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THE ACTUAL PROBLEM:
A vessel of depth H is filled with a non-homogenous liquid whose refractive index varies with y as u=(2 -(y/H)), where y is measured from bottom of the vessel. Find the apparent depth as seen by an observer from above?
(Paraxial approximation is allowed)
RELEVANT EQUATIONS:
We know in paraxial approximation
u1/x=u2/y
Where u1 is refractive index of medium 1, x is object distance from surface, u2 is refractive index of surrounding medium , y2 is apparent depth.
MY ATTEMPT:
I took a differential strip of thickness dy at a height of y from bottom.
its refractive index is
(2-(y/H)-(dy/H)) and refractive index of element just below it is (2-(y/H))
Now,
Lets say that the image of the bottom of the vessel formed by refraction from all strips below this height y be at a distance x from the bottom.
Therefore, Its distance from the strip is y-x.
So using the formulae i mentioned.
(2-y/H)/(y-x)=
[(2-(y/H)-(dy/H))]/[y-x-dx)]
So ydx/H -2dx=-ydy/H+xdy/H.
After this am struck cause I cannot integrate.
I know other ways of solving this problem(mentioned in my textbook), but I want to know what exactly am I doing wrong here?
Any help/inputs will be really appreciated.
A vessel of depth H is filled with a non-homogenous liquid whose refractive index varies with y as u=(2 -(y/H)), where y is measured from bottom of the vessel. Find the apparent depth as seen by an observer from above?
(Paraxial approximation is allowed)
RELEVANT EQUATIONS:
We know in paraxial approximation
u1/x=u2/y
Where u1 is refractive index of medium 1, x is object distance from surface, u2 is refractive index of surrounding medium , y2 is apparent depth.
MY ATTEMPT:
I took a differential strip of thickness dy at a height of y from bottom.
its refractive index is
(2-(y/H)-(dy/H)) and refractive index of element just below it is (2-(y/H))
Now,
Lets say that the image of the bottom of the vessel formed by refraction from all strips below this height y be at a distance x from the bottom.
Therefore, Its distance from the strip is y-x.
So using the formulae i mentioned.
(2-y/H)/(y-x)=
[(2-(y/H)-(dy/H))]/[y-x-dx)]
So ydx/H -2dx=-ydy/H+xdy/H.
After this am struck cause I cannot integrate.
I know other ways of solving this problem(mentioned in my textbook), but I want to know what exactly am I doing wrong here?
Any help/inputs will be really appreciated.