Edits to “Statement”, “Solution”, “Answer”
en/14.3.16.md
+5 −8
| @@ -1,19 +1,16 @@ | |||
| ### Statement | |||
| − | $14.3.16 | ||
| + | $14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with refractive index $n$. | ||
| − | ### Solution | ||
| − | |||
| − | $14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with | ||
| − | refractive index $n$. | ||
| − | |||
| $14.3.15$ How many times will the amplitude of a plane electromagnetic wave change | |||
| when it passes into a coordinate system moving at a speed $\beta c$ in the direction | |||
| of wave propagation? | |||
| + | ### Solution | ||
| + | |||
| To solve this problem we need to use the Lorentz transformation for fields, taking into account that the fields of the electromagnetic wave are | |||
| perpendicular to the velocity of the moving frame. Also, in each frame, the relation $\vec{E} = [\vec{v} \times \vec{B}]$ holds, where $\vec{v}$ is the | |||
| − | velocity vector of the electromagnetic wave; in this case $\vec{v} = \frac{c}{n}$. | ||
| + | velocity vector of the electromagnetic wave; in this case inside the dielectric media $\vec{v} = \frac{c}{n}$. | ||
| \begin{equation} | |||
| E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \frac{1-\beta n}{\sqrt{1-\beta^2}} | |||
| \end{equation} | |||
| And using the relation between fields: | |||
| \begin{equation} | |||
| B' = B \frac{1-\beta n}{\sqrt{1-\beta^2}} | |||
| \end{equation} | |||
| So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller. | |||
| @@ -29,4 +26,4 @@Solution | |||
| #### Answer | |||
| − | |||
| + | So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller. | ||
| @@ -1,19 +1,16 @@ | |||
| ### Statement | ### Statement | ||
| $14.3.16 |
$14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with refractive index $n$. | ||
| ### Solution | |||
| $14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with | |||
| refractive index $n$. | |||
| $14.3.15$ How many times will the amplitude of a plane electromagnetic wave change | $14.3.15$ How many times will the amplitude of a plane electromagnetic wave change | ||
| when it passes into a coordinate system moving at a speed $\beta c$ in the direction | when it passes into a coordinate system moving at a speed $\beta c$ in the direction | ||
| of wave propagation? | of wave propagation? | ||
| ### Solution | |||
| To solve this problem we need to use the Lorentz transformation for fields, taking into account that the fields of the electromagnetic wave are | To solve this problem we need to use the Lorentz transformation for fields, taking into account that the fields of the electromagnetic wave are | ||
| perpendicular to the velocity of the moving frame. Also, in each frame, the relation $\vec{E} = [\vec{v} \times \vec{B}]$ holds, where $\vec{v}$ is the | perpendicular to the velocity of the moving frame. Also, in each frame, the relation $\vec{E} = [\vec{v} \times \vec{B}]$ holds, where $\vec{v}$ is the | ||
| velocity vector of the electromagnetic wave; in this case $\vec{v} = \frac{c}{n}$. | velocity vector of the electromagnetic wave; in this case inside the dielectric media $\vec{v} = \frac{c}{n}$. | ||
| \begin{equation} | \begin{equation} | ||
| E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \frac{1-\beta n}{\sqrt{1-\beta^2}} | E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \frac{1-\beta n}{\sqrt{1-\beta^2}} | ||
| \end{equation} | \end{equation} | ||
| And using the relation between fields: | And using the relation between fields: | ||
| \begin{equation} | \begin{equation} | ||
| B' = B \frac{1-\beta n}{\sqrt{1-\beta^2}} | B' = B \frac{1-\beta n}{\sqrt{1-\beta^2}} | ||
| \end{equation} | \end{equation} | ||
| So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller. | So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller. | ||
| @@ -29,4 +26,4 @@Solution | |||
| #### Answer | #### Answer | ||
| So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller. | |||