Edit to “Statement”
en/12.2.11.md
+2 −1
| ### Statement | |||
| $12.2.11.$ | |||
| The drawing shows a flat glass plate with blackened ring sections. This | |||
| plate overlapped a parallel beam of monochromatic light with a wavelength | |||
| λ. It turned out that the blackened rings of the plate coincided with even | |||
| Fresnel zones for the axial point A. How did the light intensity change at this | |||
| point? | |||
| b. A parallel beam of monochromatic light was blocked by a plate in which the | |||
| @@ -11,7 +11,8 @@Statement | |||
| blackened ring sections were replaced by layers of a dielectric that changes the | |||
| phase of the passing wave by π. How did the light intensity change at point | |||
| A in this case? | |||
| − | [For problem $12.2.11$|720x224, 50%](../../img/12.2.11/1000125861.jpg) | ||
| + | |||
| + |  | ||
| ### Solution | |||
| Data: | |||
| Parallel beam of monochromatic light of wavelength $\lambda$ | |||
| Observation point A on the axis. | |||
| The plate has rings that coincide with the Fresnel zones. | |||
| It is considered that the plate covers the first m = 9 Fresnel zones. | |||
| Reference amplitude (without plate): | |||
| With the entire wavefront open, the amplitude at A is half of the first zone: | |||
| $A_{\text{free}} \approx \frac{A_1}{2}$ | |||
| and the corresponding intensity is | |||
| $ I_0 \propto \left(\frac{A_1}{2}\right)^2 = \frac{A_1^2}{4}$ | |||
| Case (a): Opaque plate on even zones | |||
| The plate has the even zones blackened out (blocked): 2, 4, 6, 8. | |||
| The odd zones remain open: 1, 3, 5, 7, 9 (five zones). | |||
| Since the amplitudes of the zones are | |||
| $approximately equal (A_1 \approx A_3 \approx A_5 \approx A_7 \approx A_9)$, the total amplitude at A is: | |||
| $A_a = 5 A_1$ | |||
| Intensity: | |||
| $I_a \propto (5 A_1)^2 = 25 A_1^2$ | |||
| Compared with the intensity without the plate: | |||
| $\frac{I_a}{I_0} = \frac{25 A_1^2}{A_1^2/4} = 25 \times 4 = 100$ | |||
| $\boxed{\text{The intensity increases by a factor of 100.}}$ | |||
| --- | |||
| Case (b): Phase plate (phase shift \pi) on even zones | |||
| The rings that were previously opaque are replaced by dielectric layers that add a phase shift of \pi to the wave passing through. | |||
| · The odd zones (1, 3, 5, 7, 9) pass with no phase change. | |||
| · The even zones (2, 4, 6, 8) would normally arrive with a phase shift of $\pi $due to the path difference, but the layer adds another \pi, so they end up in phase with the odd ones. | |||
| Now all zones (9 in total) contribute constructively: | |||
| $A_b = 9 A_1$ | |||
| Intensity: | |||
| $I_b \propto (9 A_1)^2 = 81 A_1^2$ | |||
| Compared with the intensity without the plate: | |||
| $\frac{I_b}{I_0} = \frac{81 A_1^2}{A_1^2/4} = 81 \times 4 = 324$ | |||
| $\boxed{\text{The intensity increases by a factor of 324.}}$ | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||
| unchanged lines 67 | |||
| ### Statement | ### Statement | ||
| $12.2.11.$ | $12.2.11.$ | ||
| The drawing shows a flat glass plate with blackened ring sections. This | The drawing shows a flat glass plate with blackened ring sections. This | ||
| plate overlapped a parallel beam of monochromatic light with a wavelength | plate overlapped a parallel beam of monochromatic light with a wavelength | ||
| λ. It turned out that the blackened rings of the plate coincided with even | λ. It turned out that the blackened rings of the plate coincided with even | ||
| Fresnel zones for the axial point A. How did the light intensity change at this | Fresnel zones for the axial point A. How did the light intensity change at this | ||
| point? | point? | ||
| b. A parallel beam of monochromatic light was blocked by a plate in which the | b. A parallel beam of monochromatic light was blocked by a plate in which the | ||
| @@ -11,7 +11,8 @@Statement | |||
| blackened ring sections were replaced by layers of a dielectric that changes the | blackened ring sections were replaced by layers of a dielectric that changes the | ||
| phase of the passing wave by π. How did the light intensity change at point | phase of the passing wave by π. How did the light intensity change at point | ||
