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| In S', the capacitor is at rest: | | In S', the capacitor is at rest: |
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| · Plates perpendicular to the z-axis; separation h. | | · Plates perpendicular to the z-axis; separation h. |
| Surface charge density$ \pm\sigma$ | | Surface charge density$ \pm\sigma$ |
| Volumetric leakage current j flowing in the z-direction (from the positive plate to the negative plate). | | Volumetric leakage current j flowing in the z-direction (from the positive plate to the negative plate). |
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| By symmetry (infinite plates), the magnetic field in S' is zero: | | By symmetry (infinite plates), the magnetic field in S' is zero: |
| $\mathbf{B}' = 0$ | | $\mathbf{B}' = 0$ |
| The electric field is uniform and perpendicular to the plates (according to Gauss's law): | | The electric field is uniform and perpendicular to the plates (according to Gauss's law): |
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| $\mathbf{E}' = (0,\,0,\,E_0), \qquad E_0 = \frac{\sigma}{\varepsilon_0}$ | | $\mathbf{E}' = (0,\,0,\,E_0), \qquad E_0 = \frac{\sigma}{\varepsilon_0}$ |
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| The leakage current is j in the z-direction; it does not create a magnetic field because it is uniform and infinite in the plane. | | The leakage current is j in the z-direction; it does not create a magnetic field because it is uniform and infinite in the plane. |
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| Transformation to the laboratory system S | | Transformation to the laboratory system S |
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| The capacitor moves relative to S with velocity | | The capacitor moves relative to S with velocity |
| $ \mathbf{v} = \beta c\,\hat{\mathbf{x}}$ (parallel to the plates). | | $ \mathbf{v} = \beta c\,\hat{\mathbf{x}}$ (parallel to the plates). |
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| @@ -28,9 +32,7 @@Solution |
| For a boost in the x-direction, the fields transform according to: |
| For a boost in the x-direction, the fields transform according to: |
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| |
| $\begin{aligned} |
| $\begin{aligned} |
| E_x &= E'_x, \quad & E_y &= \gamma(E'_y + v B'_z), \quad & E_z &= \gamma(E'_z - v B'_y),\\[4pt] |
| E_x &= E'_x, \quad & E_y &= \gamma(E'_y + v B'_z), \quad & E_z &= \gamma(E'_z - v B'_y) & B_x = B'_x, \quad B_y = \gamma\left(B'_y - \frac{v}{c^2}E'_z\right), \quad B_z = \gamma\left(B'_z + \frac{v}{c^2}E'_y\right)\end{aligned}$ |
| B_x &= B'_x, \quad & B_y &= \gamma\!\left(B'_y - \frac{v}{c^2}E'_z\right), \quad & B_z &= \gamma\!\left(B'_z + \frac{v}{c^2}E'_y\right), |
| |
| \end{aligned}$ |
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| |
| with |
| with |
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| |
| $\gamma = \dfrac{1}{\sqrt{1-\beta^2}}$ | | $\gamma = \dfrac{1}{\sqrt{1-\beta^2}}$ |
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| Substituting $\mathbf{E}' = (0,0,E_0)$ | | Substituting $\mathbf{E}' = (0,0,E_0)$ |
| and $ \mathbf{B}' = 0$: | | and $ \mathbf{B}' = 0$: |
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| $\mathbf{E} = (0,\,0,\,\gamma E_0), \qquad | | $\mathbf{E} = (0,\,0,\,\gamma E_0), \qquad |
| \mathbf{B} = \left(0,\,-\gamma\frac{v}{c^2}E_0,\,0\right)$ | | \mathbf{B} = \left(0,\,-\gamma\frac{v}{c^2}E_0,\,0\right)$ |
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| The electric field remains perpendicular to the plates, but its magnitude increases by a factor of$ \gamma$: | | The electric field remains perpendicular to the plates, but its magnitude increases by a factor of$ \gamma$: |
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| $\boxed{E_z = \gamma E_0 = \frac{E_0}{\sqrt{1-\beta^2}}}$ | | $\boxed{E_z = \gamma E_0 = \frac{E_0}{\sqrt{1-\beta^2}}}$ |
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| Consistency with the sources | | Consistency with the sources |
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| The total charge on each plate is invariant. | | The total charge on each plate is invariant. |
| Moving parallel to the plates, the length of the plates in the direction of motion contracts:$ L_x = L'_x/\gamma$ | | Moving parallel to the plates, the length of the plates in the direction of motion contracts:$ L_x = L'_x/\gamma$ |
| The area of each plate decreases: $A = A'/\gamma$ | | The area of each plate decreases: $A = A'/\gamma$ |
| Thus, the surface charge density in S is $\sigma_S = Q/A = \gamma\sigma$. | | Thus, the surface charge density in S is $\sigma_S = Q/A = \gamma\sigma$. |
