13: Problem and Answers
- Page ID
- 25055
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 13.1: Chapter 1
- This page covers multiple topics in electric and magnetic fields. It includes calculations of electric fields from stationary and moving charges, the interactions of magnetic dipoles, and the effects of oscillating charges on generated fields. The discussions also explore gradient and Laplacian calculations in various coordinate systems, current configurations' magnetic fields, and evaluations of line integrals of vector functions.
- 13.2: Chapter 2
- This page covers electric fields, potentials, and charge distributions through various problems involving both static and polarized systems. It examines uniform charge distributions, applies Gauss's law to free charge distributions, and explores bound charge densities for polarized materials like discs and cylinders. Key calculations include electric fields and potentials derived from surface charge densities, effects of polarization, and interaction with conductive surfaces.
- 13.3: Chapter 3
- This page covers the calculations and principles of capacitors and dielectric materials, exploring concepts such as effective dipole moments, electric potentials around dipoles, and forces within capacitors. Specific problems related to capacitance involving mylar and oil are presented, highlighting the effects of dielectric media on energy dynamics and charge interactions.
- 13.4: Chapter 4
- This page covers various topics related to magnetic fields and their calculations in different geometries and scenarios. It details the magnetic flux density in coaxial cables and coils, solenoid and loop problems, and the importance of symmetry for uniform fields. Additionally, it explores vector potentials, magnetization in materials, and effective current densities.
- 13.5: Chapter 5
- This page covers various aspects of magnetic phenomena, including the behavior of magnetic fields in solenoids, permeable materials, and superconductors. It details calculations of induced electromotive force, repulsive forces on magnetic dipoles, and field distributions in cylindrical shields. The influence of magnetic fields on charged particle motion and the derivation of magnetic fields at the nucleus are discussed, with formulas for magnetic energy and current density.
- 13.6: Chapter 6
- This page covers electromagnetism problems involving field generation with soft iron and permanent magnets. It includes calculations for the current needed to achieve a 1.0 Tesla magnetic field (10.0 Amps), the magnetic field at a gap's midpoint from permanent magnets (0.558 Teslas), and the magnetic fields between magnetized areas on a hard disk, detailing the strength at various points.
- 13.7: Chapter 7
- This page covers the physics of oscillating electric dipoles and small current loops, detailing Hertz's experiments on electric and magnetic fields, with a focus on radiation fields at various distances. It includes derivations of vector potentials for circular current loops and explores fields from oscillating magnetic dipoles, including practical applications.
- 13.8: Chapter 8
- This page covers various aspects of energy flow and electromagnetic field interactions in electrical systems. Key topics include energy radiation rates related to steady currents in wires and oscillating dipole antennas, interference of electric fields from dipoles, and vector potentials derived from time-varying polarization. The text emphasizes calculations for electric and magnetic fields, the role of the Poynting vector, and the radiation resistance of antennas.
- 13.9: Chapter 9
- This page covers various aspects of electromagnetic waves, including properties of plane waves, interactions with optical devices like wave plates, and radiation scattering from atomic configurations. Key discussions include calculating frequencies, magnetic fields, and energy transport rates for plane waves; the relationship between electric and magnetic fields via the Poynting vector; the behavior of polarized light; and the structure factor's role in scattering intensity patterns.
- 13.10: Chapter- 10
- This page covers the interaction of electromagnetic waves with different media, such as metals and dielectrics, detailing Maxwell's equations and Stokes' theorem application. Topics include wave behavior at interfaces, continuity of electric and magnetic fields, calculation of amplitudes for reflected and transmitted waves, energy transport, and absorption rates.
- 13.11: Chapter- 11
- This page covers various problems related to transmission line theory, focusing on strip-line and co-axial cables. Topics include velocity of electromagnetic waves, characteristic impedance, energy calculations, and pulse reflections under different load conditions. Key aspects involve analyzing impedance seen from a generator connected to co-axial cables of varying lengths, calculating voltage standing wave ratios (VSWR), and understanding load impedance effects.
- 13.12: Chapter- 12
- This page addresses microwave power transmission in rectangular waveguides, emphasizing the TE10 mode at 24 GHz and its electric and magnetic field expressions. It explores properties of elliptically polarized waves, attenuation, and energy flow via the Poynting vector. The behavior of a waveguide filled with a dielectric constant εr=9.00 is analyzed, including calculations for a 1 Watt signal.