Maxwell's Equations Reference Cheat Sheet

A printable reference covering Maxwell's equations, flux, circulation, electromagnetic waves, and key constants for grades 11-12.

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Maxwell's equations describe how electric and magnetic fields are created and how they change. This cheat sheet helps students connect the integral forms, field geometry, and electromagnetic wave results in one reference. It is especially useful for solving problems involving charge, current, flux, circulation, induction, and light. The goal is to make each equation meaningful, not just memorized. The core ideas are electric flux, magnetic flux, electric circulation, and magnetic circulation. Gauss's law relates electric flux to enclosed charge, while Gauss's law for magnetism says there are no isolated magnetic poles. Faraday's law connects changing magnetic flux to induced electric fields, and the Ampere-Maxwell law connects currents and changing electric flux to magnetic fields. Together, these equations predict electromagnetic waves traveling at $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ .

Key Facts

Gauss's law for electricity is $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\mathrm{enc}}}{\epsilon_0}$ , which relates electric flux through a closed surface to enclosed charge.
Gauss's law for magnetism is $\oint \vec{B} \cdot d\vec{A} = 0$ , which means the net magnetic flux through any closed surface is zero.
Faraday's law is $\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}$ , so a changing magnetic flux creates a circulating electric field.
The Ampere-Maxwell law is $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\mathrm{enc}} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ , so magnetic circulation comes from current and changing electric flux.
Electric flux is $\Phi_E = \int \vec{E} \cdot d\vec{A}$ , and for a uniform field on a flat surface it becomes $\Phi_E = EA\cos\theta$ .
Magnetic flux is $\Phi_B = \int \vec{B} \cdot d\vec{A}$ , and for a uniform field on a flat surface it becomes $\Phi_B = BA\cos\theta$ .
Electromagnetic waves in vacuum travel at $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3.00 \times 10^8\ \mathrm{m/s}$ .
For an electromagnetic wave in vacuum, the field magnitudes satisfy $E = cB$ and the energy travels in the direction of $\vec{E} \times \vec{B}$ .

Vocabulary

Electric Flux: Electric flux $\Phi_E$ measures how much electric field passes through a surface.
Magnetic Flux: Magnetic flux $\Phi_B$ measures how much magnetic field passes through a surface.
Circulation: Circulation is the loop integral of a field, such as $\oint \vec{E} \cdot d\vec{\ell}$ or $\oint \vec{B} \cdot d\vec{\ell}$ , around a closed path.
Displacement Current: Displacement current is the term $\epsilon_0 \frac{d\Phi_E}{dt}$ that lets a changing electric flux create a magnetic field.
Permittivity of Free Space: The constant $\epsilon_0$ describes how electric fields behave in vacuum and has value $8.85 \times 10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$ .
Permeability of Free Space: The constant $\mu_0$ describes how magnetic fields behave in vacuum and has value $4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}$ .

Common Mistakes to Avoid

Forgetting that flux uses the perpendicular component of the field is wrong because $\Phi = BA\cos\theta$ or $\Phi = EA\cos\theta$ depends on the angle to the area vector, not the surface itself.
Dropping the negative sign in Faraday's law is wrong because $\oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}$ represents Lenz's law and the opposition to the change in flux.
Treating $\oint \vec{B} \cdot d\vec{A} = 0$ as saying there is no magnetic field is wrong because it only says the net magnetic flux through a closed surface is zero.
Using only $\mu_0 I_{\mathrm{enc}}$ in the Ampere-Maxwell law is wrong when electric flux changes, because the term $\mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ is needed.
Confusing open surfaces with closed surfaces is wrong because Gauss's laws use closed surfaces, while many flux calculations for induction use open surfaces bounded by a loop.

Practice Questions

1 A uniform electric field of $500\ \mathrm{N/C}$ passes through a flat area of $0.20\ \mathrm{m^2}$ at an angle of $60^\circ$ to the area vector. Find the electric flux $\Phi_E$ .
2 A circular loop has area $0.030\ \mathrm{m^2}$ , and the magnetic field perpendicular to it changes from $0.20\ \mathrm{T}$ to $0.80\ \mathrm{T}$ in $0.50\ \mathrm{s}$ . Find the magnitude of the induced emf using $|\mathcal{E}| = \left|\frac{\Delta \Phi_B}{\Delta t}\right|$ .
3 In a vacuum electromagnetic wave, the magnetic field amplitude is $2.0 \times 10^{-7}\ \mathrm{T}$ . Find the electric field amplitude using $E = cB$ with $c = 3.00 \times 10^8\ \mathrm{m/s}$ .
4 Explain why Maxwell added the displacement current term to Ampere's law and how it helps electromagnetic waves exist in empty space.

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