From e3b95ba3b35c2aa086dbeb8984397eaa6994c651 Mon Sep 17 00:00:00 2001 From: Indigo5684 <159226326+Indigo5684@users.noreply.github.com> Date: Tue, 30 Sep 2025 13:19:34 -0500 Subject: [PATCH] Other WIP --- .../10-electromagnetic-waves.md | 16 ++++++++++++++-- .../homework/Problem Set 3.rnote | Bin 0 -> 1337936 bytes .../physics/thermal/homework/Homework 2.ods | Bin 0 -> 53406 bytes .../physics/thermal/homework/Homework 2.rnote | Bin 0 -> 1345112 bytes 4 files changed, 14 insertions(+), 2 deletions(-) create mode 100644 statics/physics/electrodynamics/homework/Problem Set 3.rnote create mode 100644 statics/physics/thermal/homework/Homework 2.ods create mode 100644 statics/physics/thermal/homework/Homework 2.rnote diff --git a/docs/physics/electrodynamics/10-electromagnetic-waves.md b/docs/physics/electrodynamics/10-electromagnetic-waves.md index e7cef5c..294e80e 100644 --- a/docs/physics/electrodynamics/10-electromagnetic-waves.md +++ b/docs/physics/electrodynamics/10-electromagnetic-waves.md @@ -37,7 +37,7 @@ $$\begin{align} ### Section 10.1.1 - The Wave Equation and Plane Waves -**Definition**. The equation $[\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}] f(x, t) = 0$ is well-known to mathematicians (see [Differential Equations](/todo.md)), and is known as the **wave equation**. In physics, the speed of the wave is $v = c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$. +**Definition**. The equation $[\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}] f(x, t) = 0$ is well-known to mathematicians (see [Differential Equations](../../todo.md)), and is known as the **wave equation**. In physics, the speed of the wave is $v = c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$. Consider some function $f(s)$. If $s = x - vt$ or $x + vt$, it is trivial to see that $f(x)$ satisfies the wave equation. @@ -84,5 +84,17 @@ Notably, an elliptically polarized wave can be described as follows: $$\begin{align} \vb{E}(\vb{r}, t) &= E_{x0} \vu{x} \cos(kz - \omega t) + E_{y0} \vu{y} \sin(kz - \omega t) \\ -\vb{H}(\vb{r}, t) &= \sqrt{\frac{\varepsilon}{\mu}} (E_{x0} \vu{x} \cos(kz - \omega t) + E_{y0} \vu{y} \sin(kz - \omega t)) +\vb{H}(\vb{r}, t) &= \sqrt{\frac{\varepsilon}{\mu}} (E_{x0} \vu{y} \cos(kz - \omega t) - E_{y0} \vu{x} \sin(kz - \omega t)) \end{align}$$ + +If $E_{x0} = E_{y0}$, the wave is said to be circularly polarized. + +## Section 10.2 - Reflection and Refraction of Plane Electromagnetic Waves at a a Planar Interface + +This section will focus on plane monochromatic waves incident from material 1 onto material 2, where both materials are homogenous insulators and the surface between the two materials is smooth (on the scale of the wavelength). + +In this case, we must re-consider Maxwell's equations. We know from previous sections that $\div \vb{E} = \frac{\vb{\rho_e}}{\varepsilon_0}$ and $\div \vb{H} = \frac{\vb{\rho_m}}{\mu_0}$. We also know that $\div \vb{D} = \rho_{ef}$ and $\div \vb{B} = \rho_{mf}$. + +Consider the boundary between the two materials. If we consider $\div \vb{D}$, and take the integral over a Gaussian pillbox on the boundary, we can apply divergence theorem to see that $\int_V \div \vb{D} dV = \int_{SofV} D \vdot \vu{n} dS = \rho_{efree}$. If we assume the materials are insulating, we do not expect to find any electrical charge, so $\rho_{efree} = 0$. Thus, we can say that $\int_{SofV} D \vdot \vu{n} = 0$, so $\vb{D}_1 \vdot \vu{n} + \vb{D}_2 \vdot \vu{n} = \vb{D_1} \vdot \vu{z} + \vb{D}_1 \vdot (-\vu{n}) = 0$. Then, we can say that $\vb{D}_1 \vdot{z} = \vb{D}_2 \vdot{z}$, or in simpler terms, $\vdot{D}_1^\perp = \vdot{D}_2^\perp$. + +Applying the same logic to $\vb{B}$, we see that $\vdot{B}_1^\perp = \vdot{B}_2^\perp$. 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