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| Plane capacitor fixed in the laboratory, electric field in vacuum (between plates) E. | | Plane capacitor fixed in the laboratory, electric field in vacuum (between plates) E. |
| Dielectric plate of constant $\varepsilon$ moves parallel to the plates with velocity $\mathbf{v} = \beta c\,\hat{\mathbf{x}}$ | | Dielectric plate of constant $\varepsilon$ moves parallel to the plates with velocity $\mathbf{v} = \beta c\,\hat{\mathbf{x}}$ |
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| the field in vacuum is uniform. | | the field in vacuum is uniform. |
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| Charge density on the plates | | Charge density on the plates |
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| In the laboratory system S, the plates are at rest. The relation between field and surface density in CGS is$ E = 4\pi\sigma$, hence: | | In the laboratory system S, the plates are at rest. The relation between field and surface density in CGS is$ E = 4\pi\sigma$, hence: |
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| $\sigma = \frac{E}{4\pi}$ | | $\sigma = \frac{E}{4\pi}$ |
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| Rest frame of the dielectric (S') | | Rest frame of the dielectric (S') |
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| We go to the system S' that moves with the dielectric (velocity +$\beta c $relative to S). In S': | | We go to the system S' that moves with the dielectric (velocity +$\beta c $relative to S). In S': |
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| The capacitor plates and their charges move with velocity $-\beta c$ | | The capacitor plates and their charges move with velocity $-\beta c$ |
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| The length in the direction of motion contracts; since the total charge is invariant, the surface density increases by$ \gamma = 1/\sqrt{1-\beta^2}$ | | The length in the direction of motion contracts; since the total charge is invariant, the surface density increases by$ \gamma = 1/\sqrt{1-\beta^2}$ |
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| $\sigma' = \gamma\sigma = \frac{\gamma E}{4\pi}$ | | $\sigma' = \gamma\sigma = \frac{\gamma E}{4\pi}$ |
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| · The electric field in vacuum (outside the dielectric) in S' is: | | · The electric field in vacuum (outside the dielectric) in S' is: |
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| $E' = 4\pi\sigma' = \gamma E$ | | $E' = 4\pi\sigma' = \gamma E$ |
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| In S' the dielectric plate is at rest and there is no magnetic field inside it $(\mathbf{B}'_{\text{diel}} = 0)$ | | In S' the dielectric plate is at rest and there is no magnetic field inside it $(\mathbf{B}'_{\text{diel}} = 0)$ |
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| Electric field inside the dielectric in S' | | Electric field inside the dielectric in S' |
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| The vacuum–dielectric boundary is perpendicular to the field. The boundary condition for the electric displacement $(\mathbf{D}) $in the absence of free surface charges is $D'_{\text{vac}} = D'_{\text{diel}}$. In CGS$ (\varepsilon_0 = 1)$ | | The vacuum–dielectric boundary is perpendicular to the field. The boundary condition for the electric displacement $(\mathbf{D}) $in the absence of free surface charges is $D'_{\text{vac}} = D'_{\text{diel}}$. In CGS$ (\varepsilon_0 = 1)$ |
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| $E' = \varepsilon\,E'_{\text{diel}} \quad\Longrightarrow\quad | | $E' = \varepsilon\,E'_{\text{diel}} \quad\Longrightarrow\quad |
| E'_{\text{diel}} = \frac{E'}{\varepsilon} = \frac{\gamma E}{\varepsilon}$ | | E'_{\text{diel}} = \frac{E'}{\varepsilon} = \frac{\gamma E}{\varepsilon}$ |
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| Transformation back to the laboratory (S) | | Transformation back to the laboratory (S) |
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| Now we return to S by applying a boost of velocity +$\beta c$ to the internal fields of the dielectric$ (S' \to S)$. | | Now we return to S by applying a boost of velocity +$\beta c$ to the internal fields of the dielectric$ (S' \to S)$. |
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| Since $\mathbf{B}'_{\text{diel}} = 0$ | | Since $\mathbf{B}'_{\text{diel}} = 0$ |
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| $\begin{aligned} | | $\begin{aligned} |
| E_{\text{diel}} &= \gamma\left(E'_{\text{diel}} + \beta B'_{\text{diel}}\right) = \gamma\,E'_{\text{diel}} = \frac{\gamma^2 E}{\varepsilon}, \\[4pt] | | E_{\text{diel}} &= \gamma\left(E'_{\text{diel}} + \beta B'_{\text{diel}}\right) = \gamma\,E'_{\text{diel}} = \frac{\gamma^2 E}{\varepsilon}, \\[4pt] |
| B_{\text{diel}} &= \gamma\left(B'_{\text{diel}} + \beta E'_{\text{diel}}\right) = \gamma\beta\,E'_{\text{diel}} = \frac{\beta\gamma^2 E}{\varepsilon}. | | B_{\text{diel}} &= \gamma\left(B'_{\text{diel}} + \beta E'_{\text{diel}}\right) = \gamma\beta\,E'_{\text{diel}} = \frac{\beta\gamma^2 E}{\varepsilon}. |
| \end{aligned}$ | | \end{aligned}$ |
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| Substituting $\gamma^2 = \dfrac{1}{1-\beta^2}$: | | Substituting $\gamma^2 = \dfrac{1}{1-\beta^2}$: |
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| $\boxed{\mathbf{E}_{\text{diel}} = \frac{E}{\varepsilon(1-\beta^2)}\,\hat{\mathbf{y}}},\qquad | | $\boxed{\mathbf{E}_{\text{diel}} = \frac{E}{\varepsilon(1-\beta^2)}\,\hat{\mathbf{y}}},\qquad |
| \boxed{\mathbf{B}_{\text{diel}} = \frac{\beta E}{\varepsilon(1-\beta^2)}\,\hat{\mathbf{z}}} \quad (\text{CGS system})$ | | \boxed{\mathbf{B}_{\text{diel}} = \frac{\beta E}{\varepsilon(1-\beta^2)}\,\hat{\mathbf{z}}} \quad (\text{CGS system})$ |
