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| Configuration and fields | | Configuration and fields |
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| We take the z-axis perpendicular to the wall, with the metal occupying z < 0 and the vacuum in z > 0. The wave is incident from the right. | | We take the z-axis perpendicular to the wall, with the metal occupying z < 0 and the vacuum in z > 0. The wave is incident from the right. |
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| Incident wave (traveling in the -z direction): | | Incident wave (traveling in the -z direction): |
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| $\mathbf{E}_i = E \cos(\omega t + kz)\,\hat{\mathbf{x}}, \qquad | | $\mathbf{E}_i = E \cos(\omega t + kz)\,\hat{\mathbf{x}}, \qquad |
| \mathbf{B}_i = \frac{E}{c} \cos(\omega t + kz)\,\hat{\mathbf{y}} \quad (\text{SI})$ | | \mathbf{B}_i = \frac{E}{c} \cos(\omega t + kz)\,\hat{\mathbf{y}} \quad (\text{SI})$ |
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| In CGS: $\mathbf{B}_i = E \cos(\omega t + kz)\,\hat{\mathbf{y}}$ . | | In CGS: $\mathbf{B}_i = E \cos(\omega t + kz)\,\hat{\mathbf{y}}$ . |
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| Reflected wave (from a perfect conductor): | | Reflected wave (from a perfect conductor): |
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| $\mathbf{E}_r = -E \cos(\omega t - kz)\,\hat{\mathbf{x}}, \qquad | | $\mathbf{E}_r = -E \cos(\omega t - kz)\,\hat{\mathbf{x}}, \qquad |
| \mathbf{B}_r = \frac{E}{c} \cos(\omega t - kz)\,\hat{\mathbf{y}} \quad (\text{SI})$. | | \mathbf{B}_r = \frac{E}{c} \cos(\omega t - kz)\,\hat{\mathbf{y}} \quad (\text{SI})$. |
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| In CGS:$ \mathbf{B}_r = E \cos(\omega t - kz)\,\hat{\mathbf{y}}$. | | In CGS:$ \mathbf{B}_r = E \cos(\omega t - kz)\,\hat{\mathbf{y}}$. |
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| At the surface (z = 0): | | At the surface (z = 0): |
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| $\mathbf{E}_{\text{total}} = E \cos\omega t - E \cos\omega t = 0 \quad (\text{node, electric field is zero})$. | | $\mathbf{E}_{\text{total}} = E \cos\omega t - E \cos\omega t = 0 \quad (\text{node, electric field is zero})$. |
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| $\mathbf{B}_{\text{total}} = \frac{E}{c}\cos\omega t + \frac{E}{c}\cos\omega t = \frac{2E}{c}\cos\omega t\,\hat{\mathbf{y}} \quad (\text{SI}), \qquad | | $\mathbf{B}_{\text{total}} = \frac{E}{c}\cos\omega t + \frac{E}{c}\cos\omega t = \frac{2E}{c}\cos\omega t\,\hat{\mathbf{y}} \quad (\text{SI}), \qquad |
| \mathbf{B}_{\text{total}} = 2E\cos\omega t\,\hat{\mathbf{y}} \quad (\text{CGS})$ | | \mathbf{B}_{\text{total}} = 2E\cos\omega t\,\hat{\mathbf{y}} \quad (\text{CGS})$ |
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| At the surface of a perfect conductor, the magnetic field has a discontinuity related to the surface current $\mathbf{K}$ | | At the surface of a perfect conductor, the magnetic field has a discontinuity related to the surface current $\mathbf{K}$ |
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| In SI: | | In SI: |
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| $\mathbf{n} \times (\mathbf{H}_{\text{vacuum}} - \mathbf{H}_{\text{metal}}) = \mathbf{K}$ | | $\mathbf{n} \times (\mathbf{H}_{\text{vacuum}} - \mathbf{H}_{\text{metal}}) = \mathbf{K}$ |
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| The normal vector \mathbf{n} points from the metal into the vacuum: | | The normal vector \mathbf{n} points from the metal into the vacuum: |
| $\mathbf{n} = \hat{\mathbf{z}}$. | | $\mathbf{n} = \hat{\mathbf{z}}$. |
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| Inside the metal | | Inside the metal |
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| $\mathbf{H} = 0$. | | $\mathbf{H} = 0$. |
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| In vacuum | | In vacuum |
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| $\mathbf{H} = \mathbf{B}/\mu_0$. | | $\mathbf{H} = \mathbf{B}/\mu_0$. |
