Question

for a prism prove relation i + e=a+d where the symbols have their usual meaning ​

Answer 1

If you follow the incident light ray in, it gets bent by an angle θ1=i−r1θ1=i−r1.
If you follow the light ray where it leaves the glass, it gets bent again by an angle θ2=e−r2θ2=e−r2,
so the total deviation is:

D=θ1+θ2
=i+e−(r1+r2)

For the next step look at the triangle formed by the top of the prism and the light ray, and note that the internal angles must add up to 180°. So:

A+(90−r1)+(90−r2)=180

and a quick rearrangement gives:

A=r1+r2A
=r1+r2

Now substitute for r1+r2r1+r2 in our first equation and we get:

D=i+e−AD
=i+e−A

or:

D+A=i+e
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