Using fermat's principle derive the law of refraction
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The law of reflection of light states that the angle at which a ray is incident on a reflective surface is equal to the angle at which it is reflected from the surface. In other words, in the following diagram (light travels from A to O to B), θi=θr.
reflection
Let’s try to derive this law from Fermat’s principle. The total distance that the ray of light has to travel from A to B with one reflection at O is
L=
¯
AO
+
¯
OB
=
√
a2+x2
+
√
b2+(l−x)2
Now since the speed of light is constant, minimizing the time taken by light to take the path AOB is equivalent to minimizing the path itself. To do this, we set
dL
dx
=0.
dL
dx
=0
⟹
d
dx
(
√
a2+x2
+
√
b2+(l−x)2
)=0
⟹
2x
2
√
a2+x2
+
2(l−x)(−1)
2
√
b2+(l−x)2
=0
⟹
x
√
a2+x2
=
(l−x)
√
b2+(l−x)2
⟹sinθi=sinθr
⟹θi=θr◼
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