🧲AP Physics 2

Electric Field Formulas

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Why This Matters

Electric fields are the foundation of everything you'll encounter in AP Physics 2's electromagnetism units—from understanding how charges interact at a distance to analyzing circuits and even connecting to magnetism later. You're being tested on your ability to apply these formulas to novel situations, not just plug and chug. The exam loves asking you to compare scenarios: What happens to the force if you double the distance? How does adding a dielectric change the energy stored? These questions require you to understand the mathematical relationships, not just memorize the equations.

The formulas in this guide demonstrate key principles: inverse-square relationships, superposition, energy conservation, and the connection between fields and potentials. When you see a formula, ask yourself what physical principle it represents and when you'd choose it over another. Don't just memorize $E = kq/r^2$ —know that it only works for point charges and that the field points radially. Understanding the "why" behind each formula is what separates a 3 from a 5.

Force and Field Fundamentals

These formulas establish how charges create and respond to electric fields. The key insight: fields exist in space and exert forces on any charge placed within them. Separating "what creates the field" from "what feels the force" is essential for problem-solving.

Coulomb's Law

$F = k\frac{q_1 q_2}{r^2}$ —describes the electrostatic force between two point charges, where $k \approx 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2$
Inverse-square relationship means doubling the distance reduces the force to one-fourth—a favorite exam question
Sign determines direction: like charges repel (positive force), opposite charges attract (negative force), with force acting along the line connecting the charges

Electric Field Definition

$\vec{E} = \frac{\vec{F}_E}{q}$ —defines the electric field as force per unit charge, measured in N/C or V/m
Test charge concept: we imagine placing a tiny positive charge to "probe" the field without disturbing it
Vector quantity pointing in the direction a positive test charge would accelerate—away from positive sources, toward negative sources

Electric Field of a Point Charge

$E = \frac{kq}{r^2}$ —the field magnitude at distance $r$ from a single point charge $q$
Direction is radial: outward for positive charges, inward for negative charges
Same inverse-square behavior as Coulomb's Law—this is no coincidence; the field is the force framework

Compare: Coulomb's Law vs. Point Charge Field—both use $k/r^2$ , but Coulomb's Law gives the force between two specific charges, while the field formula describes what any charge would experience at that location. FRQs often ask you to find the field first, then calculate the force on a specific charge placed there.

Superposition and Complex Configurations

When multiple charges are present, each contributes independently to the total field. This principle lets you break down complex problems into manageable pieces.

Superposition Principle

Vector sum of individual fields—the total electric field at any point equals $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + ...$
Components are essential: break each field into x and y components, sum separately, then find the resultant magnitude and direction
AP boundary: calculations are limited to configurations with four or fewer interacting point charges

Electric Dipole Field

Two equal and opposite charges separated by a small distance create a characteristic field pattern
Field decreases as $1/r^3$ along the axis—faster than a single charge because the opposite charges partially cancel at large distances
Dipole moment $p = qd$ characterizes the strength, where $d$ is the separation distance—crucial for understanding polar molecules

Compare: Point Charge Field ( $1/r^2$ ) vs. Dipole Field ( $1/r^3$ )—the dipole field falls off faster because the two opposite charges increasingly cancel at large distances. If an FRQ shows field strength dropping more rapidly than inverse-square, think dipole.

Special Charge Distributions

Certain symmetric charge arrangements produce fields with predictable, simplified forms. Recognizing these patterns saves enormous calculation time on the exam.

Infinite Line of Charge

$E = \frac{\lambda}{2\pi\varepsilon_0 r}$ —field at distance $r$ from a line with linear charge density $\lambda$ (C/m)
Inverse-first-power relationship: field decreases as $1/r$ , not $1/r^2$ —the extended charge distribution changes the geometry
Direction is radial: perpendicular to and outward from the line (for positive $\lambda$ )

Infinite Plane of Charge

$E = \frac{\sigma}{2\varepsilon_0}$ —field from a plane with surface charge density $\sigma$ (C/m²)
Constant magnitude: the field does not depend on distance from the plane—a striking result of the infinite geometry
Direction is perpendicular to the surface, pointing away from positive charge, toward negative

Field Inside a Conductor

$E = 0$ inside a conductor in electrostatic equilibrium—charges redistribute on the surface to cancel any internal field
Shielding effect: this is why Faraday cages work; external fields cannot penetrate a conducting shell
Surface charge concentrates at points of high curvature—important for understanding charge distribution

Compare: Line Charge ( $1/r$ ) vs. Plane Charge (constant)—as you extend a charge distribution from a point to a line to a plane, the field falls off more slowly. The plane's field being constant explains why parallel-plate capacitors create uniform fields.

Electric Potential and Energy

These formulas connect force (a vector) to energy (a scalar). Working with potential often simplifies problems because you don't need to track directions.

Electric Potential Energy

$U = \frac{kq_1 q_2}{r}$ —potential energy stored in a two-charge system
Sign matters: positive $U$ for like charges (energy required to bring them together), negative for opposite charges (energy released)
Reference point at infinity: we define $U = 0$ when charges are infinitely separated

Electric Potential

$V = \frac{kq}{r}$ —potential at distance $r$ from a point charge, measured in Volts (J/C)
Scalar quantity: no direction to worry about; just add potentials algebraically for multiple charges
Potential decreases with distance from a positive charge; increases as you approach a negative charge

Potential-Energy Relationship

$\Delta U_E = q\Delta V$ —change in potential energy equals charge times potential difference
Positive charges naturally move from high to low potential; negative charges move the opposite direction
Electronvolt connection: 1 eV = energy gained by an electron crossing 1 V potential difference = $1.6 \times 10^{-19}$ J

Compare: Potential Energy ( $U = kq_1q_2/r$ ) vs. Electric Potential ( $V = kq/r$ )—potential energy requires two charges and describes their interaction; potential describes what a single charge creates at each point in space. Think of $V$ as "potential energy per unit charge."

