Maxwell’s Equations

Abstract

Most students in engineering and science have heard the term “Maxwell’s equations,” and some may also have heard that Maxwell’s equations “describe all electromagnetic phenomena.” However, it is not always clear what exactly do we mean by these equations. How are they any different from what we have studied in the previous chapters? We recall discussing static and time-dependent fields and, in the process, discussed many applications. To do so, we used the definitions of the curl and divergence of the electric and magnetic fields: what we called “the postulates.” Are Maxwell’s equations different? Do they add anything to the previously described phenomena? Perhaps the best question to ask is the following: Is there any other electromagnetic phenomenon that was not discussed in the previous chapters because the definitions we used were not sufficient to do so? If so, do Maxwell’s equations define these yet unknown properties of the electromagnetic field? The answer to the latter is emphatically yes.

From a long view of the history of mankind—seen from say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American civil war will pale into provincial insignificance in comparison with this important scientific event of the same decade.

Richard P. Feynman (1918–1988)

Physicist, Nobel laureate

Problems

11.1.1 Maxwell’s Equations, Displacement Current, and Continuity

• 11.1 Displacement Current Density. A magnetic flux density, B = ŷ0.1(cos100t)(cos5z) [T] exists in a linear, isotropic, homogeneous material characterized by ε and μ. Find the displacement current density in the material if there are no source charges or current densities in the material.

• 11.2 Displacement Current in Spherical Capacitor. Determine the displacement current I _d [A] which flows between two concentric, conducting spherical shells of radii a and b [m] where b > a in free space with a voltage difference V ₀sinωt [V] applied between the spheres.

• 11.3 Application: Displacement Current in Cylindrical Capacitor. A voltage source V ₀sinωt [V] is connected between two concentric conductive cylinders r = a and r = b [A], where b > a, with length L [A]. ε = ε _r ε ₀ [F/m], μ = μ ₀ [H/m], and σ = 0 for a < r < b. Neglect any end effects and find:

  • (a) The displacement current density at any point a < r < b.

  • (b) The total displacement current I _d flowing between the two cylinders.

• 11.4 Conservation of Charge and Displacement Current. Show that the displacement current in Maxwell’s second equation (Ampere’s law) is a direct consequence of the law of conservation of charge.

• 11.5 Application: Displacement and Conduction Current Densities in Lossy Capacitor. A lossy dielectric is located between two parallel plates which are connected to an AC source (Figure 11.5). Material properties of the dielectric are ε = 9ε ₀ [F/m], μ = μ ₀ [H/m], and σ = 4 S/m. The source is given as V = 1cosωt [V]. Calculate the frequency at which the magnitude of the displacement current density is equal to the magnitude of the conduction current density. Assume all material properties are independent of frequency.

  

  Figure 11.5

• 11.6 Displacement and Conduction Current Densities. A capacitor is made of two parallel plates with a dielectric between them. The relative permittivity of the dielectric is ε _r  = 4, the distance between plates is d = 1 mm, and the area of each plate is S = 100 mm². Because of an accident, the dielectric became wet with a salt solution and therefore became conducting with conductivity σ = 10⁻³ S/m. If the capacitor is connected to an AC source of amplitude V [V] and frequency ω [rad/s], show that:

  • (a) The ratio between the amplitudes of the conduction current density and displacement current density is

    $$ \frac{J_{cond}}{J_{disp}}=\frac{\sigma }{\omega \varepsilon }. $$

  • (b) The conduction and displacement current densities are 90° out of phase.

• 11.7 Application: Lossy Capacitor. The capacitor in Problem 11.6 is connected to a 12 V DC source and charged for a long period of time. Now the source is disconnected. Find the time constant of discharge of the capacitor.

• 11.8. Displacement Current. An AC generator operating at a frequency of 1 GHz is connected with a wire to a small conducting sphere of radius a = 10 mm at some distance away (see Figure 11.6 ). If the sphere is in free space, calculate the current in the wire. Neglect any effect the ground may have. The generator generates a sinusoidal voltage of amplitude 100 V.

