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Chapter 8: Problem 2

(a)\( Generalize the boundary conditions \)(8.6 .5)\( to \)(8.6 .8)\( to include the case where a surface charge density \)\sigma=\Delta q_{\text {tree }} / \Delta S\( and a surface current of magnitude \)K=\Delta I_{\text {fres }} / \Delta l\( exist on the boundary surface, establishing the conditions $$ \text { fi } \begin{aligned} & \cdot\left(x_{22} \mathbf{E}_{2}-x_{41} \mathbf{E}_{1}\right)=\frac{\sigma}{\epsilon_{0}} \\ \text { fi } \times\left(\frac{\mathbf{B}_{2}}{k_{m 2}}-\frac{\mathbf{B}_{1}}{k_{m 1}}\right) &=\mu_{0} \mathbf{K} . \end{aligned} $$ (b) Show that the boundary conditions remain valid when the boundary is not plane and when the respective media are not homogeneous. (c) What are the boundary conditions on the \)\mathbf{D}\( and \)\mathbf{H}$ fields?

Short Answer

Expert verified
Based on the above step-by-step solution, we can form a short answer: The boundary conditions for the electric and magnetic fields with consideration of surface charge density and surface current are given as: $$ \mathbf{f1} \cdot \left(\mathbf{E}_2 - \mathbf{E}_1\right) = \frac{\sigma}{\epsilon_{0}} $$ $$ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} $$ These boundary conditions remain valid for non-plane boundaries and non-homogeneous media. And by using the constitutive relations, we can derive the boundary conditions for D and H fields as: $$ \mathbf{f1} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma $$ $$ \mathbf{f2} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K} $$

Step by step solution

01

Rewrite the given boundary conditions

The given boundary conditions are: $$ \text { f1 } \cdot\left(x_{22} \mathbf{E}_{2}-x_{41} \mathbf{E}_{1}\right)=\frac{\sigma}{\epsilon_{0}} \\ \text { f2 } \times\left(\frac{\mathbf{B}_{2}}{k_{m 2}}-\frac{\mathbf{B}_{1}}{k_{m 1}}\right) =\mu_{0} \mathbf{K}. $$ #Step 2: Generalize the boundary conditions for surface charge density and surface current#
02

Generalize the boundary conditions

In order to include the surface charge density \(\sigma\) and surface current \(K\), we can modify the boundary conditions as follows: For electric field: $$ \mathbf{f1} \cdot (\mathbf{E}_2 - \mathbf{E}_1) = \frac{\sigma}{\epsilon_{0}} $$ For magnetic field: $$ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} $$ #Step 3: Show that the boundary conditions are valid for non-plane boundaries and non-homogeneous media#
03

Analyze for non-plane boundaries and non-homogeneous media

The above generalized boundary conditions do not depend on the assumption of a plane boundary or homogeneous media. They were derived based on Maxwell's equations and the assumption that the fields are continuous across the boundary. Therefore, it can be concluded that the boundary conditions remain valid for non-plane boundaries and when the media are not homogeneous. #Step 4: Derive boundary conditions for D and H fields#
04

Find boundary conditions for D and H fields

Using the constitutive relations, we can derive the boundary conditions for D and H fields: For D field, we use \(\mathbf{D} = \epsilon \mathbf{E}\), where \(\epsilon_1 = \epsilon_0 x_{41}\) and \(\epsilon_2 = \epsilon_0 x_{22}\). $$ \mathbf{f1} \cdot (\epsilon_2 \mathbf{E}_2 - \epsilon_1 \mathbf{E}_1) = \sigma $$ Now, using the constitutive relation for the D field, this becomes: $$ \mathbf{f1} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma $$ For the H field, we use \(\mathbf{H} = \frac{1}{\mu} \mathbf{B}\), where \(\mu_1 = \mu_0 k_{m1}\) and \(\mu_2 = \mu_0 k_{m2}\). $$ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} $$ Now, using the constitutive relation for the H field, this becomes: $$ \mathbf{f2} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K} $$ The final boundary conditions on the \(\mathbf{D}\) and \(\mathbf{H}\) fields are: $$ \mathbf{f1} \cdot (\mathbf{D}_2 - \mathbf{D}_1) = \sigma $$ $$ \mathbf{f2} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{K} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Charge Density
Surface charge density is often denoted by the symbol \( \sigma \) and represents the amount of electric charge per unit area on a surface. This concept becomes crucial in electromagnetic boundary conditions as it affects the behavior of electric fields at the boundary between two different materials.
Imagine it as how densely packed the charge is on the surface.
This impacts how the electric field (\( \mathbf{E} \)) behaves just at that boundary:
  • The boundary condition that includes surface charge density is expressed as:
\[ \mathbf{f1} \cdot (\mathbf{E}_2 - \mathbf{E}_1) = \frac{\sigma}{\epsilon_{0}} \]This shows that the difference in the normal components of the electric fields (\( \mathbf{E}_2 \) and \( \mathbf{E}_1 \)) across a boundary is proportional to the surface charge density divided by the permittivity of free space \( \epsilon_0 \).
In simpler terms, this defines how electric charge on a surface modulates the electric fields crossing that surface.
The higher the surface charge density, the larger the change in the electric field components.
Surface Current
Surface current, often noted as \( K \), represents current flowing across a given surface. It is expressed in terms of current per unit length.
This concept is key in defining how magnetic fields behave at material boundaries.
  • The boundary condition that includes surface current is expressed as:
\[ \mathbf{f2} \times \left(\frac{\mathbf{B}_2}{\mu_2} - \frac{\mathbf{B}_1}{\mu_1}\right) = \mu_{0} \mathbf{K} \]Here, this equation tells us about the change in the tangential components of the magnetic fields (\( \mathbf{B}_2 \) and \( \mathbf{B}_1 \)) across a boundary. This difference is proportional to the surface current and the permeability of free space \( \mu_0 \).
In simple words, surface current affects how magnetic fields are redirected or changed at the boundary, much like how a current-disrupting coil might alter a magnetic field's path.
Maxwell's Equations
Maxwell's equations form the foundation of classical electrodynamics, optics, and electric circuits. They describe how electric and magnetic fields interact with matter.
These equations are crucial in the generalization of boundary conditions:
  • Gauss's Law for electricity and magnetism
  • Faraday's Law of induction
  • Amp猫re's Law with Maxwell's addition
  • Gauss's Law for magnetism
Boundary conditions derived from Maxwell's equations allow us to connect the behavior of electric and magnetic fields across the different media.
The generalization to include surface charge density and current is because Maxwell's equations do not assume plane or homogeneous media.
Hence, thanks to the power of Maxwell's equations, electromagnetic fields continuously adjust even when boundary conditions are introduced by surface charge or current.
Non-homogeneous Media
Non-homogeneous media refer to materials with varying properties throughout their volume.
Unlike homogeneous media, where properties are constant, non-homogeneous materials can have varying permittivity and permeability.
So, why does it matter for boundary conditions?

