($\vec{E}_{\text{internal}} = 0$). Any nonzero internal field would exert forces on mobile conduction electrons, causing ongoing current flow.
2. All excess charge resides entirely on the outer surface of the conductor.
3. The electric field immediately outside the conducting surface is perpendicular (normal) to the surface at every point.

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3. Electric Potential Energy, Electric Potential, and Equipotential Lines

Electrostatic forces are conservative forces; work performed against electrostatic forces is stored as electrostatic potential energy.

Electric Potential Energy ($U$)

The electric potential energy ($U$) of a two-charge system ($q_1$ and $q_2$) separated by distance $r$ is defined as the work required to assemble the charges from infinite separation:

$$U = \frac{k q_1 q_2}{r}$$

• Unlike Coulomb's force ($1/r^2$), potential energy exhibits an inverse-first-power ($1/r$) dependence.
• Sign Conventions:
  • Like charges ($q_1 q_2 > 0$): $U > 0$. Positive potential energy signifies a repulsive, unstable configuration. External work must be performed on the system to push like charges together.
  • Opposite charges ($q_1 q_2 < 0$): $U < 0$. Negative potential energy signifies an attractive, bound state. Energy is released spontaneously as opposite charges approach.

Electric Potential ($V$)

Electric potential ($V$), commonly called voltage, is a scalar quantity defined as the electric potential energy per unit positive test charge:

$$V = \frac{U}{q_0} = \frac{k Q}{r}$$

• Units: Volts ($\text{V}$), where $1\text{ Volt} = 1\text{ Joule per Coulomb}$ ($1\text{ J}/\text{C}$).
• Electric potential is a scalar property of the field itself. For multiple charges, net potential is the algebraic (scalar) sum:

$$V_{\text{net}} = \sum_{i=1}^n \frac{k Q_i}{r_i}$$

Potential Difference and Work

The potential difference ($\Delta V$) between initial point $A$ and final point $B$ is the work ($W$) performed per unit charge in moving from $A$ to $B$:

$$\Delta V = V_B - V_A = \frac{W_{\text{external}}}{q} = -\frac{W_{\text{field}}}{q}$$

In a uniform electric field oriented across distance $d$, electric field and voltage are related by:

$$V = E d \iff E = \frac{\Delta V}{d}$$

Positive charges naturally accelerate from high potential to low potential (spontaneous $\Delta V < 0$), decreasing potential energy ($\Delta U = q \Delta V < 0$). Conversely, negative charges accelerate from low potential to high potential (spontaneous $\Delta V > 0$), which also decreases their potential energy ($\Delta U = (-q)(+\Delta V) < 0$).

Equipotential Surfaces

An equipotential surface consists of all spatial coordinates possessing an identical electric potential ($V = \text{constant}$):

• Perpendicular Orientation: Equipotential lines are everywhere perpendicular to electric field lines.
• Zero Work Path: Moving a charge along an equipotential surface requires zero net work ($W = q \Delta V = q(0) = 0$), because the displacement is orthogonal to the electrostatic force vector.

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Vector versus Scalar Electrostatic Quantities

The single most common electrostatics error on the MCAT is adding potentials as vectors or fields as scalars. Fix the character of each quantity before doing any arithmetic.