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Light falls perpendicular on one side of a glass prism with refractive index n. The light is totally reflected on the right side.

1.1: Determine the angle of incidence using the figure below.

1.2: What's the smallest value n can have?

1.3: When the prism is immersed into a liquid with refractive index 1.15 there is still total internal reflection. But when it's immersed into water with refractive index 1.33 then the total internal reflection disappears. Give the boundaries of the possible values for n.

enter image description here

I'm new to optics and found this one in some textbook I found in the library.

Here were my ideas:

1.1: Wouldn't the angle of incidence would just be $45°$ then?

1.2: I looked up some formulas on wiki and found this one:

$n_1\sin{\alpha_1}=n_2\sin{\alpha_2}$, meaning I would need to plugin $45°$ into $\alpha_1$ and $\alpha_2$, but that doesn't get me in anywhere.

1.3: I have no idea how to approach this one.

Can anyone more knowledgeable in optics help me out here?

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Rafa Fafa
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1 Answers1

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As long as this is not homework:

1.1: Yes.

1.2: If you are taking a class, then you had better learn this equation well, as it is Snell's Law, or the law of refraction. $\alpha_1$ is indeed the angle of incidence from the surface normal. $\alpha_2$ is the angle of refraction from the surface normal. In this case, $\alpha_1 = 45^{\circ}$. You find $\alpha_2$ given $n_1$ and $n_2$:

$$\sin{\alpha_2} = \frac{n_1}{n_2} \sin{\alpha_1} $$

Note that, if $n_1 \le n_2$, then there will always be light refracted, no matter what the angle of incidence is. However, if $n_1 \gt n_2$, then there will be some critical angle $\alpha_c$ at which $\sin{\alpha_2} =1$, beyond which the light does not refract into the surrounding medium but instead reflects back.

In the case pictured, the light reflects back at an incidence of $\alpha_1=45^{\circ}$, $n_2=1$. Thus, the smallest value $n_1$ can have is whatever makes $\sin{\alpha_2}=1$, or $n_1 = 1/\sin{\alpha_1} = \sqrt{2} \approx 1.414$.

1.3: You can get the minimum value of $n_1$ the same way I did in 1.2 above for the fluid at $n_2=1.15$. The maximum value of $n_1$ results using the same condition at $n_2=1.33$.

Ron Gordon
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