Maxwell's Equations

The Four Maxwell's Equations

1. Gauss's Law (Electric)

Integral form: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$

Differential form: $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$

Electric field diverges from charges.

2. Gauss's Law for Magnetism

Integral form: $\oint \vec{B} \cdot d\vec{A} = 0$

Differential form: $\nabla \cdot \vec{B} = 0$

No magnetic monopoles; magnetic field lines are closed loops.

3. Faraday's Law

Integral form: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

Differential form: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

Changing magnetic field creates electric field (circulation).

4. Ampere-Maxwell Law

Integral form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0\epsilon_0\frac{d\Phi_E}{dt}$

Differential form: $\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$

Current and changing electric field create magnetic field (circulation).

Displacement Current

Maxwell's addition to Ampere's law:

$I_d = \epsilon_0\frac{d\Phi_E}{dt}$

Displacement current density: $\vec{J}_d = \epsilon_0\frac{\partial \vec{E}}{\partial t}$

Why needed:

In charging capacitor, no conduction current between plates, but changing $\vec{E}$ creates magnetic field as if current flowed.

Ensures $\nabla \cdot \vec{J} + \frac{\partial \rho}{\partial t} = 0$ (charge conservation).

Symmetry in Maxwell's Equations

In vacuum ($\rho = 0$, $\vec{J} = 0$):

$\nabla \cdot \vec{E} = 0, \quad \nabla \cdot \vec{B} = 0$

$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla \times \vec{B} = \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$

Nearly symmetric! Differences:

• No magnetic monopoles (no $\rho_m$)
• Factor $\mu_0\epsilon_0$ in Ampere-Maxwell

Poynting Vector

Energy flux in electromagnetic field:

$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$

Units: W/m² (power per area)

Direction: Direction of energy propagation

Intensity: $I = |\vec{S}|_{avg}$

Energy Density

Electric field: $u_E = \frac{1}{2}\epsilon_0 E^2$

Magnetic field: $u_B = \frac{1}{2\mu_0}B^2$

Total: $u = u_E + u_B = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$

Continuity Equation

Energy conservation:

$\frac{\partial u}{\partial t} + \nabla \cdot \vec{S} = -\vec{J} \cdot \vec{E}$

Rate of energy change + energy flux out = work done by field on charges.

Historical Significance

Maxwell's equations unified electricity and magnetism, predicted electromagnetic waves, led to:

• Radio
• Radar
• Modern telecommunications
• Understanding of light as electromagnetic wave