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leright
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I am looking for a mathematical derivation for the idea that symmetry of the light beam path through a prism gives rise to a minimum deviation.
Thanks.
Thanks.
Integral said:May I ask, deviation of what?
jtbell said:I think he's referring to this:
http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/prism.html#c2
leright, are you asking why it is that the path shown in the diagram (the one in which section AB is parallel to the base of the prism) must be the one with minimum deviation ([itex]\delta[/itex])?
I think the only rigorous way to do it is to find an equation for [itex]\delta[/itex] in terms of the incident angle (or some other convenient angle), then find the minimum via the usual calculus technique: take the derivative and set it equal to zero. See for example
http://scienceworld.wolfram.com/physics/Prism.html
Some books use a "symmetry argument" which goes something like this: Suppose for the sake of argument that the minimum deviation occurs when the entrance and exit angles are not equal. In a ray diagram, you can always reverse the direction of a light ray and get another valid light ray. In this case, reversing the ray switches the values of the entrance and exit angles. So there are two different values for the entrance angle that give minimum deviation. But if there's only one minimum, this can't be true. Therefore the initial supposition must be false, and the entrance and exit angles must be equal at minimum deviation.
Of couse, in order to make the assumption that I've put in boldface above, you have to know something in advance about how the deviation angle varies with entrance angle, for example by measuring it experimentally and making a graph of deviation angle versus entrance angle. Otherwise, how do you know the graph isn't actually W-shaped, with two minima?
The minimum deviation angle for a prism is the angle at which the incident light ray is deviated by the prism at the minimum angle possible.
The minimum deviation angle for a prism can be calculated using the formula: δm = (A + D)/2 - i, where δm is the minimum deviation angle, A is the angle of the prism, D is the angle of the prism's apex, and i is the angle of incidence.
The minimum deviation angle for a prism is significant because it determines the angle at which the light will be refracted and the amount of dispersion that will occur. It is also used in various optical devices, such as spectrometers, to accurately measure the refractive index of different materials.
The shape of the prism does not affect the minimum deviation angle, as long as the angle of the prism and the angle of the prism's apex remain constant. However, the shape of the prism can affect the amount of dispersion that occurs.
Yes, the minimum deviation angle for a prism can be changed by altering the angle of incidence or by changing the refractive index of the material the prism is made of. It can also be changed by adjusting the angle of the prism or the angle of its apex.