| a) along the direction of the plates? | | a) along the direction of the plates? |
| b) perpendicular to the plates? | | b) perpendicular to the plates? |
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| ### Solution | | ### Solution |
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| The charge on the plates, \( Q \), remains constant in all reference frames because charge is an invariant quantity in the theory of relativity. That is, | | The charge on the plates, \( Q \), remains constant in all reference frames because charge is an invariant quantity in the theory of relativity. That is, |
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| \[ | | \[ |
| Q = CU \quad \Longleftrightarrow \quad \text{Invariant} | | Q = CU \quad \Longleftrightarrow \quad \text{Invariant} |
| \] | | \] |
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| Let the Lorentz factor be denoted as: | | Let the Lorentz factor be denoted as: |
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| \[ | | \[ |
| \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} = \frac{1}{\sqrt{1 - \beta^2}} | | \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} = \frac{1}{\sqrt{1 - \beta^2}} |
| \] | | \] |
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| #### a) Motion Along the Plates | | #### a) Motion Along the Plates |
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| In this case, relativistic effects manifest as follows: | | In this case, relativistic effects manifest as follows: |
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| - The distance between the plates \( d \) remains unchanged, since length contraction only affects dimensions along the direction of motion. | | - The distance between the plates \( d \) remains unchanged, since length contraction only affects dimensions along the direction of motion. |
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| - The area of the plates decreases due to contraction in one dimension: | | - The area of the plates decreases due to contraction in one dimension: |
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| \[ | | \[ |
| S' = \frac{S}{\gamma} | | S' = \frac{S}{\gamma} |
| \] | | \] |
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| - The capacitance becomes: | | - The capacitance becomes: |
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| \[ | | \[ |
| C' = \frac{\varepsilon_0 S'}{d} = \frac{\varepsilon_0 \frac{S}{\gamma}}{d} = \frac{C}{\gamma} | | C' = \frac{\varepsilon_0 S'}{d} = \frac{\varepsilon_0 \frac{S}{\gamma}}{d} = \frac{C}{\gamma} |
| \] | | \] |
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| Using charge invariance \( Q = C U = C' U' \), we find that the potential difference increases by a factor of \( \gamma \): | | Using charge invariance \( Q = C U = C' U' \), we find that the potential difference increases by a factor of \( \gamma \): |
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| \[ | | \[ |
| U' = \gamma U = \frac{1}{\sqrt{1 - \beta^2}} U | | U' = \gamma U = \frac{1}{\sqrt{1 - \beta^2}} U |
| \] | | \] |
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| Thus, the potential difference increases by a factor of \( \frac{1}{\sqrt{1 - \beta^2}} \). | | Thus, the potential difference increases by a factor of \( \frac{1}{\sqrt{1 - \beta^2}} \). |
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| #### b) Motion Perpendicular to the Plates | | #### b) Motion Perpendicular to the Plates |
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| Here, the relativistic effects are: | | Here, the relativistic effects are: |
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| - The distance between the plates contracts: | | - The distance between the plates contracts: |
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| \[ | | \[ |
| d' = \frac{d}{\gamma} | | d' = \frac{d}{\gamma} |
| \] | | \] |
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| - The area of the plates remains the same: | | - The area of the plates remains the same: |
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| \[ | | \[ |
| S' = S | | S' = S |
| \] | | \] |
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| - The capacitance becomes: | | - The capacitance becomes: |
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| \[ | | \[ |
| C' = \frac{\varepsilon_0 S}{d'} = \frac{\varepsilon_0 S}{\frac{d}{\gamma}} = \gamma C | | C' = \frac{\varepsilon_0 S}{d'} = \frac{\varepsilon_0 S}{\frac{d}{\gamma}} = \gamma C |
| \] | | \] |
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| Again using charge invariance \( Q = C U = C' U' \), the potential difference decreases by a factor of \( \gamma \): | | Again using charge invariance \( Q = C U = C' U' \), the potential difference decreases by a factor of \( \gamma \): |
| | | |
| \[ | | \[ |
| U' = \frac{U}{\gamma} | | U' = \frac{U}{\gamma} |
| \] | | \] |
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| So, the potential difference decreases by a factor of \( \frac{1}{\sqrt{1 - \beta^2}} \). | | So, the potential difference decreases by a factor of \( \frac{1}{\sqrt{1 - \beta^2}} \). |
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| ### Answer | | ### Answer |
