Различия
Показаны различия между двумя версиями страницы.
| Предыдущая версия справа и слева Предыдущая версия Следующая версия | Предыдущая версия | ||
| optics:res2.3 [2020/02/10 07:46] – root | optics:res2.3 [2020/02/10 08:26] (текущий) – root | ||
|---|---|---|---|
| Строка 44: | Строка 44: | ||
| Складывая, | Складывая, | ||
| $$ | $$ | ||
| - | \left[div\mathbf{B}\right]_{\mathbf{k}}=\left(\nabla\cdot\mathbf{B}\right)_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}=i\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right) | + | \left[\text{div}\mathbf{B}\right]_{\mathbf{k}}=\left(\nabla\cdot\mathbf{B}\right)_{\mathbf{k}}=\left[\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}\right]_{\mathbf{k}}=i\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right) |
| . | . | ||
| $$ | $$ | ||
| Строка 51: | Строка 51: | ||
| $$ | $$ | ||
| - | \left[rot\mathbf{B}\right]_{\mathbf{k}}=\left[\nabla\times\mathbf{B}\right]_{\mathbf{k}}=i\left[\mathbf{k}\times\mathbf{B}_{\mathbf{k}}\right] | + | \left[\text{rot}\mathbf{B}\right]_{\mathbf{k}}=\left[\nabla\times\mathbf{B}\right]_{\mathbf{k}}=i\left[\mathbf{k}\times\mathbf{B}_{\mathbf{k}}\right] |
| . | . | ||
| $$ | $$ | ||
| Строка 59: | Строка 59: | ||
| .$$ | .$$ | ||
| - | \textemdash{} | + | Тогда окончательно запишем разложение по монохроматическим волнам: |
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | $$\text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} \mapsto \left[\nabla\times\mathbf{E}_{\omega}\right]=\frac{i\omega}{c}\mathbf{B}_{\omega} |
| - | $\mapsto$ $\left[\nabla\times\mathbf{E}_{\omega}\right]=\frac{i\omega}{c}\mathbf{B}_{\omega}$ | + | ,$$ |
| + | $$ | ||
| + | \text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} \mapsto | ||
| + | \left[\nabla\times\mathbf{H}_{\omega}\right]=\frac{4\pi}{c}\mathbf{j}_{\omega}+\frac{i\omega}{c}\mathbf{D}_{\omega} | ||
| + | ,$$ | ||
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | $$\text{div}\mathbf{D}=4\pi\rho \mapsto \left(\nabla\cdot\mathbf{D}_{\omega}\right)=4\pi\rho_{\omega} |
| - | $\mapsto$ $\left[\nabla\times\mathbf{H}_{\omega}\right]=\frac{4\pi}{c}\mathbf{j}_{\omega}+\frac{i\omega}{c}\mathbf{D}_{\omega}$ | + | ,$$ |
| + | $$ | ||
| + | \text{div}\mathbf{B}=0 \mapsto \left(\nabla\cdot\mathbf{B}_{\omega}\right)=0. | ||
| + | $$ | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $\left(\nabla\cdot\mathbf{D}_{\omega}\right)=4\pi\rho_{\omega}$ | + | Разложение по плоским волнам: |
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\nabla\cdot\mathbf{B}_{\omega}\right)=0$ | + | $$ |
| + | \text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} | ||
| + | \mapsto | ||
| + | i\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}}\right]=-\frac{1}{c}\frac{\partial\mathbf{B_{\mathbf{k}}}}{\partial t} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} | ||
| + | \mapsto | ||
| + | i\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}}\right]=\frac{4\pi}{c}\mathbf{j}_{\mathbf{k}}+\frac{1}{c}\frac{\partial\mathbf{D}_{\mathbf{k}}}{\partial t} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{div}\mathbf{D}=4\pi\rho | ||
| + | \mapsto | ||
| + | i\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}}\right)=4\pi\rho_{\mathbf{k}} | ||
| + | ,$$ | ||
| + | $$ | ||
| + | \text{div}\mathbf{B}=0 \mapsto \left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right)=0 | ||
| + | . | ||
| + | $$ | ||
| - | \textemdash{} | + | Разложение по плоским и монохроматическим волнам: |
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | $$ |
| - | $\mapsto$ $i\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}}\right]=-\frac{1}{c}\frac{\partial\mathbf{B_{\mathbf{k}}}}{\partial t}$ | + | \text{rot}\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} |
| - | + | \mapsto | |
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | \left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}\omega}\right]=\frac{\omega}{c}\mathbf{B}_{\mathbf{k}\omega},$$ |
| - | $\mapsto$ $i\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}}\right]=\frac{4\pi}{c}\mathbf{j}_{\mathbf{k}}+\frac{1}{c}\frac{\partial\mathbf{D}_{\mathbf{k}}}{\partial t}$ | + | |
| - | + | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $i\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}}\right)=4\pi\rho_{\mathbf{k}}$ | + | |
| - | + | ||
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}}\right)=0$ | + | |
| - | + | ||
| - | \textemdash{} | + | |
| - | + | ||
| - | $rot\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ | + | |
| - | $\mapsto$ $\left[\mathbf{k}\times\mathbf{E}_{\mathbf{k}\omega}\right]=\frac{\omega}{c}\mathbf{B}_{\mathbf{k}\omega}$ | + | |
| - | + | ||
| - | $rot\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}$ | + | |
| - | $\mapsto$ $\left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}\omega}\right]=-\frac{i4\pi}{c}\mathbf{j}_{\mathbf{k}\omega}+\frac{\omega}{c}\mathbf{D}_{\mathbf{k}\omega}$ | + | |
| - | + | ||
| - | $div\mathbf{D}=4\pi\rho$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}\omega}\right)=-i4\pi\rho_{\mathbf{k}\omega}$ | + | |
| - | + | ||
| - | $div\mathbf{B}=0$ $\mapsto$ $\left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}\omega}\right)=0$ | + | |
| + | $$\text{rot}\mathbf{H}=\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t} | ||
| + | \mapsto | ||
| + | \left[\mathbf{k}\times\mathbf{H}_{\mathbf{k}\omega}\right]=-\frac{i4\pi}{c}\mathbf{j}_{\mathbf{k}\omega}+\frac{\omega}{c}\mathbf{D}_{\mathbf{k}\omega}, | ||
| + | $$ | ||
| + | \text{div}\mathbf{D}=4\pi\rho \mapsto \left(\mathbf{k}\cdot\mathbf{D}_{\mathbf{k}\omega}\right)=-i4\pi\rho_{\mathbf{k}\omega}, | ||
| + | $$ | ||
| + | $$\text{div}\mathbf{B}=0 \mapsto \left(\mathbf{k}\cdot\mathbf{B}_{\mathbf{k}\omega}\right)=0 | ||
| + | .$$ | ||