consider a light ray in medium 1 with index of refraction n1=c/v1 striking a pla

Question

consider a light ray in medium 1 with index of refraction n1=c/v1 striking a planar surface at angle?1 with respect to the planar normal. the planar surface marks the boundray into meduim 2 with an index of refraction n2 = c/v2. the light ray in meduim 2 forms an angle of?2 with respect to the planar normal in meduim 2. choose an arbitrary point A on the light ray in meduim 1 and an arbitry point B on the light ray in medium 2.

a. write an equation for the propagation time t1 from point A on the light ray in meduim 1 to point P where the light ray strikes the planar boundary.

b.write an equation for the propagation time t2 from point P where the light ray strikes the planar boundary to point Bon the light ray in medium 2.

c. Based on above equations, what constraning relationship must exist if the propagation time from point A to point B is to be minimized?

Explanation / Answer

let the distance of the point on the line be l

t1=l/v1

t2=l/v2

T=t1+t2

also c*sintheta1/v1=c*sintheta2/v2

hence v1=v2*sintheta1/sintheta2

put this in the T eq

and diss wrt v2

T=l(sintheta2/sintheta1*v2 +1/v2)

0=sintheta2/sintheta1+1

sintheta2= -sintheta 1

implies the points must be in a straight line,hence proof of fermets theorm

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