Maxwell’s Equations
11.9 Maxwell’s Equations. What value of A and β are required if the two fields:
E ¼ ^y120πcos 106πt βx
V=m
H ¼ ^zAπcos 106πt βx
A=m
satisfy Maxwell’s equations in a linear, isotropic, homogeneous medium with εr ¼ μr ¼ 4 and σ ¼ 0? Assume there
are no current or charge densities in space.
11.10 Dependency in Maxwell’s Equations. Show that Eq. (11.8) (∇B ¼ 0) can be derived from Eq. (11.5) and,
therefore, is not an independent equation.
11.11 Dependency in Maxwell’s Equations. Show that Eq. (11.7) (∇D ¼ ρv) can be derived from Eq. (11.6) with the
use of the continuity equation [Eq. (11.13)] and, therefore, is not an independent equation.
11.12 The Lorenz Condition (Gauge). Show that the Lorenz condition in Eq. (11.49) leads to the continuity equation.
Hint: Use the expression for electric potential due to a general volume charge distribution and the expression for the
magnetic vector potential due to a general current density in a volume.
11.13 Maxwell’s Equations. Maxwell’s equations in Eqs. (11.24) through (11.27) are equivalent to eight scalar equations.
Find these equations by writing the vector fields explicitly in Cartesian coordinates and equating components.
11.14 Maxwell’s Equations in Cylindrical Coordinates. Write Maxwell’s equations explicitly in cylindrical coordinates
by expanding the expressions in Eqs. (11.24) through (11.27).
11.15 Maxwell’s Equations. A time-dependent magnetic field is given as B ¼ ^x20e jð104tþ104zÞ ½T in a material with
properties εr ¼ 9 and μr ¼ 1. Assume there are no sources in the material. Using Maxwell’s equations:
(a) Calculate the electric field intensity in the material.
(b) Calculate the electric flux density and the magnetic field intensity in the material.
11.16 Maxwell’s Equations. A time-dependent electric field intensity is given as E ¼ ^x10πcos 106t 50z
½V=m. The
field exists in a material with properties εr ¼ 4 and μr ¼ 1. Given that J ¼ 0 and ρv ¼ 0, calculate the magnetic field
intensity and magnetic flux density in the material.
Potential Functions
11.17 Current Density as a Primary Variable in Maxwell’s Equations. Given: Maxwell’s equations in a linear, isotropic,
homogeneous medium. Assume that there are no source current densities and no charge densities anywhere in the
solution space. An induced current density Je [A/m2] exists in conducting materials. Assume the whole space is
conducting, with a very low conductivity, σ [S/m]. Rewrite Maxwell’s equations in terms of the current density
Je ¼ σE. In other words, assume you need to solve for Je directly.
11.18 Magnetic Scalar Potential. Write an equation, equivalent to Maxwell’s equations in terms of a magnetic scalar
potential in a linear, isotropic, homogeneous medium. State the conditions under which this can be done:
(a) Show that Maxwell’s equations reduce to a second-order partial differential equation. What are the assumptions
necessary for this equation to be correct?
(b) What can you say about the relation between the electric and magnetic field intensities under the given
conditions?
11.19 Magnetic Vector Potential. Given: Maxwell’s equations and the vector B ¼ ∇ A, in a linear, isotropic,
homogeneous medium. Assume that E ¼ 0 for static fields:
(a) By neglecting the displacement currents, show that Maxwell’s equations reduce to a second-order partial
differential equation in A alone.
(b) What is the electric field intensity?
(c) Show that by using the Coulomb’s gauge, the equation in (a) is a simple Poisson equation.