Equivalence principle aka Babinet's Principle
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
Solution
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Exercise:
Suppose that you have a beam of light that propagates along the z-axis of a coordinate system along the +z-direction. Ase that at the plane z d the ensity outside a circle of radius R centered at x y is immeasurably small. Also ase that at the plane z we can place an aperture with an arbitrary shape but that blocks all light greater than a distance a from the z-axis. Let Ix y d be the ensity pattern measured at z d with the aperture in place. Suppose now that we replace the aperture with its complement: If we let txy be the transmission function of the original aperture we now exchange this aperture for a new one with a new aperture with transmission function tcx y -tx y. In other words all places where the original aperture blocked light will now allow light to pass and vice versa. We now measure a new ensity pattern I_cx y d at the detector plane. Show using the principle of superposition that Ix y d I_cx y d for x_ + y_ R_.
Solution:
In physics Babinet’s Principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam ensity. It was formulated in the s by French physicist Jacques Babinet. Ase B is the original diffracting body and B' is its complement i.e. a body that is transparent where B is opaque and opaque where B is transparent. We can think that the superposition of an object and its complement is equivalent to having empty space thus the of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In this case we define Uxyd as the field from the original aperture and U_cxyd as the field from the complement. Outside the circular of radius R we know that according to the principle of superposition U + U_c . Therefore U_c -U i.e. U_c and U have opposite phases. Moreover we can write I_cxyd |U_c|^ |U|^ Ixyd.
Suppose that you have a beam of light that propagates along the z-axis of a coordinate system along the +z-direction. Ase that at the plane z d the ensity outside a circle of radius R centered at x y is immeasurably small. Also ase that at the plane z we can place an aperture with an arbitrary shape but that blocks all light greater than a distance a from the z-axis. Let Ix y d be the ensity pattern measured at z d with the aperture in place. Suppose now that we replace the aperture with its complement: If we let txy be the transmission function of the original aperture we now exchange this aperture for a new one with a new aperture with transmission function tcx y -tx y. In other words all places where the original aperture blocked light will now allow light to pass and vice versa. We now measure a new ensity pattern I_cx y d at the detector plane. Show using the principle of superposition that Ix y d I_cx y d for x_ + y_ R_.
Solution:
In physics Babinet’s Principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam ensity. It was formulated in the s by French physicist Jacques Babinet. Ase B is the original diffracting body and B' is its complement i.e. a body that is transparent where B is opaque and opaque where B is transparent. We can think that the superposition of an object and its complement is equivalent to having empty space thus the of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In this case we define Uxyd as the field from the original aperture and U_cxyd as the field from the complement. Outside the circular of radius R we know that according to the principle of superposition U + U_c . Therefore U_c -U i.e. U_c and U have opposite phases. Moreover we can write I_cxyd |U_c|^ |U|^ Ixyd.
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Exercise:
Suppose that you have a beam of light that propagates along the z-axis of a coordinate system along the +z-direction. Ase that at the plane z d the ensity outside a circle of radius R centered at x y is immeasurably small. Also ase that at the plane z we can place an aperture with an arbitrary shape but that blocks all light greater than a distance a from the z-axis. Let Ix y d be the ensity pattern measured at z d with the aperture in place. Suppose now that we replace the aperture with its complement: If we let txy be the transmission function of the original aperture we now exchange this aperture for a new one with a new aperture with transmission function tcx y -tx y. In other words all places where the original aperture blocked light will now allow light to pass and vice versa. We now measure a new ensity pattern I_cx y d at the detector plane. Show using the principle of superposition that Ix y d I_cx y d for x_ + y_ R_.
Solution:
In physics Babinet’s Principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam ensity. It was formulated in the s by French physicist Jacques Babinet. Ase B is the original diffracting body and B' is its complement i.e. a body that is transparent where B is opaque and opaque where B is transparent. We can think that the superposition of an object and its complement is equivalent to having empty space thus the of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In this case we define Uxyd as the field from the original aperture and U_cxyd as the field from the complement. Outside the circular of radius R we know that according to the principle of superposition U + U_c . Therefore U_c -U i.e. U_c and U have opposite phases. Moreover we can write I_cxyd |U_c|^ |U|^ Ixyd.
Suppose that you have a beam of light that propagates along the z-axis of a coordinate system along the +z-direction. Ase that at the plane z d the ensity outside a circle of radius R centered at x y is immeasurably small. Also ase that at the plane z we can place an aperture with an arbitrary shape but that blocks all light greater than a distance a from the z-axis. Let Ix y d be the ensity pattern measured at z d with the aperture in place. Suppose now that we replace the aperture with its complement: If we let txy be the transmission function of the original aperture we now exchange this aperture for a new one with a new aperture with transmission function tcx y -tx y. In other words all places where the original aperture blocked light will now allow light to pass and vice versa. We now measure a new ensity pattern I_cx y d at the detector plane. Show using the principle of superposition that Ix y d I_cx y d for x_ + y_ R_.
Solution:
In physics Babinet’s Principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam ensity. It was formulated in the s by French physicist Jacques Babinet. Ase B is the original diffracting body and B' is its complement i.e. a body that is transparent where B is opaque and opaque where B is transparent. We can think that the superposition of an object and its complement is equivalent to having empty space thus the of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In this case we define Uxyd as the field from the original aperture and U_cxyd as the field from the complement. Outside the circular of radius R we know that according to the principle of superposition U + U_c . Therefore U_c -U i.e. U_c and U have opposite phases. Moreover we can write I_cxyd |U_c|^ |U|^ Ixyd.
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