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| In the system S' the solenoid is at rest: | | In the system S' the solenoid is at rest: |
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| The solenoid is "long and flat": its turns define a plane; the magnetic moment $\mathbf{M}$ | | The solenoid is "long and flat": its turns define a plane; the magnetic moment $\mathbf{M}$ |
| is perpendicular to that plane (normal direction). Let us take | | is perpendicular to that plane (normal direction). Let us take |
| take $\mathbf{M}' =M\,\hat{\mathbf{z}}$ | | take $\mathbf{M}' =M\,\hat{\mathbf{z}}$ |
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| There is no electric moment | | There is no electric moment |
| $\mathbf{p}' = 0$. | | $\mathbf{p}' = 0$. |
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| In the laboratory system S, the solenoid moves with velocity $\mathbf{v} = v\,\hat{\mathbf{x}}$ | | In the laboratory system S, the solenoid moves with velocity $\mathbf{v} = v\,\hat{\mathbf{x}}$ |
| parallel to the flat surfaces (hence, perpendicular to $\mathbf{M})$ | | parallel to the flat surfaces (hence, perpendicular to $\mathbf{M})$ |
| | | |
| Under a boost with velocity | | Under a boost with velocity |
| $ \mathbf{v}$ | | $ \mathbf{v}$ |
| the electric dipole moment $\mathbf{p}$ and magnetic dipole moment $\mathbf{m}$ transform according to: | | the electric dipole moment $\mathbf{p}$ and magnetic dipole moment $\mathbf{m}$ transform according to: |
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| $\begin{aligned} | | $\begin{aligned} |
| \mathbf{p} &= \gamma\left( \mathbf{p}' + \frac{\mathbf{v}}{c} \times \mathbf{m}' \right) - \frac{\gamma^2}{\gamma+1}\,\frac{\mathbf{v}}{c}\left( \frac{\mathbf{v}}{c} \cdot \mathbf{p}' \right),\\ | | \mathbf{p} &= \gamma\left( \mathbf{p}' + \frac{\mathbf{v}}{c} \times \mathbf{m}' \right) - \frac{\gamma^2}{\gamma+1}\,\frac{\mathbf{v}}{c}\left( \frac{\mathbf{v}}{c} \cdot \mathbf{p}' \right),\\ |
| \mathbf{m} &= \gamma\left( \mathbf{m}' - \frac{\mathbf{v}}{c} \times \mathbf{p}' \right) - \frac{\gamma^2}{\gamma+1}\,\frac{\mathbf{v}}{c}\left( \frac{\mathbf{v}}{c} \cdot \mathbf{m}' \right). | | \mathbf{m} &= \gamma\left( \mathbf{m}' - \frac{\mathbf{v}}{c} \times \mathbf{p}' \right) - \frac{\gamma^2}{\gamma+1}\,\frac{\mathbf{v}}{c}\left( \frac{\mathbf{v}}{c} \cdot \mathbf{m}' \right). |
| \end{aligned}$ | | \end{aligned}$ |
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| In our case:$ \mathbf{p}' = 0$ and $\mathbf{v} \cdot \mathbf{m}' = 0 $(because they are perpendicular) | | In our case:$ \mathbf{p}' = 0$ and $\mathbf{v} \cdot \mathbf{m}' = 0 $(because they are perpendicular) |
| Therefore: | | Therefore: |
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| $\boxed{\mathbf{p} = \frac{\gamma}{c}\,\mathbf{v} \times \mathbf{M}}, \qquad | | $\boxed{\mathbf{p} = \frac{\gamma}{c}\,\mathbf{v} \times \mathbf{M}}, \qquad |
| \mathbf{m} = \gamma\,\mathbf{M}$ | | \mathbf{m} = \gamma\,\mathbf{M}$ |
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| The magnetic moment in S increases by a factor of $\gamma$ | | The magnetic moment in S increases by a factor of $\gamma$ |
| and an electric moment appears perpendicular to $\mathbf{v}$ and to $\mathbf{M}$ | | and an electric moment appears perpendicular to $\mathbf{v}$ and to $\mathbf{M}$ |
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| For $ v \ll c (\gamma \approx 1)$ | | For $ v \ll c (\gamma \approx 1)$ |
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| In CGS: $\displaystyle \mathbf{p} \approx \frac{1}{c}\,\mathbf{v} \times \mathbf{M}$. | | In CGS: $\displaystyle \mathbf{p} \approx \frac{1}{c}\,\mathbf{v} \times \mathbf{M}$. |
| | | |
| In SI: the factor is | | In SI: the factor is |
| $1/c^2$ | | $1/c^2$ |
| $\displaystyle \mathbf{p} = \frac{1}{c^2}\,\mathbf{v} \times \mathbf{M} $ | | $\displaystyle \mathbf{p} = \frac{1}{c^2}\,\mathbf{v} \times \mathbf{M} $ |
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| (with $\mathbf{M}$ in $A·m²$) | | (with $\mathbf{M}$ in $A·m²$) |
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| Magnitude (if $\mathbf{v} \perp \mathbf{M})$: | | Magnitude (if $\mathbf{v} \perp \mathbf{M})$: |
| $ p = \frac{v}{c}\,M (CGS)$ | | $ p = \frac{v}{c}\,M (CGS)$ |
| or $ p = \frac{v}{c^2}\,M$(SI) | | or $ p = \frac{v}{c^2}\,M$(SI) |
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| Direction: | | Direction: |
| perpendicular to$ \mathbf{v} $and to $\mathbf{M}$, | | perpendicular to$ \mathbf{v} $and to $\mathbf{M}$, |
| that is, parallel to the flat surfaces but perpendicular to the motion. | | that is, parallel to the flat surfaces but perpendicular to the motion. |
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| #### Answer | | #### Answer |
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| $\boxed{\mathbf{p} = \frac{\gamma}{c}\,\mathbf{v} \times \mathbf{M} \;\;(\text{CGS}), \qquad | | $\boxed{\mathbf{p} = \frac{\gamma}{c}\,\mathbf{v} \times \mathbf{M} \;\;(\text{CGS}), \qquad |
| \mathbf{p} = \frac{\gamma}{c^2}\,\mathbf{v} \times \mathbf{M} \;\;(\text{SI})}$ | | \mathbf{p} = \frac{\gamma}{c^2}\,\mathbf{v} \times \mathbf{M} \;\;(\text{SI})}$ |
| | | |
| In the low-velocity limit $(\gamma \to 1)$: | | In the low-velocity limit $(\gamma \to 1)$: |
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| $p \approx \frac{v}{c}M \;\;(\text{CGS}), \qquad | | $p \approx \frac{v}{c}M \;\;(\text{CGS}), \qquad |
| p \approx \frac{v}{c^2}M \;\;(\text{SI})$ | | p \approx \frac{v}{c^2}M \;\;(\text{SI})$ |
