Edits to “Statement”, “Solution”, “Answer”
en/12.1.22.md
+6 −5
| @@ -1,13 +1,12 @@ | |||
| ### Statement | |||
| − | $12.1.22.$ [Insert the problem statement] | ||
| − | |||
| − | ### Solution | ||
| − | |||
| $12.1.22$ The amplitude of the electric field strength of a plane sinusoidal wave is equal | |||
| to $E_0$. What is the average pressure exerted by this wave on a flat metal wall | |||
| when it is normally incident on it? | |||
| + | ### Solution | ||
| + | |||
| + | |||
| The energy per unit area per unit time striking the metal wall is equal to the Poynting vector: | |||
| \begin{equation} | |||
| \vec{S} = c^2 \epsilon_0 (\vec{E} \times \vec{B}) | |||
| \end{equation} | |||
| For this case: | |||
| \begin{equation} | |||
| \vec{E} = \hat{y} E_0 \cos(\omega t) \quad \text{and} \quad \vec{B} = \hat{z} \frac{E_0}{c} \cos(\omega t) | |||
| \end{equation} | |||
| The pressure exerted by the incoming flow on the wall is: | |||
| \begin{equation} | |||
| P = \frac{\left\langle S\right\rangle_T }{c} = c \epsilon_0 \left\langle \vec{E} \times \vec{B}\right\rangle_T = \frac{\epsilon_0 E_0^2}{2} | |||
| \end{equation} | |||
| In this equation we have taken the time average of the cross product. | |||
| This is half of the answer, because there is a pressure exerted by the incoming flow and another equal pressure from the outgoing flow; our answer must be twice that for this reason: | |||
| \begin{equation} | |||
| P_{\text{total}} = \epsilon_0 E_0^2 | |||
| \end{equation} | |||
| @@ -33,4 +32,6 @@Solution | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | \begin{equation} | ||
| + | P_{\text{total}} = \epsilon_0 E_0^2 | ||
| + | \end{equation} | ||
| @@ -1,13 +1,12 @@ | |||
| ### Statement | ### Statement | ||
| $12.1.22.$ [Insert the problem statement] | |||
| ### Solution | |||
| $12.1.22$ The amplitude of the electric field strength of a plane sinusoidal wave is equal | $12.1.22$ The amplitude of the electric field strength of a plane sinusoidal wave is equal | ||
| to $E_0$. What is the average pressure exerted by this wave on a flat metal wall | to $E_0$. What is the average pressure exerted by this wave on a flat metal wall | ||
| when it is normally incident on it? | when it is normally incident on it? | ||
| ### Solution | |||
| The energy per unit area per unit time striking the metal wall is equal to the Poynting vector: | The energy per unit area per unit time striking the metal wall is equal to the Poynting vector: | ||
| \begin{equation} | \begin{equation} | ||
| \vec{S} = c^2 \epsilon_0 (\vec{E} \times \vec{B}) | \vec{S} = c^2 \epsilon_0 (\vec{E} \times \vec{B}) | ||
| \end{equation} | \end{equation} | ||
| For this case: | For this case: | ||
| \begin{equation} | \begin{equation} | ||
| \vec{E} = \hat{y} E_0 \cos(\omega t) \quad \text{and} \quad \vec{B} = \hat{z} \frac{E_0}{c} \cos(\omega t) | \vec{E} = \hat{y} E_0 \cos(\omega t) \quad \text{and} \quad \vec{B} = \hat{z} \frac{E_0}{c} \cos(\omega t) | ||
| \end{equation} | \end{equation} | ||
| The pressure exerted by the incoming flow on the wall is: | The pressure exerted by the incoming flow on the wall is: | ||
| \begin{equation} | \begin{equation} | ||
| P = \frac{\left\langle S\right\rangle_T }{c} = c \epsilon_0 \left\langle \vec{E} \times \vec{B}\right\rangle_T = \frac{\epsilon_0 E_0^2}{2} | P = \frac{\left\langle S\right\rangle_T }{c} = c \epsilon_0 \left\langle \vec{E} \times \vec{B}\right\rangle_T = \frac{\epsilon_0 E_0^2}{2} | ||
| \end{equation} | \end{equation} | ||
| In this equation we have taken the time average of the cross product. | In this equation we have taken the time average of the cross product. | ||
| This is half of the answer, because there is a pressure exerted by the incoming flow and another equal pressure from the outgoing flow; our answer must be twice that for this reason: | This is half of the answer, because there is a pressure exerted by the incoming flow and another equal pressure from the outgoing flow; our answer must be twice that for this reason: | ||
| \begin{equation} | \begin{equation} | ||
| P_{\text{total}} = \epsilon_0 E_0^2 | P_{\text{total}} = \epsilon_0 E_0^2 | ||
| \end{equation} | \end{equation} | ||
| @@ -33,4 +32,6 @@Solution | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | \begin{equation} | ||
| P_{\text{total}} = \epsilon_0 E_0^2 | |||
| \end{equation} | |||