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| ### Statement |
| ### Statement |
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| $12.1.13.$ [Insert the problem statement] |
| $12.1.13.$ |
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| A charged flat capacitor is swung by moving it parallel to the plates. As the os- |
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| cillation frequency ν increases, the average intensity I of the electromagnetic |
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| waves emitted by the capacitor first increases, then decreases to zero, then |
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| increases again, and so on. At what frequencies does the capacitor not emit |
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| energy? Estimate the frequency at which the 1st and $kth$ emission maxima |
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| are observed. |
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| ### Solution |
| ### Solution |
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| The capacitor has two parallel plates with opposite charges $\pm Q$ and area S. When oscillating along the x-axis (parallel to the plates) with velocity $v(t)$, each plate is equivalent to a surface current: |
| The capacitor has two parallel plates with opposite charges $\pm Q$ and area S. When oscillating along the x-axis (parallel to the plates) with velocity $v(t)$, each plate is equivalent to a surface current: |
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| $K_+ = \sigma v(t)\,\hat{x}, \qquad K_- = -\sigma v(t)\,\hat{x}, \qquad \sigma = \frac{Q}{S}$. |
| $K_+ = \sigma v(t)\,\hat{x}, \qquad K_- = -\sigma v(t)\,\hat{x}, \qquad \sigma = \frac{Q}{S}$ |
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| These time-varying currents generate electromagnetic fields that propagate: the capacitor becomes a transmitting antenna. |
| These time-varying currents generate electromagnetic fields that propagate: the capacitor becomes a transmitting antenna. |
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| Each plate emits electromagnetic waves. For a distant observer in the direction perpendicular to the plates (z-axis), the waves emitted by the front and back plates travel different paths. The path difference is the separation between the plates, d. |
| Each plate emits electromagnetic waves. For a distant observer in the direction perpendicular to the plates (z-axis), the waves emitted by the front and back plates travel different paths. The path difference is the separation between the plates, d. |
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| If the wavelength is $\lambda = c/\nu$ |
| If the wavelength is |
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| $\lambda = c/\nu$ |
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| the phase difference between the two waves at the observation point is: |
| the phase difference between the two waves at the observation point is: |
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| $\Delta\phi = \frac{2\pi}{\lambda}\,d = \frac{2\pi\nu}{c}\,d$. |
| $\Delta\phi = \frac{2\pi}{\lambda}\,d = \frac{2\pi\nu}{c}\,d$ |
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| Constructive interference (intensity maximum): |
| Constructive interference (intensity maximum): |
| It occurs when the waves arrive in phase: |
| It occurs when the waves arrive in phase: |
| $ \Delta\phi = 2\pi k \quad\Rightarrow\quad \frac{2\pi\nu d}{c} = 2\pi k \quad\Rightarrow\quad$ |
| $ \Delta\phi = 2\pi k \quad\Rightarrow\quad \frac{2\pi\nu d}{c} = 2\pi k \quad\Rightarrow\quad$ |
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| $ \boxed{\nu_k' = \frac{c}{d}\,k},\quad k = 1,2,3,\dots$ | | $ \boxed{\nu_k' = \frac{c}{d}\,k},\quad k = 1,2,3,\dots$ |
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| These are the frequencies of the emission maxima. | | These are the frequencies of the emission maxima. |
| The first maximum $(k = 1) is \nu_1' = c/d$ | | The first maximum $(k = 1) is \nu_1' = c/d$ |
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| Destructive interference (zero intensity): | | Destructive interference (zero intensity): |
| It occurs when the waves arrive in opposite phase: | | It occurs when the waves arrive in opposite phase: |
| $ \Delta\phi = (2k+1)\pi \quad\Rightarrow\quad \frac{2\pi\nu d}{c} = (2k+1)\pi \quad\Rightarrow\quad$ | | $ \Delta\phi = (2k+1)\pi \quad\Rightarrow\quad \frac{2\pi\nu d}{c} = (2k+1)\pi \quad\Rightarrow\quad$ |
| $ \boxed{\nu_k'' = \frac{c}{d}\left(k + \frac{1}{2}\right)},\quad k = 0,1,2,\dots$ | | $ \boxed{\nu_k'' = \frac{c}{d}\left(k + \frac{1}{2}\right)},\quad k = 0,1,2,\dots$ |
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| At these frequencies, the contributions from the two plates cancel out and the capacitor emits no energy (in that main direction). That is why the average intensity drops to zero. | | At these frequencies, the contributions from the two plates cancel out and the capacitor emits no energy (in that main direction). That is why the average intensity drops to zero. |
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| As the frequency \nu increases from zero, the phase difference \Delta\phi grows. For very low frequencies, | | As the frequency \nu increases from zero, the phase difference \Delta\phi grows. For very low frequencies, |
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| $\Delta\phi \approx 0$ | | $\Delta\phi \approx 0$ |
| and there is reinforcement. When reaching | | and there is reinforcement. When reaching |
| $\Delta\phi = \pi $(first cancellation,$ k = 0)$ | | $\Delta\phi = \pi $(first cancellation,$ k = 0)$ |
| the intensity goes to zero. Then, upon reaching $ \Delta\phi = 2\pi (k = 1)$, | | the intensity goes to zero. Then, upon reaching $ \Delta\phi = 2\pi (k = 1)$, |
| it becomes maximum again, and so on periodically. The intensity oscillates between high values and zeros, giving rise to the observed alternation. | | it becomes maximum again, and so on periodically. The intensity oscillates between high values and zeros, giving rise to the observed alternation. |
