Fundamental Electromagnetic Equations

Why This Matters

Electromagnetic equations aren't just formulas to memorize—they're the mathematical language describing how charges create fields, how fields exert forces, and how changing fields generate each other. You're being tested on your ability to apply these equations to real scenarios: calculating forces between charges, determining electric fields using symmetry, predicting induced currents, and understanding how energy flows through circuits and space. These concepts connect directly to circuits, capacitors, inductors, and electromagnetic waves.

The key insight is that electromagnetism builds logically: Coulomb's Law gives you forces, Gauss's Law simplifies field calculations, Faraday's Law and Ampère's Law show how electric and magnetic fields create each other, and Maxwell's Equations unify everything. Don't just memorize the equations—know when to use each one and what physical principle it represents.

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Electrostatic Foundations: Charges and Fields

These equations describe how stationary charges create electric fields and forces. The inverse-square relationship appears repeatedly because field lines spread out over spherical surfaces.

Coulomb's Law

• $F = k\frac{q_1 q_2}{r^2}$—the electrostatic force between two point charges, where $k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2$
• Inverse-square dependence means doubling the distance reduces the force to one-fourth; this mirrors gravitational force mathematically
• Vector nature requires attention to sign—like charges repel, opposite charges attract along the line connecting them

Electric Field Equation

• $\vec{E} = \frac{\vec{F}}{q}$—defines the electric field as force per unit positive test charge, measured in N/C or V/m
• Vector quantity points in the direction a positive charge would accelerate; negative charges experience force opposite to the field direction
• Superposition principle allows you to add field contributions from multiple charges vectorially to find the net field

Gauss's Law

• $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$—relates electric flux through a closed surface to the enclosed charge
• Symmetry is essential—only useful when you can construct a Gaussian surface where $E$ is constant or zero; works beautifully for spheres, infinite planes, and cylinders
• Enclosed charge only matters; charges outside the Gaussian surface contribute zero net flux

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Energy and Storage: Potential and Capacitance

These equations connect fields to energy, showing how work is done on charges and how systems store electrical energy. Potential is the bridge between field concepts and circuit analysis.

Electric Potential Equation

• $V = -\int \vec{E} \cdot d\vec{r}$—electric potential is the work done per unit charge against the electric field
• Scalar quantity (no direction), making it easier to calculate than fields; add potentials algebraically, not vectorially
• Potential difference ($\Delta V$) drives current in circuits and determines energy gained or lost by charges: $\Delta U = q\Delta V$

Capacitance Equation

• $C = \frac{Q}{V}$—capacitance measures charge stored per volt of potential difference, in farads (F)
• Geometry-dependent: for a parallel-plate capacitor, $C = \frac{\epsilon_0 A}{d}$; larger plates and smaller separation increase capacitance
• Energy storage in a capacitor is $U = \frac{1}{2}CV^2$—crucial for understanding RC circuits and energy transfer

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Magnetostatics: Currents Creating Fields

These laws describe how moving charges (currents) generate magnetic fields. Unlike electric fields that begin and end on charges, magnetic field lines always form closed loops.

Biot-Savart Law

• $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \hat{r}}{r^2}$—gives the magnetic field contribution from a small current element
• Cross product means the field is perpendicular to both the current direction and the displacement vector; use the right-hand rule
• Integration required for total field—essential for calculating fields from current loops, where $B_{center} = \frac{\mu_0 I}{2R}$

Ampère's Law

• $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$—the line integral of magnetic field around a closed path equals $\mu_0$ times the enclosed current
• Symmetry-dependent like Gauss's Law—most useful for long straight wires ($B = \frac{\mu_0 I}{2\pi r}$), solenoids ($B = \mu_0 n I$), and toroids
• Amperian loop must be chosen so $B$ is constant along the path; the current must pass through the loop, not just near it

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Electromagnetic Induction: Changing Fields

These laws describe how changing magnetic fields create electric fields (and vice versa). This is where electricity and magnetism become truly unified—changing one field generates the other.

Faraday's Law of Induction

• $\mathcal{E} = -\frac{d\Phi_B}{dt}$—a changing magnetic flux through a loop induces an EMF (voltage)
• Magnetic flux $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta$ for uniform fields; change B, A, or θ to induce EMF
• Foundation of generators and transformers—rotating coils in magnetic fields convert mechanical energy to electrical energy

Lenz's Law

• Negative sign in Faraday's Law—the induced current creates a magnetic field that opposes the change in flux
• Conservation of energy is the underlying principle; if induced currents aided the flux change, you'd get infinite energy
• Practical application: when a magnet approaches a loop, the induced current creates a field to repel it; when the magnet retreats, the current reverses to attract it

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The Complete Picture: Maxwell's Equations

Maxwell's Equations unify all electromagnetic phenomena into four elegant statements. They reveal that light itself is an electromagnetic wave—oscillating electric and magnetic fields propagating through space.

Maxwell's Equations

• Four equations: Gauss's Law ($\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$), Gauss's Law for Magnetism ($\nabla \cdot \vec{B} = 0$), Faraday's Law ($\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$), and Ampère-Maxwell Law ($\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$)
• Displacement current ($\epsilon_0 \frac{\partial \vec{E}}{\partial t}$) was Maxwell's key addition—a changing electric field acts like a current, completing the symmetry between $\vec{E}$ and $\vec{B}$
• Electromagnetic waves emerge naturally: the equations predict waves traveling at $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \, \text{m/s}$—the speed of light

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Quick Reference Table

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Self-Check Questions

1. Which two laws both calculate electric fields but require different problem-solving approaches? When would you choose one over the other?

2. A bar magnet is pushed toward a conducting loop. Using both Faraday's Law and Lenz's Law, describe what happens and explain why the induced current flows in the direction it does.

3. Compare Coulomb's Law and the Biot-Savart Law: what mathematical similarities do they share, and what fundamental difference exists between electric and magnetic field lines?

4. If an FRQ asks you to find the electric field inside a uniformly charged sphere, which equation should you use and why? What Gaussian surface would you construct?

5. How does Maxwell's addition of displacement current complete the symmetry between electric and magnetic fields, and why was this necessary to explain electromagnetic wave propagation?