| The charge is at rest at the origin: | | The charge is at rest at the origin: |
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| $\mathbf{E}' = \frac{q}{r'^3}\,\mathbf{r}' \quad (\text{CGS}),\qquad \mathbf{B}' = 0$ | | $\mathbf{E}' = \frac{q}{r'^3}\,\mathbf{r}' \quad (\text{CGS}),\qquad \mathbf{B}' = 0$ |
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| Transformation to the laboratory (S) | | Transformation to the laboratory (S) |
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| The system S' moves with velocity $\mathbf{v} = \beta c\,\hat{\mathbf{z}}$ | | The system S' moves with velocity $\mathbf{v} = \beta c\,\hat{\mathbf{z}}$ |
| relative to S. For a boost along z, the parallel and perpendicular components transform as: | | relative to S. For a boost along z, the parallel and perpendicular components transform as: |
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| $E_z = E'_z, \quad \mathbf{E}_\perp = \gamma\,\mathbf{E}'_\perp, \qquad | | $E_z = E'_z, \quad \mathbf{E}_\perp = \gamma\,\mathbf{E}'_\perp, \qquad |
| B_z = 0, \quad \mathbf{B}_\perp = \gamma\,\frac{\mathbf{v}}{c}\times\mathbf{E}'$ | | B_z = 0, \quad \mathbf{B}_\perp = \gamma\,\frac{\mathbf{v}}{c}\times\mathbf{E}'$ |
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| Change of coordinates | | Change of coordinates |
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| The positions are related by$ x' = x,\; y' = y,\; z' = \gamma(z - vt)$ | | The positions are related by$ x' = x,\; y' = y,\; z' = \gamma(z - vt)$ |
| The distance in S' as a function of the coordinates in S is: | | The distance in S' as a function of the coordinates in S is: |
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| $r' = r\,\frac{\sqrt{1-\beta^2\sin^2\alpha}}{\sqrt{1-\beta^2}}$ | | $r' = r\,\frac{\sqrt{1-\beta^2\sin^2\alpha}}{\sqrt{1-\beta^2}}$ |
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| where$ \alpha$ is the angle between $\mathbf{v}$ and$ \mathbf{r} in S$ | | where$ \alpha$ is the angle between $\mathbf{v}$ and$ \mathbf{r} in S$ |
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| Electric field in S | | Electric field in S |
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| Substituting the expression for r' into $\mathbf{E}'$ and applying the transformation of components, one obtains a radial field from the instantaneous position of the charge: | | Substituting the expression for r' into $\mathbf{E}'$ and applying the transformation of components, one obtains a radial field from the instantaneous position of the charge: |
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| $\boxed{\mathbf{E} = \frac{q}{r^3}\,\frac{1-\beta^2}{\bigl(1-\beta^2\sin^2\alpha\bigr)^{3/2}}\;\mathbf{r}} \quad (\text{CGS})$ | | $\boxed{\mathbf{E} = \frac{q}{r^3}\,\frac{1-\beta^2}{\bigl(1-\beta^2\sin^2\alpha\bigr)^{3/2}}\;\mathbf{r}} \quad (\text{CGS})$ |
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| In SI, q is replaced by $\dfrac{q}{4\pi\varepsilon_0}$ | | In SI, q is replaced by $\dfrac{q}{4\pi\varepsilon_0}$ |
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| Magnetic field in S | | Magnetic field in S |
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| $\boxed{\mathbf{B} = \frac{\mathbf{v}}{c}\times\mathbf{E}} \quad (\text{CGS}),\qquad | | $\boxed{\mathbf{B} = \frac{\mathbf{v}}{c}\times\mathbf{E}} \quad (\text{CGS}),\qquad |
| \boxed{\mathbf{B} = \frac{1}{c^2}\,\mathbf{v}\times\mathbf{E}} \quad (\text{SI})$ | | \boxed{\mathbf{B} = \frac{1}{c^2}\,\mathbf{v}\times\mathbf{E}} \quad (\text{SI})$ |
