10: Plane Waves II
- Page ID
- 22726
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An investigation of the behaviour of plane waves incident on a plane interface between two media having different optical properties.
- 10.1: Normal Incidence
- This page examines the interaction of plane waves at a vacuum-isotropic material interface, emphasizing the impact of a frequency-dependent dielectric constant on wavevector behavior. It applies Maxwell's equations to establish conditions for wave solutions influenced by complex parameters.
- 10.2: Boundary Conditions
- This page examines the continuity of electric and magnetic field components at material boundaries, applying Maxwell's equations and Stokes’ theorem. It states that the tangential components of electric fields (\(E_t\)) and magnetic fields (\(H_t\)) must match at interfaces, resulting in equations \(E_{t1} = E_{t2}\) and \(H_{t1} = H_{t2}\).
- 10.3: Application of the Boundary Conditions to a Plane Interface
- This page examines wave interactions at a plane interface, emphasizing boundary conditions for the continuity of electric and magnetic fields. It derives relationships for total fields through Maxwell's equations and formulates reflection and transmission coefficients based on material properties.
- 10.4: Reflection from a Metal at Radio Frequencies
- This page covers the behavior of metals at low frequencies, focusing on DC conductivity and copper's charge carrier dynamics at 300K. It discusses electromagnetic wave propagation, field attenuation, and boundary conditions at metal-vacuum interfaces. The importance of surface current density in perfect conductors is highlighted, indicating zero internal magnetic fields and specific relationships for electric fields.
- 10.5: Oblique Incidence
- When a plane wave falls upon the plane interface between two media the incident and reflected wave-vectors define the plane of incidence. The direction of the electric field vector in the incident wave may make an arbitrary angle with the plane of incidence. The general case may be treated as the sum of two special cases: an electric vector perpendicular to the plane of incidence (s-polarized light) and an electric vector which lies in the plane of incidence (p-polarized light).
- 10.6: Example- Copper
- This page explores the refractive index and reflectivity of room temperature copper for S- and P-polarized light at 0.5145 microns, noting limited angular dependence of these properties. It reveals that the real part of reflectivity for P-polarized light vanishes near 69°, accompanied by a substantial phase shift, while S-polarized light experiences a smaller phase shift under similar angles. The discussion is supported by referenced figures that illustrate these characteristics.
- 10.7: Example- Crown Glass
- This page examines how reflectivity varies with the angle of incidence for Crown glass and copper at 0.5145 microns, focusing on S- and P-polarized light. It discusses Brewster's angle, where P-polarized light reflection ceases, resulting in all reflected light being S-polarized. Additionally, it highlights the connection between the angles of reflected and transmitted beams at Brewster's angle and provides formulas linking refractive indices and angles for determining Brewster's angle.
- 10.8: Metals at Radio Frequencies
- This page covers the interaction of electromagnetic waves with metallic surfaces, focusing on DC conductivity and the response to electric fields at low frequencies. It introduces key Maxwell's equations and derives S-polarization field equations, emphasizing energy dissipation through Joule heating.
Thumbnail: The wavefronts of a plane wave traveling in 3-space. (Public Domain; Quibik via Wikipedia)