Maxwell’s Equations (Math View)

Key idea

Vector calculus form of electromagnetism.

In depth

Four equations in vector calculus: ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −B_t, ∇×B = μ₀J + μ₀ε₀E_t. Combining them yields the wave equation for E and B, predicting electromagnetic waves at c = 1/√(μ₀ε₀).

Worked example

In vacuum, ∇×(∇×E) leads to E_tt = c²ΔE.

Real-world uses

• Designing antennas, Wi-Fi and microwave ovens from electromagnetic waves

Common mistake

Mixing up divergence and curl equations.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Differential Equations, Abstract Algebra & Applied Math

• Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
• Runge–Kutta Methods – Higher-accuracy numerical solvers.
• Second-Order Homogeneous Equations – Use the characteristic equation.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.