  

  Figure 11.6

11.1.2 Maxwell’s Equations

• 11.9 Maxwell’s Equations. What value of A and β are required if the two fields:

  $$ \begin{array}{l}\mathbf{E}=\widehat{\mathbf{y}}120\pi \cos \left({10}^6\pi t-\beta x\right)\kern1em \Big[\mathrm{V}/\mathrm{m}\Big]\\ {}\mathbf{H}=\widehat{\mathbf{z}} A\pi \cos \left({10}^6\pi t-\beta x\right)\kern1em \Big[\mathrm{A}/\mathrm{m}\Big]\end{array} $$

  satisfy Maxwell’s equations in a linear, isotropic, homogeneous medium with ε _r  = μ _r  = 4 and σ = 0? Assume there are no current or charge densities in space.

• 11.10 Dependency in Maxwell’s Equations. Show that Eq. ( 11.8 ) (∇ ⋅ B = 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 = ρ _v ) 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 $\mathbf{B}=\widehat{\mathbf{x}}20{e}^{j\left({10}^4 t+{10}^{-4} z\right)}\;\left[\mathrm{T}\right]$ in a material with properties ε _r  = 9 and μ _r  = 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 $\mathbf{E}=\widehat{\mathbf{x}}10\pi \cos \left({10}^6 t-50 z\right)\ \left[\mathrm{V}/\mathrm{m}\right]$. The field exists in a material with properties ε _r  = 4 and μ _r  = 1. Given that J = 0 and ρ _v  = 0, calculate the magnetic field intensity and magnetic flux density in the material.

11.1.3 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 J _e [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 J _e  = σ E. In other words, assume you need to solve for J _e 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 = ∇ × A, in a linear, isotropic, homogeneous medium. Assume that E = 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.

• 11.20 An Electric Vector Potential. A vector potential may be derived as ∇ × F = –D where D is the electric flux density:

  • (a) What must be the static magnetic field intensity (other than H = 0 or H = C, where C is a constant vector) if we know that in the static case, ∇ × H = 0?

  • (b) Find a representation of Maxwell’s equations in terms of the vector potential F in a current-free region (i.e., a region without source currents).

  • (c) What might the divergence of F be for the representation in (b) to be useful? Explain.

• 11.21 Modified Magnetic Vector Potential. A modified vector potential may be defined as F = A + ∇ψ, where A is the magnetic vector potential as defined in Eq. ( 11.40 ) and ψ is any scalar function:

  • (a) Show that this is a correct definition of the vector potential.

  • (b) Find an expression of Maxwell’s equations in terms of F alone.

  • (c) How would you name the two potentials F and ψ?

11.1.4 Interface Conditions for General Fields

• 11.22 Displacement Current Density in a Dielectric. A time-dependent electric field intensity is applied on a dielectric as shown in Figure 11.7. The electric field intensity in free space is given as $\mathbf{E}=\widehat{\mathbf{z}}{E}_0 \cos \omega t\kern1em \left[\mathrm{V}/\mathrm{m}\right]$. The relative permittivity of the material is ε _r  = 25. For E ₀ = 100 V/m and ω = 10⁹ rad/s, calculate the peak displacement current density in the dielectric (there are no surface charges at the interface between air and material).

  

  Figure 11.7

• 11.23 The Hertz Potential. In a linear, isotropic, homogeneous medium devoid of sources, one can derive the fields from a single potential called the Hertz potential, π, as follows:

  $$ \mathbf{A}= j\omega \mu \varepsilon \boldsymbol{\uppi}, \kern1.5em V=-\nabla \cdot \boldsymbol{\uppi} $$

  where A is the magnetic vector potential and V the electric scalar potential.

  Find the expressions for the electric and magnetic field intensities to show that they are dependent on π alone.

• 11.24 The Use of a Gauge. In a linear, isotropic, homogeneous medium devoid of sources, one can define the magnetic Hertz potential π _m as

  $$ \mathbf{E}=-j\omega \mu \nabla \times {\boldsymbol{\uppi}}_m\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  Show that one can write Maxwell’s equations in the frequency domain in terms of π _m alone provided a proper gauge is defined. What is that gauge?