| A in this case? | A in this case? | ||
| [For problem $12.2.11$|720x224, 50%](../../img/12.2.11/1000125861.jpg) |
|||
|  | |||
| ### Solution | ### Solution | ||
| Data: | Data: | ||
| Parallel beam of monochromatic light of wavelength $\lambda$ | Parallel beam of monochromatic light of wavelength $\lambda$ | ||
| Observation point A on the axis. | Observation point A on the axis. | ||
| The plate has rings that coincide with the Fresnel zones. | The plate has rings that coincide with the Fresnel zones. | ||
| It is considered that the plate covers the first m = 9 Fresnel zones. | It is considered that the plate covers the first m = 9 Fresnel zones. | ||
| Reference amplitude (without plate): | Reference amplitude (without plate): | ||
| With the entire wavefront open, the amplitude at A is half of the first zone: | With the entire wavefront open, the amplitude at A is half of the first zone: | ||
| $A_{\text{free}} \approx \frac{A_1}{2}$ | $A_{\text{free}} \approx \frac{A_1}{2}$ | ||
| and the corresponding intensity is | and the corresponding intensity is | ||
| $ I_0 \propto \left(\frac{A_1}{2}\right)^2 = \frac{A_1^2}{4}$ | $ I_0 \propto \left(\frac{A_1}{2}\right)^2 = \frac{A_1^2}{4}$ | ||
| Case (a): Opaque plate on even zones | Case (a): Opaque plate on even zones | ||
| The plate has the even zones blackened out (blocked): 2, 4, 6, 8. | The plate has the even zones blackened out (blocked): 2, 4, 6, 8. | ||
| The odd zones remain open: 1, 3, 5, 7, 9 (five zones). | The odd zones remain open: 1, 3, 5, 7, 9 (five zones). | ||
| Since the amplitudes of the zones are | Since the amplitudes of the zones are | ||
| $approximately equal (A_1 \approx A_3 \approx A_5 \approx A_7 \approx A_9)$, the total amplitude at A is: | $approximately equal (A_1 \approx A_3 \approx A_5 \approx A_7 \approx A_9)$, the total amplitude at A is: | ||
| $A_a = 5 A_1$ | $A_a = 5 A_1$ | ||
| Intensity: | Intensity: | ||
| $I_a \propto (5 A_1)^2 = 25 A_1^2$ | $I_a \propto (5 A_1)^2 = 25 A_1^2$ | ||
| Compared with the intensity without the plate: | Compared with the intensity without the plate: | ||
| $\frac{I_a}{I_0} = \frac{25 A_1^2}{A_1^2/4} = 25 \times 4 = 100$ | $\frac{I_a}{I_0} = \frac{25 A_1^2}{A_1^2/4} = 25 \times 4 = 100$ | ||
| $\boxed{\text{The intensity increases by a factor of 100.}}$ | $\boxed{\text{The intensity increases by a factor of 100.}}$ | ||
| --- | --- | ||
| Case (b): Phase plate (phase shift \pi) on even zones | Case (b): Phase plate (phase shift \pi) on even zones | ||
| The rings that were previously opaque are replaced by dielectric layers that add a phase shift of \pi to the wave passing through. | The rings that were previously opaque are replaced by dielectric layers that add a phase shift of \pi to the wave passing through. | ||
| · The odd zones (1, 3, 5, 7, 9) pass with no phase change. | · The odd zones (1, 3, 5, 7, 9) pass with no phase change. | ||
| · The even zones (2, 4, 6, 8) would normally arrive with a phase shift of $\pi $due to the path difference, but the layer adds another \pi, so they end up in phase with the odd ones. | · The even zones (2, 4, 6, 8) would normally arrive with a phase shift of $\pi $due to the path difference, but the layer adds another \pi, so they end up in phase with the odd ones. | ||
| Now all zones (9 in total) contribute constructively: | Now all zones (9 in total) contribute constructively: | ||
| $A_b = 9 A_1$ | $A_b = 9 A_1$ | ||
| Intensity: | Intensity: | ||
| $I_b \propto (9 A_1)^2 = 81 A_1^2$ | $I_b \propto (9 A_1)^2 = 81 A_1^2$ | ||
| Compared with the intensity without the plate: | Compared with the intensity without the plate: | ||
| $\frac{I_b}{I_0} = \frac{81 A_1^2}{A_1^2/4} = 81 \times 4 = 324$ | $\frac{I_b}{I_0} = \frac{81 A_1^2}{A_1^2/4} = 81 \times 4 = 324$ | ||
| $\boxed{\text{The intensity increases by a factor of 324.}}$ | $\boxed{\text{The intensity increases by a factor of 324.}}$ | ||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | [Insert a concise answer or boxed result] | ||
| unchanged lines 67 | |||