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Chapter 8: Problem 4

The text following (8.2.10) refers to low-frequency (or dc) laboratory measurements of \(\epsilon_{0}\) and \(\mu_{0}\). How could you determine these constants? What logical chain of definitions and calibrations would be needed?

Short Answer

Expert verified
Question: Explain the process of determining vacuum permittivity and permeability constants in a laboratory at low-frequency or direct current (dc) conditions. Answer: To determine vacuum permittivity and permeability constants in a laboratory at low-frequency or direct current (dc) conditions, you can follow these steps: 1. Understand the physical constants (\(\epsilon_{0}\) and \(\mu_{0}\)) that describe the electric and magnetic properties of free space. 2. Measure capacitance (C) using a parallel-plate capacitor and calculate it from the stored charge (Q) and applied voltage (V). 3. Calculate vacuum permittivity (\(\epsilon_{0}\)) using the formula: \(\epsilon_{0}\) = \(\frac{C \cdot d}{A}\). 4. Measure inductance (L) using a solenoid or toroidal coil and appropriate measurement techniques like a bridge circuit or an impedance analyzer. 5. Calculate vacuum permeability (\(\mu_{0}\)) using the formula: \(\mu_{0}\) = \(\frac{L \cdot l}{N^2A}\). 6. Calibrate measurement equipment and account for factors that may affect the results, such as temperature and humidity, and repeat measurements to ensure accuracy.

Step by step solution

01

Understand the physical constants

Permittivity (\(\epsilon_{0}\)) and permeability (\(\mu_{0}\)) are fundamental constants that describe the electric and magnetic properties of free space (vacuum). They are important constants in various electromagnetic phenomena and can be experimentally measured using certain electrical devices.
02

Measure capacitance

To determine the vacuum permittivity, you can use a parallel-plate capacitor to measure the capacitance (C). The capacitance is related to the vacuum permittivity through the formula: C = \(\frac{\epsilon_{0}A}{d}\) Where A is the area of the plates and d is the distance between them. You can calculate the capacitance by applying a known voltage (V) across the capacitor and measuring the stored charge (Q). The capacitance is then: C = \(\frac{Q}{V}\)
03

Calculate vacuum permittivity

With the measured capacitance (C), you can now calculate the vacuum permittivity (\(\epsilon_{0}\)) using the formula: \(\epsilon_{0}\) = \(\frac{C \cdot d}{A}\)
04

Measure inductance

To determine the vacuum permeability, you can use a solenoid or toroidal coil to measure the inductance (L). The inductance is related to the vacuum permeability through the formula: L = \(\frac{\mu_{0}N^2A}{l}\) Where N is the number of turns, A is the cross-sectional area of the coil, and l is the length of the coil. You can measure the inductance using a bridge circuit or an impedance analyzer.
05

Calculate vacuum permeability

With the measured inductance (L), you can now calculate the vacuum permeability (\(\mu_{0}\)) using the formula: \(\mu_{0}\) = \(\frac{L \cdot l}{N^2A}\)
06

Calibration and measurement accuracy

In order to obtain accurate results for the vacuum permittivity and permeability, it is essential to calibrate the measurement equipment and account for any factors that may affect the results, such as temperature, humidity, and the effect of nearby objects. Moreover, repeat measurements and cross-checking with other measurement techniques can be helpful to ensure the accuracy of the determined constants. By following this logical chain of definitions and calibrations, you can accurately determine the vacuum permittivity and permeability constants, \(\epsilon_{0}\) and \(\mu_{0}\), in a laboratory at low-frequency or direct current (dc) conditions.

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Most popular questions from this chapter

(a) From \((8.7 .10)\) and \((8.7 .13)\), show that the guided ave impedance \(\left(E_{x}{ }^{2}+E_{y}{ }^{2}\right)^{1 / 2} /\) \(\left(H_{x}{ }^{2}+H_{z}{ }^{2}\right)^{1 / 2}\) is \(Z_{\mathrm{TE}}=\frac{Z_{0}}{\left[1-\left(\lambda_{0} / \lambda_{e}\right)^{2}\right]^{1 / 2}} \quad\) TE modes \(Z_{\text {T? }}=Z_{0}\left[1-\left(\lambda_{0} / \lambda_{c}\right)^{2}\right]^{1 / 2} \quad\) TM modes, where \(Z_{0}\) is the unbounded wave impedance \((8.3 .10)\) or, more generally, \((8.3 .12) .(b)\) For the TE o dominant mode in rectangular waveguide, show that the peak potential difference between opposite points in the cross section is $$ V_{0}=\left[\int_{0}^{b} E_{y}\left(x=\frac{1}{2} a\right) d y\right]_{p e s k}=b E_{0} $$ and that the peak axial current flowing in the top wall is $$ I_{0} \equiv\left[\frac{1}{\mu_{0}} \int_{0}^{a} B_{x}(y=b) d x\right]_{\text {pak }}=\frac{2 a E_{0}}{\pi Z_{\mathrm{TE}}} $$ Since the result of Prob. 8.7.10b can be written $$ \bar{P}=\frac{a b}{4} \frac{E_{0}{ }^{2}}{Z_{\mathrm{TB}}} $$ we can define three other (mode-dependent) waveguide impedances as follows: $$ \begin{aligned} &Z_{V, I}=\frac{V_{0}}{I_{0}}=\frac{\pi}{2}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\ &Z_{P, V}=\frac{V_{0}{ }^{2}}{2 P}=2\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \\\ &Z_{P, I}=\frac{2 \bar{P}}{I_{0}{ }^{2}}=\frac{\pi^{2}}{8}\left(\frac{b}{a} Z_{\mathrm{TE}}\right) \end{aligned} $$ which differ by small numerical factors. Only systems supporting a TEM mode (e.g., Sec. 8.1), have a unique impedance.

