- #1
YogiBear
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Mirage: we consider the x-y plane describing vertical y and horizontal x directions, with an inhomogeneous index of refraction n(y). In this case, using calculus of variations, Fermat’s principle for the trajectory of a ray of light may be re-written as n(y)/√1+(dy/dx)^2 = A. Where A is a real constant, to be determined by boundary conditions. We consider the index of refraction to be n(y) = n0(1 + αy) where n0 and α are real parameters.
Show that the trajectory y(x) of a ray of light is given by
y = − 1/α + A/n0α*cosh [ (n0α/A) * (x − x0) ]
Limits for integration are not given.
What i have done so far: Well i separated variables and then used f(x) = cosh^-1(x) f'(x) = 1/(x^2 -a^2)^1/2 to get pretty close to the solution. However i don't see where n0α/A comes from within the cosh bracket. Also I used x and x0, y and 0, as limits for integration. Huge thanks in advance
Show that the trajectory y(x) of a ray of light is given by
y = − 1/α + A/n0α*cosh [ (n0α/A) * (x − x0) ]
Limits for integration are not given.
What i have done so far: Well i separated variables and then used f(x) = cosh^-1(x) f'(x) = 1/(x^2 -a^2)^1/2 to get pretty close to the solution. However i don't see where n0α/A comes from within the cosh bracket. Also I used x and x0, y and 0, as limits for integration. Huge thanks in advance