• 11.25 Interface Conditions for General Materials. Two dielectrics meet at an interface (see Figure 11.8 ) at x = 0. A sinusoidal electric field intensity of peak value 5 V/m and frequency 1 kHz exists in dielectric (1). For x < 0, ε = 2ε ₀ [F/m], and μ = μ ₀ [H/m]. For x > 0, ε = 3ε ₀, and μ = 2μ ₀. If the electric field intensity vector is incident at 30^◦ from the normal, find the magnitudes of E and D on each side of the interface. Assume no current or charge densities exist at the interface.

  

  Figure 11.8

• 11.26 Calculation of Fields Across Interfaces. A region, denoted as region (1), occupies the space x < 0 and has relative permeability μ _r1 = 6. The magnetic field intensity in region (1) is ${\mathbf{H}}_1=\widehat{\mathbf{x}}4+\widehat{\mathbf{y}}-\widehat{\mathbf{z}}2$ [A/m]. Region (2) is defined as x > 0 with μ _r2 = 5.0. No current exists at the interface. Find B in region (2).

• 11.27 Interface Conditions for Permeable Materials. An interface between free space and a perfectly permeable material exists. In free space (1), μ = μ ₀ [H/m], ε = ε ₀ [F/m], and σ =0. In the permeable material (2), μ = ∞, σ = 0, and ε = ε ₀. Define the interface conditions at the interface between the two materials.

• 11.28 Surface Current Density at Interfaces. Two magnetic materials meet at an interface as shown in Figure 11.9. Material (1) has relative permeability of 4 and material (2) has relative permeability of 2. The interface is at z = 0. The magnetic flux density in material (1) is given as $\mathbf{B}=\widehat{\mathbf{x}}0.1+\widehat{\mathbf{y}}0.2+\widehat{\mathbf{z}}0.1\;\left[\mathrm{T}\right]$. In material (2), it is known that all tangential components of H are zero.

  • (a) Calculate the surface current density that must exist on the interface for this condition to be satisfied.

  • (b) Calculate the magnetic flux density in material (2).

    

    Figure 11.9

• 11.29 Simulated Surface Current Density. It is possible to simulate a current sheet at an interface by placing thin parallel wires at the interface. Suppose two materials meet at an interface on the x–y plane at z = 0. Both materials are the same, with relative permeability μ _r  = 2. The magnetic field intensity in material (1) (z > 0) is given as ${\mathbf{H}}_1=\widehat{\mathbf{x}}{10}^5+\widehat{\mathbf{y}}2\times {10}^5+\widehat{\mathbf{z}}{10}^4$ [A/m]. Suppose now that wires are placed on the interface such that the current in the wire points at 45° to the x axis, as shown in Figure 11.10. The current in each wire is 0.1 A and there are two wires per mm length. Calculate:

  • (a) The magnetic field intensity in material (1) and in material (2) before the current in the wires is added.

  • (b) The magnetic field intensity in both materials after the current is switched on.

  

  Figure 11.10

• 11.30 Simulated Surface Current Density. Suppose that in the previous problem, the magnetic field intensity in material (1) with the same current sheet (i.e., the total field in material (1) due to all sources, including the current sheet) is given as ${\mathbf{H}}_1=\widehat{\mathbf{x}}{10}^5+\widehat{\mathbf{y}}2\times {10}^5+\widehat{\mathbf{z}}{10}^4$ [A/m]:

  • (a) What is now the magnetic field intensity in material (2).

  • (b) Discuss the difference between the solution to this problem and the previous problem.

11.1.5 Time-Harmonic Equations/Phasors

• 11.31 Vector Operations on Phasors. Two complex vectors are given as A = a + j b and B = c + j d, where a, b, c, and d are real vectors. Calculate (* indicates complex conjugate):

  $$ \begin{array}{cccc}\hfill \mathbf{A} \cdot \mathbf{A}\hfill & \hfill \mathbf{A} \cdot {\mathbf{A}}^{*}\hfill & \hfill \mathbf{A} \cdot \mathbf{B}\hfill & \hfill \mathbf{A} \cdot {\mathbf{B}}^{*}\hfill \\ {}\hfill \mathbf{A}\times \mathbf{A}\hfill & \hfill \mathbf{A}\times {\mathbf{A}}^{*}\hfill & \hfill \mathbf{A}\times \mathbf{B}\hfill & \hfill \mathbf{A}\times {\mathbf{B}}^{*}\hfill \end{array} $$

• 11.32 Conversion of Phasors to the Time Domain. A magnetic field intensity is given as $\mathbf{H}=\widehat{\mathbf{y}}5{e}^{-j\beta z}$ [A/m]. Write the time-dependent magnetic field intensity.

