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What fraction of light reaches one of the ends of a long glass cylinder if there is a point source emitting monochromatic light isotropically at the midpoint of the cylinder's axis? Absorption in the glass can be considered negligible.

I'm having difficulty gaining any sort of tractable solution. If we let theta be the angle made with respect to the axis normal to the surface of the cylinder (so that it is orthogonal to the cylinder's axis), then for angles greater than $\theta_c = 41.8$ degrees, total internal refraction will occur, so that all light will reach one of the ends of the cylinder. This is given by $arcsin(\frac{n_2}{n_1})$, where $n_2 = 1.0$ and $n_1 = 1.5$

Below the critical angle, some light will be reflected, and some refracted. The reflected light will be reflected at the same angle which it was incident. For this reason, it will be partially refracted every time it hits the cylinder's surface. As the cylinder was assumed to be long, it's safe to say it will undergo a fairly large number of reflections/refractions, losing some of intensity on the reflected light each time. Because of this, anything below the critical angle will be completely refracted by the time it reaches the end of the cylinder.

Anything contained within the cone of radius $R$ and height $Rtan(\theta_c)$ will make it to the end (here, I am calling the radius of the cylinder $R$).

The volume of this cone is $\pi R^2 \frac{h}{3} = \pi \frac{R^3}{3} tan(\theta_c) $, and the volume of the cylinder that contains it is $\pi R^2 h = \pi R^3 tan(\theta_c)$, making the ratio $\frac{1}{3}$, regardless of the critical angle.

Does my line of reasoning follow here, or is there a better way? Thank you in advance!

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I would be surprised if the fraction did not depend on the critical angle.
Think of what would happen if the critical angle was very small and very large.

You need to consider to consider the areas, at right angles to the direction of travel of the light, through which the light would pass through rather than volumes.

It will help you if you know the area of a spherical cap?

Update

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind Nov 21 '16 at 14:42

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