The boundary conditions derived from Maxwell's equations remain valid even if the media aren't homogeneous.
  • This means they still apply across boundaries of vastly differing material compositions.
For example, imagine transitioning from air into graded materials where these properties change gradually.
Electromagnetic fields will still adhere to the boundary rules, ensuring a seamless transition that honors the material's innate properties.
This ensures accurate predictions in practical applications where materials are rarely purely homogeneous.

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Most popular questions from this chapter

Consider an electric dipole consisting of a charge \(-e\) oscillating sinusoidally in position about a stationary charge \(+e\). Show that the instantaneous total power radiated can be written in the form $$ \frac{d W}{d t}=\frac{e^{2}[a]^{2}}{6 \pi \epsilon_{0} c^{3}} $$ where \([a]=a_{0} e^{i(\omega t-k n)}\) is the instantaneous (retarded) acceleration of the moving charge Since this result does not depend upon the oscillator frequency, and since by Fourier analysis, an arbitrary motion can be described by superposing many sinusoidal motions of proper frequency, amplitude, and phase, this rate-of-radiation formula has general validity for any accelerated charge (in the nonrelativistic limit \(\diamond \ll c\) ).

Substitute (8.9.3) in (8.9.1) to find the spherical wave corresponding to an oscillating magnetic dipole (current loop) of moment \(m_{\rho} e^{j \omega t}\), namely, $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa v)} \\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$

Show that the skin depth (attenuation distance) for a high-frequency wave \(\left(\omega>\omega_{p}\right)\) is approximately $$ \delta \equiv-\frac{1}{\kappa_{i}} \approx \frac{c}{\omega_{p}}\left(\frac{2 \omega^{2}}{\nu \omega_{p}}\right)\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1 / 2} $$

Postulate wave fields of the form $$ \begin{aligned} &\mathbf{E}=\mathbf{i} f(z-c t)+\mathbf{j} g(z-c t)+\mathbf{k} h(z-c t) \\ &\mathbf{B}=\mathbf{i} q(z-c t)+\mathbf{j} r(z-c t)+\mathbf{k} s(z-c t) \end{aligned} $$ where \(f, g, h, q, r, s\) are arbitrary (nonsinusoidal) functions, independent of \(x\) and \(y .\) Show that such waves are a solution of the wave equations \((8.2 .8)\) and \((82.9)\) and that Maxwell's equations (8.2.1) to \((8.2 .4)\) require $$ \begin{aligned} &h=s=0 \\ &f=c r \\ &g=-c q \end{aligned} $$ that is, that only two of the six functions are really arbitrary.

Consider an ionized gas of uniform electron density \(n\). Regard the positive ions as a smeared-out continuous fluid which renders the gas macroscopically neutral and through which the electrons can move without friction. Now assume that, by some external means, each electron is shifted in the \(x\) direction by the displacement \(\xi=\xi(x)\), a function of its initial, unperturbed location \(x\). (a) Show from Gauss' law (8.2.1) that the resulting electric field is \(E_{x}=n e \xi / e_{0} .(b)\) Show that each electron experiences a linear (Hooke's law) restoring force such that when the external forces are removed, it oscillates about the equilibrium position \(\xi=0\) with simple harmonic motion at the angular frequency $$ \omega_{p}=\left(\frac{n e^{2}}{\epsilon_{0} m}\right)^{1 / 2} $$ which is known as the electron plasma frequency. 鈥� For fuller discussion, see J. A. Ratcliffe, "The Magneto-ionic Theory and Its Applications to the Ionosphere," Cambridge University Press, New York, 1959 ; M. A. Heald and C. B. Wharton, "Plasma Diagnostics with Microwaves," John Wiley \& Sons Inc., New York, 1965; I. P. Shkarofsky, T. W. Johnson, and M. P. Bachynski, "Particle Kinetics of Plasma," Addison-Wesley Publishing Company, Reading, Mass., \(1965 .\)
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