Field-Potential Connection

The electric field and potential are two descriptions of the same physical situation. Understanding their relationship is crucial for FRQs.

Gradient Relationship

$E = -\frac{dV}{dx}$ —the electric field equals the negative rate of change of potential with position
Field points "downhill": from high potential toward low potential, like a ball rolling down a slope
Uniform field special case: between parallel plates, $E = \frac{\Delta V}{d}$ where $d$ is the plate separation

Equipotential Surfaces

Perpendicular to field lines: equipotential surfaces (constant $V$ ) are always at right angles to electric field vectors
No work on equipotentials: moving a charge along an equipotential surface requires zero work since $\Delta V = 0$
Closer spacing = stronger field: where equipotential lines bunch together, the field magnitude is larger

Compare: The field-potential relationship works both ways—if you know $V(x)$ , take the derivative to find $E$ ; if you know $E$ , integrate to find $\Delta V$ . FRQs often give you one and ask for the other.

Capacitance and Energy Storage

Capacitors store energy in the electric field between their plates. These formulas connect geometry to electrical properties.

Capacitance Definition

$C = \frac{Q}{V}$ —capacitance is the ratio of stored charge to potential difference, measured in Farads (F)
Property of the device: capacitance depends on geometry and materials, not on how much charge is actually stored
Typical values: most capacitors are measured in μF, nF, or pF—a 1 F capacitor is enormous

Parallel Plate Capacitor

$C = \frac{\varepsilon_0 A}{d}$ —capacitance increases with plate area $A$ and decreases with separation $d$
Uniform field between plates: $E = \frac{\sigma}{\varepsilon_0} = \frac{V}{d}$ , making parallel plates ideal for controlled fields
Permittivity $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m—this constant appears throughout electrostatics

Energy Stored in a Capacitor

$U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$ —three equivalent forms; choose based on what's given
Quadratic in voltage: doubling $V$ quadruples the stored energy—important for energy calculations
Energy stored in the field: the energy physically resides in the electric field between the plates

Dielectric Effect

$C = \kappa C_0$ —inserting a dielectric material multiplies capacitance by the dielectric constant $\kappa$
Dielectric reduces field: the material polarizes, creating an opposing field that partially cancels the original
More charge at same voltage: dielectrics allow capacitors to store more energy without breakdown

Compare: $U = \frac{1}{2}CV^2$ vs. $U = \frac{Q^2}{2C}$ —if voltage is held constant (battery connected), adding a dielectric increases both $C$ and $U$ . If charge is held constant (isolated capacitor), adding a dielectric increases $C$ but decreases $U$ . This distinction appears frequently on FRQs.

Gauss's Law

This powerful tool relates electric flux through a closed surface to the enclosed charge. Use it when symmetry makes the math tractable.

Gauss's Law Statement

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$ —total electric flux through a closed surface equals enclosed charge divided by $\varepsilon_0$
Choose Gaussian surfaces wisely: spheres for point charges, cylinders for line charges, boxes for plane charges
Symmetry is essential: Gauss's Law simplifies calculations only when $E$ is constant over the surface or perpendicular to it

Compare: Gauss's Law vs. Direct Calculation—for a point charge, both give $E = kq/r^2$ , but Gauss's Law derives it in one step using a spherical surface. For complex but symmetric distributions, Gauss's Law is far more efficient.

Quick Reference Table

Concept	Best Examples
Inverse-square relationships	Coulomb's Law ( $F \propto 1/r^2$ ), Point charge field ( $E \propto 1/r^2$ ), Potential ( $V \propto 1/r$ )
Superposition	Vector addition of fields, Algebraic addition of potentials
Field-potential connection	$E = -dV/dx$ , Equipotential surfaces, $\Delta U = q\Delta V$
Conductor properties	$E = 0$ inside, Charge on surface, Shielding
Capacitor formulas	$C = Q/V$ , $C = \varepsilon_0 A/d$ , $U = \frac{1}{2}CV^2$
Dielectric effects	$C = \kappa C_0$ , Reduced field strength, Increased charge storage
Special geometries	Line charge ( $1/r$ ), Plane charge (constant), Parallel plates
Energy conservation	$U = kq_1q_2/r$ , $\Delta U_E = q\Delta V$ , Work-energy theorem

Self-Check Questions

If you triple the distance between two point charges, by what factor does the electric force change? What about the electric potential energy?
Two formulas both contain $k/r^2$ : Coulomb's Law and the point charge field. When would you use each one, and what does each calculate?
Compare the electric field of an infinite line charge to that of an infinite plane charge. Which falls off with distance, and why does the plane's field remain constant?
A parallel-plate capacitor is connected to a battery, and then a dielectric is inserted. What happens to $C$ , $V$ , $Q$ , and $U$ ? How would your answers change if the capacitor were disconnected from the battery first?
Explain why the electric field inside a conductor in electrostatic equilibrium must be zero. What would happen if it weren't?