Geometrical Optics

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Problem Statement: Two prisms with the same angle but different indices of refraction are put together to form a parallel sided block of glass (see the figure). The index of the first prism is n1 = 1.48 and that of the second prism is n2 = 1.72. A laser beam is normally incident on the first prism. What angle will the emerging beam make with the incident beam?

Eqn:

Lens Law,
GEOMETRY

Solution:

It hits on the normal the first time so it keeps going straight down that normal after entering first index of refraction. It hits the inclined plane between the two indices at to the normal to that plane:
$$\phi=tan^{-1}(1/4)$$
and using lens' law it exits to the normal to the same plane:
$$\beta=sin^{-1}(\frac{n_1}{n_2}sin(\phi))$$
then it keeps going and hits the last one at an angle
$$\alpha=2\beta-\phi$$ (measuring clockwise from normal)
and leaves at an angle
$$\gamma=sin^{-1}(n_2sin(\alpha))$$

The geometry took forever but I got it wrong

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