# Light through a tube - Solid Angle (Oblique Cone)

• Sachin Vaidya
In summary, the conversation discusses a physics problem involving the calculation of the solid angle subtended by an oblique cone, in order to determine the fraction of light that is absorbed by a cardboard cylinder placed perpendicular to a 2D disk emitting light in a growing hemisphere. The conversation also explores different equations and approaches for calculating this solid angle, ultimately arriving at the fundamental equation of radiative transfer as the most efficient method.
Sachin Vaidya
For a physics problem, I need to calculate the solid angle subtended by an oblique cone (cone where the apex does not lie on the line perpendicular to the cone's base from the center of the base).

Consider the following problem:
I have a 2D disk which emits light in an ever growing hemisphere (spanning a solid angle of ##2\pi##). Let the initial intensity of light be ##I##. Then I put a cardboard cylinder perpendicular to the disk with radius equal to the radius of the disk. This cylinder absorbs light when it hits the walls. The question is then what is the fraction of light that comes out of the cylinder.

I am only interested in the fraction and not in the absolute number so this is what I did:

Consider a diameter on the disk. Pick an arbitrary point at a distance ##x## from the top having a thickness ##dx##.
The fraction of light that comes out from this element will simply be the ratio of the solid angle subtended by the cone (whose base is the other circular base of the cylinder) and the total angle of emission ##2\pi##.

I now need to calculate the solid angle due to this oblique cone which I cannot seem to do. So since it is given that the length of the cylinder is much longer compared to the radius of its base, I thought I could brake the oblique cone into 2 half right cones and add up the solid angles due to them. I'm not sure if this is correct.

(Check attachment for diagram)

Also, the reason I only consider a single diameter is because of my interest in the fraction of light that gets through and not the absolute number (of photons) so if I calculate the fraction from one diameter, it will be the same for every other diameter and hence for the whole surface.

Here is how I proceeded:

Solid angle subtended by a right cone is given by:

\Omega = 2\pi (1-\cos \theta)

Solid angle subtended by the upper half right cone:

\Omega_1 = \pi \left( 1 - \frac{l}{\sqrt{l^2+x^2}}\right)

Solid angle subtended by the lower half right cone:

\Omega_2 = \pi \left( 1 - \frac{l}{\sqrt{l^2+(d-x)^2}}\right)

Total solid angle:

\Omega_T=\Omega_1 + \Omega_2 = \pi \left(2 - \frac{l}{\sqrt{l^2+x^2}} - \frac{l}{\sqrt{l^2+(d-x)^2}}\right)

Fraction $fr_x$ of useful atoms is the ratio of $\Omega_T$ and the total solid angle $2\pi$.

fr_x=\frac{1}{2\pi}\cdot \pi \left(2 - \frac{l}{\sqrt{l^2+x^2}} - \frac{l}{\sqrt{l^2+(d-x)^2}}\right)

If the emissions per unit length is $\kappa$, the number of photons that will come out from the element ##dx## is ##fr_x\cdot \kappa dx##.

On integrating, this gives the total number of photons that come out from the whole diameter ##d## in 3 dimensions.

\int_0^d \frac{1}{2}\left(2 - \frac{l}{\sqrt{l^2+x^2}} - \frac{l}{\sqrt{l^2+(d-x)^2}}\right) \kappa dx

The total number of photons emitted by the diameter ##d## is ##\kappa d##.

Therefore, the fraction of photons (light) emitted by the whole length ##d## will be:

F(l,d)=\frac{\int_0^d \frac{1}{2}\left(2 - \frac{l}{\sqrt{l^2+x^2}} - \frac{l}{\sqrt{l^2+(d-x)^2}}\right) \kappa dx}{\kappa d}

On integration, this gives:

\boxed{F(l,d)=1-\left( \frac{l}{d}\right) \sinh^{-1}\left( \frac{d}{l}\right)}

I'm not sure this line of thought is completely correct.

#### Attachments

• solidanglefigure.jpg
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## 1. What is the concept of solid angle?

The solid angle is a unit used to measure the amount of space that a cone-shaped beam of light occupies. It is defined as the three-dimensional angle formed by the intersection of two planes at a point. In simpler terms, it is the measure of the size of an object as seen from a particular point in space.

## 2. How is solid angle related to light through a tube?

In the context of light through a tube, solid angle is used to measure the amount of light that passes through a specific area of the tube. As the light travels through the tube, it expands and forms a cone-shaped beam, and the solid angle determines the size of this beam. This can be useful in understanding the distribution of light in a particular space or when designing optical systems.

## 3. What is the formula for calculating solid angle?

The formula for calculating solid angle is Ω = A/r^2, where Ω is the solid angle in steradians (sr), A is the area of the base of the cone, and r is the distance from the apex of the cone to the point where the solid angle is measured. This formula is based on the fact that solid angle is directly proportional to the area of the base of the cone and inversely proportional to the square of the distance from the apex.

## 4. How is solid angle different from regular angle?

Regular angle (measured in degrees or radians) is a two-dimensional concept that describes the rotation of a line or shape. Solid angle, on the other hand, is a three-dimensional concept that describes the amount of space an object occupies in three dimensions. Additionally, while regular angle can have infinite values, solid angle is always measured in steradians, which has a maximum value of 4π sr.

## 5. What is the practical application of solid angle in science?

Solid angle has various applications in science, particularly in the fields of optics and astronomy. It is used to calculate the amount of light that reaches a particular point in a system, which is crucial in designing efficient lighting systems. In astronomy, solid angle is used to measure the size of celestial objects such as stars and galaxies, as well as to determine the amount of light emitted by these objects.

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