The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular spread of visible light passing through a prism of apex angle 60.0° if the angle of incidence is 52.0°? See figure below.

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+2 votes

Best answer

This problem can be solved using a combination of snell's law and the geometry of the triangle.

We label the angles of incidence and refraction at the surface of entry with

(90° − *θ*_{2}) + Φ + (90° − *θ*_{3}) = 180°,

so

*θ*_{3} = Φ − *θ*_{2}.

Using snell's law we can solve for *θ*_{2} first, and then *θ*_{3} band *θ*_{4}.

To solve for *θ*_{2}, we use the following formula given by snell's law:

**sin θ_{1} = n sin θ_{2}** (*since index of refraction of air is 1, we are left with only

solving for ** θ_{2,}** we must solve it twice, once for the value of n (violet light) and then n (red light).

*θ*_{2} = inverse sin ( sin *θ*_{1 } / n)

n (violet) = 1.66

n (red) = 1.62

and so

*θ*_{2 (violet)} = **inverse sin ( sin θ_{1 }**

*θ*_{2 (red)} = **inverse sin ( sin θ_{1 } / 1.62)**

Now we can solve for *θ*_{3}

*θ*_{3} = Φ − *θ*_{2}

Φ = 60

so

** θ_{3 (violet)} = **60 -

** θ_{3 (red)} = **60 -

To solve for *θ*_{4 }** ,**

Use snell's law once again:

**n sin θ_{3} = sin θ_{4}**

so

*θ*_{4}* *= inverse sin( **n sin θ_{3}**

taking into account the two different values of n, we get:

θ4 (violet) = inverse sin( 1.66 * sin θ3 (violet) )

θ4 (red) = inverse sin( 1.62 * sin θ3 (red) )

And so the angular spread is:

*θ*_{4}* ***= ***θ*_{4}* *** (violet) - ***θ*_{4}* *** (red)**

...