### Maxwell Equations

Maxwell’s equations describe all (classical) electromagnetic phenomena:

$\nabla \times E =−\dfrac{\partial B}{\partial t}$ $\nabla \times H = J + \dfrac{\partial D}{\partial t}$ $\nabla \cdot D = \rho$ $\nabla \cdot B = 0$

The first is Faraday’s law of induction, the second is Ampere’s law as amended by Maxwell to include the displacement current $\dfrac{\partial D}{\partial t}$, the third and fourth are Gauss’ laws for the electric and magnetic fields.
The displacement current term $\dfrac{\partial D}{\partial t}$ in Ampere’s law is essential in predicting the existence of propagating electromagnetic waves. Its role in establishing charge conservation.
Maxwell equations are in SI units. The quantities E and H are the electric and magnetic field intensities and are measured in units of [volt/m] and [ampere/m], respectively.
The quantities D and B are the electric and magnetic flux densities and are in units of [coulomb/m$^2$] and [Weber/m$^2$], or [tesla].D is also called the electric displacement, and B, the magnetic induction. The quantities ρ and J are the volume charge density and electric current density (Charge flux) of any external charges (that is, not including any induced polarization charges and currents.) They are measured in units of [coulomb/m$^3$] and [ampere/m$^2$].
The right-hand side of the fourth equation is zero because there are no magnetic mono-pole charges. The charge and current densities $\rho$,J may be thought of as the sources of the electro-magnetic fields. For wave propagation problems, these densities are localized in space; for example, they are restricted to flow on an antenna. The generated electric and magnetic fields are radiated away from these sources and can propagate to large distances to the receiving antennas.

Maxwell’s equations take the simpler form: $\nabla \times E =− \dfrac{\partial B}{\partial t}$ $\nabla \times H = \dfrac{\partial D}{\partial t}$ $\nabla \cdot D = \rho$ $\nabla \cdot B = 0$

For example, a time-varying current J on a linear antenna generates a circulating and time-varying magnetic field H, which through Faraday’s law generates a circulating electric field E, which through Ampere’s law generates a magnetic field, and so on. The cross-linked electric and magnetic fields propagate away from the current source.

### Solved Example:

#### 9993-01

The Maxwell’s equation, $\nabla \times \overline{H} = \overline{J} + \dfrac{\partial \overline{D}}{\partial t}$ is based on: (GATE ECE 1998)

Solution:

Ampere’s law states that the magneto motive force around a closed path is equal to the current enclosed by the path,

For steady electric fields, $\oint \overline{H} \cdot \overline{dl} = I = \int \bar{J} \cdot \bar{da}$

$\overline{J} = \bar{\sigma E}$ is the conduction Current density (amp/m$^2$)

For time – varying electric fields:

$\oint \overline{H} \cdot \overline{dl} = \int ({\overline{J} + \overline{J_d}}) \cdot \overline{d_a}$

Where $\overline{J_d}$ is the displacement current density $\dfrac{\partial \overline{D}}{\partial t}$

By Stroke’s theorem $\oint \overline{H} \cdot \overline{dl} = \iint (\nabla \times \overline{H}) \cdot \overline{da}$

So $\nabla \times \overline{H} = \overline{J} + \dfrac{\partial \overline{D}}{\partial t}$

$\overline{J} + \dfrac{\partial \overline{D}}{\partial t}$ is the total current density (Conduction current density + displacement current density)

### Solved Example:

#### 9993-02

Maxwell's equation in ______ form gives information at points of discontinuity is electromagnetic field. (UPPSC Polytechnic Lecturer Electrical Nov 2021 Paper I)