Aragos spot
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
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Exercise:
Consider a po source of monochromatic light at a distance s from a po detector. In between the source and detector is an opaque thin circular disk with radius R with R gg lambda positioned such that its center is on the midpo of the line connecting the source and detector. The disk is oriented such that the surface is normal to this line. Show using Huygens’ Principle that a non-zero ensity of light reaches the detector and that the part of the phase difference that deps on the positions of the source detector and disk is exactly the phase acquired by glqq least pathgrqq light paths as would be predicted by Fermat’s principle. H: To analytically solve the problem the following approximation beyond Huygens’ Principle is required. For large values of a we can write: _a^inftyfrace^ixxddxapprox fraciae^ia
Solution:
According to Huygen's principle every po on the perimeter of the disc and beyond radiates a spherical wave. In this specific case let rsqrtx^+y^ be the distance from the center of the opaque disk s the distance between the source and the disk equal to the distance from disk to detector and theta the angle from the vertical direction. We can thus write in cylindrical coordinates: UP &propto A__R^inftyrddr_^piddthetafrace^iksqrts^+r^sqrts^+r^frace^iksqrts^+r^sqrts^+r^ pi A_ _R^inftyfracre^iksqrts^+r^s^+r^ddr Now using the substitution uksqrts^+r^ and thus ddufrackrsqrts^+r^ddr we obtain UP &propto pi A_ _ksqrts^+R^^inftyfracue^iuddu For ksqrts^+R^ gg true in view of the requirement Rgg lambda the amplitude UP simplifies to UPpropto frac-ipi A_ksqrts^+R^e^iksqrts^+R^neq Therefore a finite Amplitude UP can be measured/imaged in the center of the shadow area the Arago Spot. Following Fermat's principle the minimum path is sqrts^+R^. The light travelling this optical path accumulates a phase difference of phi e^isqrts^+R^fracpilambda e^iksqrts^+R^ quod erat demonstrandum.
Consider a po source of monochromatic light at a distance s from a po detector. In between the source and detector is an opaque thin circular disk with radius R with R gg lambda positioned such that its center is on the midpo of the line connecting the source and detector. The disk is oriented such that the surface is normal to this line. Show using Huygens’ Principle that a non-zero ensity of light reaches the detector and that the part of the phase difference that deps on the positions of the source detector and disk is exactly the phase acquired by glqq least pathgrqq light paths as would be predicted by Fermat’s principle. H: To analytically solve the problem the following approximation beyond Huygens’ Principle is required. For large values of a we can write: _a^inftyfrace^ixxddxapprox fraciae^ia
Solution:
According to Huygen's principle every po on the perimeter of the disc and beyond radiates a spherical wave. In this specific case let rsqrtx^+y^ be the distance from the center of the opaque disk s the distance between the source and the disk equal to the distance from disk to detector and theta the angle from the vertical direction. We can thus write in cylindrical coordinates: UP &propto A__R^inftyrddr_^piddthetafrace^iksqrts^+r^sqrts^+r^frace^iksqrts^+r^sqrts^+r^ pi A_ _R^inftyfracre^iksqrts^+r^s^+r^ddr Now using the substitution uksqrts^+r^ and thus ddufrackrsqrts^+r^ddr we obtain UP &propto pi A_ _ksqrts^+R^^inftyfracue^iuddu For ksqrts^+R^ gg true in view of the requirement Rgg lambda the amplitude UP simplifies to UPpropto frac-ipi A_ksqrts^+R^e^iksqrts^+R^neq Therefore a finite Amplitude UP can be measured/imaged in the center of the shadow area the Arago Spot. Following Fermat's principle the minimum path is sqrts^+R^. The light travelling this optical path accumulates a phase difference of phi e^isqrts^+R^fracpilambda e^iksqrts^+R^ quod erat demonstrandum.
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Exercise:
Consider a po source of monochromatic light at a distance s from a po detector. In between the source and detector is an opaque thin circular disk with radius R with R gg lambda positioned such that its center is on the midpo of the line connecting the source and detector. The disk is oriented such that the surface is normal to this line. Show using Huygens’ Principle that a non-zero ensity of light reaches the detector and that the part of the phase difference that deps on the positions of the source detector and disk is exactly the phase acquired by glqq least pathgrqq light paths as would be predicted by Fermat’s principle. H: To analytically solve the problem the following approximation beyond Huygens’ Principle is required. For large values of a we can write: _a^inftyfrace^ixxddxapprox fraciae^ia
Solution:
According to Huygen's principle every po on the perimeter of the disc and beyond radiates a spherical wave. In this specific case let rsqrtx^+y^ be the distance from the center of the opaque disk s the distance between the source and the disk equal to the distance from disk to detector and theta the angle from the vertical direction. We can thus write in cylindrical coordinates: UP &propto A__R^inftyrddr_^piddthetafrace^iksqrts^+r^sqrts^+r^frace^iksqrts^+r^sqrts^+r^ pi A_ _R^inftyfracre^iksqrts^+r^s^+r^ddr Now using the substitution uksqrts^+r^ and thus ddufrackrsqrts^+r^ddr we obtain UP &propto pi A_ _ksqrts^+R^^inftyfracue^iuddu For ksqrts^+R^ gg true in view of the requirement Rgg lambda the amplitude UP simplifies to UPpropto frac-ipi A_ksqrts^+R^e^iksqrts^+R^neq Therefore a finite Amplitude UP can be measured/imaged in the center of the shadow area the Arago Spot. Following Fermat's principle the minimum path is sqrts^+R^. The light travelling this optical path accumulates a phase difference of phi e^isqrts^+R^fracpilambda e^iksqrts^+R^ quod erat demonstrandum.
Consider a po source of monochromatic light at a distance s from a po detector. In between the source and detector is an opaque thin circular disk with radius R with R gg lambda positioned such that its center is on the midpo of the line connecting the source and detector. The disk is oriented such that the surface is normal to this line. Show using Huygens’ Principle that a non-zero ensity of light reaches the detector and that the part of the phase difference that deps on the positions of the source detector and disk is exactly the phase acquired by glqq least pathgrqq light paths as would be predicted by Fermat’s principle. H: To analytically solve the problem the following approximation beyond Huygens’ Principle is required. For large values of a we can write: _a^inftyfrace^ixxddxapprox fraciae^ia
Solution:
According to Huygen's principle every po on the perimeter of the disc and beyond radiates a spherical wave. In this specific case let rsqrtx^+y^ be the distance from the center of the opaque disk s the distance between the source and the disk equal to the distance from disk to detector and theta the angle from the vertical direction. We can thus write in cylindrical coordinates: UP &propto A__R^inftyrddr_^piddthetafrace^iksqrts^+r^sqrts^+r^frace^iksqrts^+r^sqrts^+r^ pi A_ _R^inftyfracre^iksqrts^+r^s^+r^ddr Now using the substitution uksqrts^+r^ and thus ddufrackrsqrts^+r^ddr we obtain UP &propto pi A_ _ksqrts^+R^^inftyfracue^iuddu For ksqrts^+R^ gg true in view of the requirement Rgg lambda the amplitude UP simplifies to UPpropto frac-ipi A_ksqrts^+R^e^iksqrts^+R^neq Therefore a finite Amplitude UP can be measured/imaged in the center of the shadow area the Arago Spot. Following Fermat's principle the minimum path is sqrts^+R^. The light travelling this optical path accumulates a phase difference of phi e^isqrts^+R^fracpilambda e^iksqrts^+R^ quod erat demonstrandum.
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