Mathematical derivation for the minimum deviation angle for a prism

In summary: I'll give that a try. In summary, he is looking for a mathematical derivation for the symmetry of the light beam path through a prism giving rise to a minimum deviation.
  • #1
leright
1,318
19
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.
 
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  • #2
and I just put this in the WRONG Forum. Sorry about that.
 
  • #3
I think this forum is fine for such a question.

May I ask, deviation of what?
 
  • #5
Integral said:
May I ask, deviation of what?

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?
 
Last edited:
  • #6
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?

Yeah, I figured the best way to do it would be to find the derivative of the deviation angle function, which is a function of the angle of incidence and the apex angle and then set it equal to zero, and then solve for the angle of incidence.

Thanks a lot.
 

1. What is the minimum deviation angle for a prism?

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.

2. How is the minimum deviation angle for a prism calculated?

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.

3. What is the significance of the minimum deviation angle for a prism?

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.

4. How does the shape of the prism affect the minimum deviation angle?

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.

5. Can the minimum deviation angle for a prism be changed?

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.

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