Maxwell's Equations | Physics - Wyatt's Notes

Maxwell’s equations are the foundation of classical electromagnetism. In SI units:

Integral Form:

\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\mathrm{enc}}{\varepsilon_0} \quad \mathrm{(Gauss's\ Law)}}

$\oint_S \mathbf{B} \cdot d\mathbf{A} = 0 \quad \mathrm{(Gauss's\ Law\ for\ Magnetism)}$

$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \quad \mathrm{(Faraday's\ Law)}$

$\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\mathrm{enc} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt} \quad \mathrm{(Ampere{-}Maxwell\ Law)}}$

Differential Form:

$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \quad \mathrm{(Gauss's\ Law)}$

$\nabla \cdot \mathbf{B} = 0 \quad \mathrm{(Gauss's\ Law\ for\ Magnetism)}$

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \mathrm{(Faraday's\ Law)}$

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \quad \mathrm{(Ampere{-}Maxwell\ Law)}$

Where $\rho$ is the charge density, $\mathbf{J}$ is the current density, $\varepsilon_0$ is the permittivity Of free space, and $\mu_0$ is the permeability of free space.

Gauss’s Law. Apply the divergence theorem to the integral form:

$\oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{E})\, dV = \frac{1}{\varepsilon_0}\int_V \rho\, dV$

Since this holds for any volume $V$: $\nabla \cdot \mathbf{E} = \rho / \varepsilon_0$.

Faraday’s Law. Apply Stokes’ theorem:

$\oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{A} = -\int_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}$

Since this holds for any surface $S$: $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$.

Gauss’s Law for Magnetism. By the divergence theorem:

$\oint_S \mathbf{B} \cdot d\mathbf{A} = \int_V (\nabla \cdot \mathbf{B})\, dV = 0$

Since $V$ is arbitrary: $\nabla \cdot \mathbf{B} = 0$. This expresses the absence of magnetic monopoles.

Ampere-Maxwell Law. Apply Stokes’ theorem:

$\oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{B}) \cdot d\mathbf{A} = \mu_0 \int_S \mathbf{J} \cdot d\mathbf{A} + \mu_0 \varepsilon_0 \frac{d}{dt}\int_S \mathbf{E} \cdot d\mathbf{A}$

Since $S$ is arbitrary: $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0\, \partial \mathbf{E}/\partial t$.

Taking the divergence of the Ampere-Maxwell law:

$\nabla \cdot (\nabla \times \mathbf{B}) = 0 = \mu_0 \nabla \cdot \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial}{\partial t}(\nabla \cdot \mathbf{E})$

Using Gauss’s law: $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$.

This is the continuity equation, expressing conservation of charge.

At an interface between two linear media (labelled 1 and 2) with surface normal $\hat{\mathbf{n}}$ Pointing from 2 into 1, Maxwell’s equations impose four boundary conditions.

Normal component of $\mathbf{D}$. Apply Gauss’s law for $\mathbf{D}$ to a thin pillbox Straddling the interface:

$\oint \mathbf{D} \cdot d\mathbf{A} = \sigma_f A \implies D_{1n} - D_{2n} = \sigma_f$

Tangential component of $\mathbf{E}$. Apply Faraday’s law to a rectangular loop Perpendicular to the interface. As the loop height $\Delta h \to 0$The flux through the Loop vanishes:

$\oint \mathbf{E} \cdot d\mathbf{l} = 0 \implies E_{1t} = E_{2t}$

In vector form: $\hat{\mathbf{n}} \times (\mathbf{E}_1 - \mathbf{E}_2) = \mathbf{0}$.

Normal component of $\mathbf{B}$. Apply Gauss’s law for $\mathbf{B}$ to a pillbox:

$B_{1n} = B_{2n}$

Tangential component of $\mathbf{H}$. Apply Ampere’s law for $\mathbf{H}$ to a loop Perpendicular to the interface:

$\hat{\mathbf{n}} \times (\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{K}_f$

Where $\mathbf{K}_f$ is the free surface current density.

Summary (no free charges or currents, $\sigma_f = 0$, $\mathbf{K}_f = \mathbf{0}$):

The electric field is a force landscape: at every point in space, it assigns a vector representing the force that a positive test charge would experience at that location. Near a positive charge, the field points outward — the “hill” slopes away from the source. Near a negative charge, the field points inward — the “valley” slopes toward the source. The field lines are the contour lines of this landscape, and their density indicates the strength of the force.

Gauss’s law says that the total “outflow” of the electric field through any closed surface equals the enclosed charge divided by the permittivity of free space. Physically, charge is a source (or sink) of field lines. Faraday’s law says that a changing magnetic field creates a circulating electric field — the force landscape twists and swirls when the magnetic environment changes. The beauty of Maxwell’s equations is that they unify electricity and magnetism into a single field description: changing electric fields create magnetic fields (Ampere-Maxwell law) and changing magnetic fields create electric fields (Faraday’s law), allowing electromagnetic waves to propagate through empty space as self-sustaining oscillations of the field landscape.

Problem. Starting from Maxwell’s equations in free space ($\rho = 0$, $\mathbf{J} = \mathbf{0}$), Derive the wave equations for $\mathbf{E}$ and $\mathbf{B}$ And show that the wave speed is $c = 1/\sqrt{\mu_0 \varepsilon_0}$.

Problem. An infinitely long line charge has linear charge density $\lambda$. Use Gauss’s law to find the electric field at a distance $r$ from the line.

Problem. A circular loop of radius 0.1 m is placed in a magnetic field that varies as $B(t) = B_0 \sin(\omega t)$ where $B_0 = 0.5$ T and $\omega = 100$ rad/s. Find the induced EMF in the loop.

Problem. A solenoid has $n = 1000$ turns per meter and carries a current $I = 2$ A. Find the magnetic field inside the solenoid.

Mistake 1: Confusing the sources of electric and magnetic fields Electric fields are produced by electric charges ($\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$), while magnetic fields are produced by currents and changing electric fields ($\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \partial\mathbf{E}/\partial t$). There are no magnetic monopoles ($\nabla \cdot \mathbf{B} = 0$). Students sometimes assume magnetic fields are produced by magnetic charges analogous to electric charges.

Mistake 2: Forgetting the displacement current term in Ampere’s law The original Ampere’s law $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ is inconsistent with the continuity equation. Maxwell’s addition of the displacement current $\mu_0\varepsilon_0 \partial\mathbf{E}/\partial t$ fixes this and predicts electromagnetic waves. Omitting this term leads to incorrect predictions for time-varying fields, such as the charging of a capacitor.

Mistake 3: Confusing integral and differential forms The integral form of Gauss’s law $\oint \mathbf{E} \cdot d\mathbf{A} = Q_{\text{enc}}/\varepsilon_0$ applies to specific symmetric configurations, while the differential form $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$ is the general statement. Students often apply the integral form without verifying that the symmetry assumptions (spherical, cylindrical, or planar) are satisfied.

flowchart TD
    A[1_Maxwell S Equations] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• Magnetostatics: Magnetostatics is the static limit of Maxwell’s equations where time derivatives vanish, describing steady currents and magnetic fields.

• Electrodynamics: Electrodynamics extends Maxwell’s equations to time-varying fields, with Faraday’s law and the displacement current.

• The Wave Equation: Electromagnetic waves are solutions to Maxwell’s equations in free space, with speed $c = 1/\sqrt{\mu_0 \varepsilon_0}$.

• Calculus

• Linear Algebra

• Vector Calculus