Use Gauss' and Stokes' theorems (Appendix A) to convert Maxwell's differential equations for vacuum, \((82.1)\) to \((8.2 .4)\), to their integral form $$ \begin{aligned} &\oint_{S} \mathbf{E} \cdot d \mathbf{S}=\frac{q}{\epsilon_{0}} \\ &\oint_{L} \mathbf{E} \cdot d \mathbf{l}=-\frac{d \Phi_{m}}{d t} \\ &\oint_{S} \mathbf{B} \cdot d \mathbf{S}=0 \\ &\oint_{L} \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I+\mu_{0} \frac{d \Phi_{*}}{d t} \end{aligned} $$ † See Sec. \(5.4\) and Prob. 8.2.4. where the closed surface \(S\) contains the net charge \(q\) and the closed line (loop) \(L\) is linked by the net current \(I\), the magnetic flux \(\Phi_{m}=\int \mathbf{B} \cdot d \mathbf{S}\), and the electric flux \(\Phi_{e}=\epsilon_{0} \int \mathbf{E} \cdot d \mathbf{S}\). Note: The corresponding equations for a general electromagnetic medium are developed in Prob. \(8.6 .1 .\)

Consider a plasma of electron density \(n\) immersed in a uniform static magnetic field \(\mathbf{B}_{0 .}\) Let \(\mathbf{B}_{0}\) be in the \(z\) direction. Revise the equation of motion (8.8.2) to include the Lorentz force \(q\left(\mathbf{v} \times \mathbf{B}_{0}\right)\) on the electrons (but drop the collision term for simplicity); write out the resulting equation in cartesian components. (a) Show that plane waves propagating in the \(x\) direction, say, but with the electric field polarized parallel to \(\mathbf{B}_{0}\), are unaffected by the presence of \(\mathrm{B}_{0}\). (b) Show that circularly polarized plane waves (see Prob. \(8.3 .5\) ) can propagate in the \(z\) direction with wave numbers $$ K=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}}{\omega\left(\omega \pm \omega_{b}\right)}\right]^{1 / 2} $$ where \(\omega_{b} \equiv c B_{0} / m\) is the cyclotron frequency. (c) Show that a TM wave can propagate in the \(x\) direction with the wave magnetic field polarized parallel to \(\mathbf{B}_{\theta}\), with the wave number $$ k=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}\left(\omega^{2}-\omega_{p}^{2}\right)}{\omega^{2}\left(\omega^{2}-\omega_{p}^{2}-\omega_{b}^{2}\right)}\right]^{1 / 2} $$

Use the results of Prob. 8.2.3 to compute the Poynting vector for a coaxial transmission line. Integrate it over the annular area between conductors and show that the power carried down the line by the wave is $$ P=i^{2} Z_{0}=\frac{v^{2}}{Z_{0}}, $$ where \(i\) and \(v\) are the instantaneous current and voltage and \(Z_{0}\) is the characteristic impedance \((8.1 .9)\), that is, just the result one would expect from elementary circuit analysis.

Consider two unbounded plane waves whose vector wave numbers \(k_{1}\) and \(\kappa_{2}\left(\left.\right|_{1,2} \mid=\right.\) \(\omega / c)\) define a plane and whose electric fields are polarized normal to the plane. (a) Show that the superposition of these two plane waves is a wave traveling in the direction bisecting the angle \(\alpha\) between \({ }_{1}{ }_{1}\) and \({ }_{k}\) and that the \(\mathbf{E}\) field vanishes on a set of nodal planes spaced \(a=\) \(\lambda_{0} / 2 \sin \frac{1}{2} \alpha\) apart. \((b)\) Show that plane conducting walls can be placed at two adjacent nodal planes without violating the electromagnetic boundary conditions and likewise that a second pair of conducting walls of arbitrary separation \(b\) can be introduced to construct a rectangular waveguide of cross section \(a\) by \(b\), propagating the \(T E_{10}\) mode. Thus establish that the TE \(_{10}\) mode (more generally, the TE \(_{10}\) modes) may be interpreted as the superposition of two plane waves making the angle \(\frac{1}{2} \alpha\) with the waveguide axis and undergoing multiple reflections from the sidewalls. Note: The situation is directly analogous to that discussed in Sec. 2.4. Higherorder TE modes \((m>0)\) and TM modes may be described similarly as a superposition of four plane waves.
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