Electrodynamics
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)
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. (UPPSC Polytechnic Lecturer Electrical Nov 2021 Paper I)
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. (UPRVUNL AE EE 2016)
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
Solved Example: 9911-01
The ratio of the amplitudes of the electric field and magnetic field strengths has the same dimensions as that of: (AAI ATC Junior Executive 2018 Shift I)
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: (SDSC (ISRO) Tech Asst Electronics April 2018)
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: (ISRO Scientist ECE 2013)
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: (AAI ATC Junior Executive 2018 Shift I)
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: (VSSC (ISRO) Technician B: Electronic Mechanic Dec 2017)
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: (DFCCIL Jr. Executive Electrical Sep 2021)
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: (ESE Electrical 2015 Paper I)
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: (UPPCL AE EC Nov 2019)
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. (ISRO Scientist ECE: 2020)
A. $a_x$
B. $-a_z$
C. $\dfrac{{\left( {{a_x} + {a_y}} \right)}}{{\sqrt 2 }}$
D. $a_y$
Correct Answer: B