Question (a) In a medium with refractive index n, light travels at speed c/n. Use this fact to show that the total time it takes light to travel from 1 to 2 along a path given by =(), as shown in the figure, is given by TL 11+(??n(2)d, dia where I' 15 marks] (b) Hence show that the path that minimises the time taken satisfies dz A?n(z)2 - 1, dz where A is a constant 5 marks (c) Give an expression for A for a path that just grazes the ground (= = 0), as shown below. [3 marks (d) Assume that the refractive index at a height : above the ground is given by (2) = n(1 + az), where Tg and a are constants. Assume also that az <1. Show that for the grazing path, 202. (6 marks (@) Hence show that, for an observer at height h, the grazing ray appears to come from a point on the ground a distance d=Vh/2a away. (6 marks)
the are length of the path taken is:ds=sqrt((dx)^(2)+(dz)^(2))=sqrt(1+((dx)/(dz))^(2))dzTime taken by light is :" Time, "T=(" distance ")/(" speed ")small time taken by light in kavelling distance ds :dT=(ds)/(c//n)=(dsn)/(c){:[dT=(dsn(z))/(c)=(1)/(c)sqrt(1+((dx)/(dz))^(2)dzn)],[dT=(1)/(c)sqrt(1+(x^('))^(2))n(z)dz]:}Integrating from z_(1) to z_(2)==>T=(1)/(c)int_(z_(1))^(z_(2))sqrt(1+(x^(1))^(2))n(z)dzatb) -:the lagrangian of above functional is :-L=(1)/(c)sqrt(1+(x^('))^(2))n(2)euler-logrange equation :-=>quad(d)/(dz)((del L)/(delx^(')))=(del L)/(del x){:[(del L)/(delx^('))=(1)/(c)(1)/(2)[1+(x^('))^(2)]^('2)2x^(')n(2)],[(del L)/(delx^('))=(x^(')n(2))/(csqrt(1+(x^('))^(2)))],[d","(del L)/(del x)=0],[:.(d)/(d2)((del L)/(delx^(')))=(del L)/(del x)],[=>(d)/(d2)((x^(')n(2))/(csqrt(1+(x^('))^(2))))=0],[=>(x^(')n(2))/(sqrt(1+(x^('))^(2)))=" constant "=A" (2ay) "],[=>x^(')n(2)=],[=>Asqrt(1+(x^('))^(2))]:}{:[x^('2)n^(2)(2)=A^(2)(1+(x^('))^(2))],[x^('2)n^(2)(2)-A^(2)x^('2)=A^(2)],[x^('2)[n^(2)(2)-A^(2)]=A^(2)],[x^('2)=(A^(2))/(n^(2)(2)-A ... See the full answer