Light Source in Cylinder

Classical Mechanics Level 4

There is an isotropic light source at the center of a hollow right circular cylinder of height $2h$ and radius $R$ .

If half of the radiant power is incident upon the two circular end pieces combined, determine the ratio $\frac{R}{h}$ to 3 decimal places.

Assumption: Neglect reflections.

Hint: Make sure that when you integrate over the entire cylinder, you get the source power.

The answer is 1.732.

3 solutions

Mark Hennings
Nov 28, 2016

Each circular end-piece subtends a solid angle $\int_0^{\tan^{-1}\lambda} \,d\theta \int_0^{2\pi} \,d\phi\, \sin\theta \; = \; 2\pi\left(1 - \tfrac{1}{\sqrt{\lambda^2+1}}\right)$ where $\lambda = \tfrac{R}{h}$ . This solid angle should be equal to $\pi$ for the required condition to be true, so that $\sqrt{\lambda^2+1} = 2$ , and hence $\lambda = \boxed{\sqrt{3}}$ .

I integrated the power density for the same result (see posted solution). Your approach is even more succinct.

Steven Chase - 4 years, 6 months ago

That's a wonderful part of Brilliant, learning from each other's ideas and approaches.

Calvin Lin Staff - 4 years, 6 months ago

Steven Chase
Nov 28, 2016

Divyc Divc
Jan 1, 2017

Can be solved using solid angle- $p(1-cos(\theta$ ) = p/2) so (theta)\ is 60 hence R/h is sqrt(3)

Light Source in Cylinder

The answer is 1.732.

3 solutions

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