Electrodynamics
Maxwell Equations
Learning Objectives:
- Define Maxwell's equations and explain their significance in understanding electromagnetism.
- Apply vector calculus to solve problems related to Maxwell's equations, including gradient, divergence, and curl operations.
- Discuss the practical applications of Maxwell's equations in electrical and computer engineering, including the design of antennas, communication systems, and electromagnetic interference (EMI) mitigation.
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.
Euisudyi, CC BY-SA 4.0, via Wikimedia Commons
Solved Example: 9993-01
The Maxwell’s equation, \[\nabla \times \overline{H} = \overline{J} + \dfrac{\partial \overline{D}}{\partial t}\] is based on:
A. Ampere’s law
B. Gauss’s law
C. Faraday’s law
D. Coulomb’s law
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)
Correct Answer: A
Solved Example: 9993-02
Maxwell's equation in ______ form gives information at points of discontinuity is electromagnetic field.
A. Differential
B. Integral
C. Algebraic
D. None of these
Correct Answer: B
Solved Example: 9993-03
The Maxwell's equations are written as given below. Select the erroneous (INCORRECT) equation.
A. $\nabla .D = \rho$
B. $\nabla .B = \mu$
C. $\nabla \times H = J_c$
D. $\nabla \times E = - \dfrac{{\partial B}}{{\partial t}}$
Correct Answer: B
Wave Propagation
- Electrical energy that has escaped into free space
- Travel in a straight line at approximately the speed of light and are made up of magnetic and electric fields that are right angles to each other and at right angles to the direction of propagation
- Essential properties: Frequency, Intensity, Direction of Travel, Plane of Polarization
- A form of electromagnetic radiation similar to light and heat
- Differ from other radiations in the manner in which they area generated and detected and in frequency range Consists of traveling electric and magnetic fields with the energy evenly divided between two types of fields
Becarlson, CC BY-SA 4.0, via Wikimedia Commons
Solved Example: 9911-01
The ratio of the amplitudes of the electric field and magnetic field strengths has the same dimensions as that of:
A. Permittivity
B. Inductance
C. Capacitance
D. Impedance
Correct Answer: D
Solved Example: 9911-02
The wave length ($\lambda$) in meters of an electromagnetic wave is related to its frequency (f) in MHz as:
A. $\lambda = \dfrac{3\times10^8}{f}$
B. $\lambda = \dfrac{3\times10^{10}}{f}$
C. $\lambda = \dfrac{300}{f}$
D. None of the above
Correct Answer: C
Solved Example: 9911-03
The electric field intensity E and magnetic field intensity H are coupled and propagating in free space in x and y direction respectively, the Poynting vector is given by:
A. $EH \hat {a_x}$
B. $EH \hat {a_y}$
C. $EH \hat {a_x}\hat {a_y}$
D. None of the above
Correct Answer: D
Solved Example: 9911-04
The group velocity of matter waves associated with a moving particle is:
A. The same as phase velocity
B. Less than the particle velocity
C. Equal to the particle velocity
D. More than the particle velocity
Correct Answer: C
Solved Example: 9911-05
In electromagnetic spectrum visible light lies in between:
A. X-rays and UV
B. Infrared and microwave
C. Microwaves and radio waves
D. UV and infrared
Correct Answer: D
Solved Example: 9911-06
A long cylindrical wire of radius r and length l is carrying a current of magnitude i. When the ends are across potential difference V, the pointing vector on the surface of the wire will be:
A. $\dfrac {Vi}{2\pi r l}$
B. $\dfrac {Vi}{\pi r^2 l}$
C. $\dfrac {Vi}{2\pi r^3 + 2 \pi rl}$
D. $\dfrac {Vi}{2\pi r^2 l}$
Correct Answer: A
Solved Example: 9911-07
Brewster angle is the angle when a wave is incident on the surface of a perfect dielectric at which there is no reflected wave and the incident wave is:
A. Parallely polarized
B. Perpendicularly polarized
C. Normally polarized
D. None of the above
Correct Answer: A
Solved Example: 9911-08
The uniform plane wave is:
A. X-directed
B. Neither longitudinal nor transverse
C. Longitudinal in nature
D. Transverse in nature
Correct Answer: D
Solved Example: 9911-09
The plane wave propagating through the dielectric has the magnetic field component as $H = 20 e^{-ax} \cos (\omega t - 0.25x) a_y A/m$ ($a_x$, $a_y$, $a_z$ are unit vectors along x, y, and z-axis respectively). Determine the Polarization of the wave.
A. $a_x$
B. $-a_z$
C. $\dfrac{{\left( {{a_x} + {a_y}} \right)}}{{\sqrt 2 }}$
D. $a_y$
Correct Answer: B