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| $\mathbf{H}_{\text{vacuum}} = \frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{y}}$. | | $\mathbf{H}_{\text{vacuum}} = \frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{y}}$. |
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| Then: | | Then: |
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| $\mathbf{K} = \hat{\mathbf{z}} \times \left( \frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{y}} \right) = -\frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{x}}.$ | | $\mathbf{K} = \hat{\mathbf{z}} \times \left( \frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{y}} \right) = -\frac{2E}{\mu_0 c}\cos\omega t\,\hat{\mathbf{x}}.$ |
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| The amplitude (linear current density) is: | | The amplitude (linear current density) is: |
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| $\boxed{i = \frac{2E}{\mu_0 c} = 2\varepsilon_0 c E \quad (\text{SI}).}$ | | $\boxed{i = \frac{2E}{\mu_0 c} = 2\varepsilon_0 c E \quad (\text{SI}).}$ |
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| In CGS: | | In CGS: |
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| $\mathbf{n} \times (\mathbf{B}_{\text{vacuum}} - \mathbf{B}_{\text{metal}}) = \frac{4\pi}{c}\mathbf{K}$ | | $\mathbf{n} \times (\mathbf{B}_{\text{vacuum}} - \mathbf{B}_{\text{metal}}) = \frac{4\pi}{c}\mathbf{K}$ |
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| With | | With |
| $\mathbf{B}_{\text{vacuum}} = 2E\cos\omega t\,\hat{\mathbf{y}}, \mathbf{B}_{\text{metal}} = 0, \mathbf{n} = \hat{\mathbf{z}}$: | | $\mathbf{B}_{\text{vacuum}} = 2E\cos\omega t\,\hat{\mathbf{y}}, \mathbf{B}_{\text{metal}} = 0, \mathbf{n} = \hat{\mathbf{z}}$: |
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| $\hat{\mathbf{z}} \times (2E\cos\omega t\,\hat{\mathbf{y}}) = -\frac{4\pi}{c}\mathbf{K} \quad\Rightarrow\quad \mathbf{K} = -\frac{cE}{2\pi}\cos\omega t\,\hat{\mathbf{x}}$ | | $\hat{\mathbf{z}} \times (2E\cos\omega t\,\hat{\mathbf{y}}) = -\frac{4\pi}{c}\mathbf{K} \quad\Rightarrow\quad \mathbf{K} = -\frac{cE}{2\pi}\cos\omega t\,\hat{\mathbf{x}}$ |
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| Amplitude: | | Amplitude: |
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| $\boxed{i = \frac{cE}{2\pi} \quad (\text{CGS}).}$ | | $\boxed{i = \frac{cE}{2\pi} \quad (\text{CGS}).}$ |
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| The average pressure on a perfect reflector is$ p = 2I/c$ where I is the average incident intensity. | | The average pressure on a perfect reflector is$ p = 2I/c$ where I is the average incident intensity. |
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| In SI: | | In SI: |
| Average intensity: $I = \frac{1}{2}\varepsilon_0 c E^2$ | | Average intensity: $I = \frac{1}{2}\varepsilon_0 c E^2$ |
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| $p = \frac{2I}{c} = \varepsilon_0 E^2$ | | $p = \frac{2I}{c} = \varepsilon_0 E^2$ |
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| $\boxed{p = \varepsilon_0 E^2 \quad (\text{SI}).}$ | | $\boxed{p = \varepsilon_0 E^2 \quad (\text{SI}).}$ |
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| In CGS: | | In CGS: |
| Average intensity: | | Average intensity: |
| $ I = \frac{c}{8\pi} E^2$ | | $ I = \frac{c}{8\pi} E^2$ |
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| p = \frac{2I}{c} = \frac{E^2}{4\pi}. | | p = \frac{2I}{c} = \frac{E^2}{4\pi}. |
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| $\boxed{p = \frac{E^2}{4\pi} \quad (\text{CGS}).}$ | | $\boxed{p = \frac{E^2}{4\pi} \quad (\text{CGS}).}$ |
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| #### Answer | | #### Answer |
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| $\boxed{p = \varepsilon_0 E^2 \quad (\text{SI}).}$ | | $\boxed{p = \varepsilon_0 E^2 \quad (\text{SI}).}$ |
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| $\boxed{p = \frac{E^2}{4\pi} \quad (\text{CGS}).}$ | | $\boxed{p = \frac{E^2}{4\pi} \quad (\text{CGS}).}$ |
