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Problem 6.97:A ray of light initially travelling horizontally,  is incident on the reflecting surface of a hemispherical object and gets deflected vertically upward as shown in the diagram. Find the value of height, $H$ at which the ray hit the object.

Solution:

The problem can be solved using laws of reflection which says that the angle of incidence is equal to angle of reflection.

Angle between incident and reflected rays will be always equal to twice the angle of incidence.

Here incident ray is horizontal and reflected ray is vertical.

Hence angle between incident and reflected rays will be $90^o$.

So angle of incidence will be $\frac{90^o}{2}\:=\:45^o$.

From the second diagram, angle of incidence is $\theta$.

Also, $tan\theta\:=\:\frac{H}{R}$

Therefore, $tan\:45^o\:=\:\frac{H}{R}$.

But $tan\:45^o\:=\:1$.

Then, $\frac{H}{R}\:=\:1$.

or, $H\:=\:R$.

So the horizontal ray can get reflected vertically upward if it hits the hemispherical surface at a height $H\:=\:R$ from the base.