• 11.33 Conversion to Phasors. The following magnetic field intensity is given in a domain 0 ≤ x ≤ a, 0 ≤ y ≤ b:

  $$ H\left(x,y,z,t\right)={H}_0\kern-0.15em \sin \frac{m\pi x}{a} \cos \frac{n\pi y}{b} \cos \left(\omega t-kz\right)\kern1em \left[\mathrm{A}/\mathrm{m}\right] $$

  where x, y, and z are the space variables, m and n are integers, and k is a constant. Find the rectangular, polar, and exponential phasor representations of the field.

• 11.34 Conversion to Phasors. An electric field intensity is given as

  $$ E\left( z, t\right)={E}_1 \cos \left(\omega t- kz+\psi \right)+{E}_2 \cos \left(\omega t+ kz+\psi \right)\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  Write the phasor form of E in polar and exponential forms.

• 11.35 Conversion of Phasors to the Time Domain. A phasor is given as

  $$ E\left(x,z\right)={E}_0{e}^{-j{\beta}_0\left(x\kern-0.1em \sin \kern-0.11em {\theta}_i+z\kern-0.1em \cos \kern-0.11em {\theta}_i\right)}\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  where x and z are variables and β ₀ and θ _i are constants. Find the time-dependent form of the field E.

• 11.36 Conversion to Phasors. The electric field intensity in a domain is given as

  $$ {E}_x\left( z, t\right)={E}_0 \cos \left(\omega t- kz+\phi \right)\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  Find:

  • (a) The phasor representation of the field in exponential form.

  • (b) The first-order time derivative of the phasor.

• 11.37 Time-Harmonic Fields. The electric field intensity

  $$ \mathbf{E}=\widehat{\mathbf{x}}10\pi \cos \left({10}^6 t-50 z\right)+\widehat{\mathbf{y}}10\pi \cos \left({10}^6 t-50 z\right)\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  is given in a linear, isotropic, homogeneous medium of permeability μ ₀ [H/m] and permittivity ε ₀ [F/m]. Write the magnetic field intensity and the magnetic flux density:

  • (a) In terms of the time-dependent electric field intensity.

  • (b) In terms of the time-harmonic electric field intensity.

• 11.38 Time-Harmonic Fields. The magnetic field intensity in free space is given as $\mathbf{H}=\left(\widehat{\mathbf{x}}{H}_x+\widehat{\mathbf{y}}{H}_y+\widehat{\mathbf{z}}{H}_z\right){e}^{j\beta z}{e}^{j\phi }$ [A/m], where H _x , H _y and H _z are complex numbers given as H _x  = h _x  + jg _x , H _y  = h _y  + jg _y and H _z  = h _z  + jg _z :

  • (a) What is the time-dependent magnetic field intensity H in air?

  • (b) Write the magnetic field intensity in terms of amplitude and phase.

• 11.39 Two vector fields are given in phasor form as

  $$ {\mathbf{E}}_1=\widehat{\mathbf{x}}\left(20+ j20\right){e}^{j0.3\pi z}+\widehat{\mathbf{y}}\left(10- j20\right){e}^{j0.3\pi z},\kern1em {\mathbf{E}}_2=-\widehat{\mathbf{x}}\left(20- j10\right){e}^{j0.3\pi z}+\widehat{\mathbf{y}}\left(20+ j20\right){e}^{j0.3\pi z}\kern1em \left[\mathrm{V}/\mathrm{m}\right] $$

  Calculate:

  • (a) The time domain representation of the two fields.

  • (b) The sum E ₁ + E ₂ in phasor form and in the time domain.

  • (c) The difference E ₁ – E ₂ in phasor form and in the time domain.

  • (d) The vector product of the two fields in the time domain and in phasor form.

  • (e) The scalar product of the two fields in the time domain and in phasor form.