The Fermat principle says that the light travels from a point to a point along the...

 

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The Fermat principle says that the light travels from a point to a point along the path providing the minimal traveling time. Suppose we have two different media separated by the surface of the unit sphere x + y + 2 = 1 so that the speed of light is c inside the sphere and c outside the sphere. The light travels from a point A (x1, y1, 21) inside the sphere to a point A2 (22, 32, 22) outside it and crosses the surface of the sphere at a point B(x, y, z). The light goes from A to B along a straight line then goes from B to A2 along a different line due to refraction, see the picture. B C2 C1 (a) Find how long does it take to the light to get from A to A2. Your answer will be a function T(x, y, z) depending on the coordinates of B (b) Write (but do not solve) the Lagrange equations for minimizing T(x, y, z) where (x, y, z) lies on the unit sphere. (c) Suppose we have a fixed point A(ZA, YA, ZA) in space. Suppose D is the function which associates to a point X in space the distance |AX|. Show that if X + A, then VD(X) that is VD(X) is the unit vector in the direction of AX. (d) Deduce from the Largange equations and the previous part that the vector = AX |AX| AB AB CAB CAB| + 1 is parallel to the normal to the sphere at B. (e) Denote by n the normal to the sphere at B. Denote by a the acute angle between AB and n. Likewise, let a2 be the actute angle between AB and n. Find the relationship between sin 0, sin a2, c and c2. By doing so, you will re-discover one of the most famous physical laws called Snell's law The Fermat principle says that the light travels from a point to a point along the path providing the minimal traveling time. Suppose we have two different media separated by the surface of the unit sphere x + y + 2 = 1 so that the speed of light is c inside the sphere and c outside the sphere. The light travels from a point A (x1, y1, 21) inside the sphere to a point A2 (22, 32, 22) outside it and crosses the surface of the sphere at a point B(x, y, z). The light goes from A to B along a straight line then goes from B to A2 along a different line due to refraction, see the picture. B C2 C1 (a) Find how long does it take to the light to get from A to A2. Your answer will be a function T(x, y, z) depending on the coordinates of B (b) Write (but do not solve) the Lagrange equations for minimizing T(x, y, z) where (x, y, z) lies on the unit sphere. (c) Suppose we have a fixed point A(ZA, YA, ZA) in space. Suppose D is the function which associates to a point X in space the distance |AX|. Show that if X + A, then VD(X) that is VD(X) is the unit vector in the direction of AX. (d) Deduce from the Largange equations and the previous part that the vector = AX |AX| AB AB CAB CAB| + 1 is parallel to the normal to the sphere at B. (e) Denote by n the normal to the sphere at B. Denote by a the acute angle between AB and n. Likewise, let a2 be the actute angle between AB and n. Find the relationship between sin 0, sin a2, c and c2. By doing so, you will re-discover one of the most famous physical laws called Snell's law