What is Solid angle: Definition and 68 Discussions

In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.
In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere,

4
π

{\displaystyle 4\pi }
. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.
A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse.

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1. A Convention of units for densities in cosmology

I have a table of densities of galaxies : Expected number density of galaxies for photometric survey per unit area and redshift intervals, ##\mathrm{d} N / \mathrm{d} \Omega \mathrm{d} z\left[\mathrm{sr}^{-1}\right]## and the corresponding density of galaxies per ##\operatorname{arcmin}^2## for...
2. Solid angle in an optics problem (artificial irradiance of the Earth)

The actual problem can be found as #2 on this link: https://ocw.mit.edu/courses/mechanical-engineering/2-71-optics-spring-2009/assignments/MIT2_71S09_ups1.pdf I rewrote the problem above with the solar irradiance data that they give. My interpretation is of a square 1 m x 1 m plane sitting...
3. Calculating the Solid Angle of a Nebula from Earth

If I assume the nebula is a circle, than the length of arc viewed from Earth is a half of the circumference. So, here $$l = \frac{1}{2} \pi D$$ From the problem, ##D = 125 000 ly##. Because the distance of nebula is much larger than the diameter; I try to approximate R with the distance of...
4. I How Does Solid Angle Affect Power Calculation in Radiometry?

I have a doubt regarding the role of the solid angle when calculating the power(W) with the brightness of the source I'm observing with area Asource. I was given the definition: with A the Aantenna. If now I take as an example the picture below to calculate W, we conisder as solid angle the one...
5. A What is the meaning of ##d\Omega## in solid angle integration?

Anyone have any idea how to perform the following two integrals? ##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}## where the n is a unit vector.
6. I Where does this solid angle come from?

I found this paper https://arxiv.org/abs/quant-ph/0412216 We have an interferometer with to arms. The firsr has a couple of HWP's inclened by an angle theta and the second has the crossed couple. A mixed state is in input. i look to the figure withe the Bloch sphere. i see 2 paths on it. one...
7. Using the solid angle to simplify an integral when deriving Gauss' Law

I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.
8. How to calculate a solid angle in steradians from arcminutes?

I tried $$\Omega = \pi(\frac{\theta}{2})^{\frac{1}{2}}$$
9. A Calculating Light Intensity on a Target Using Lamp Setup

Hi, I am trying to simulate necessary flux values on a car due to solar radiance. I'm trying to attain the necessary flux values using a lamp setup. I have the lamp specifications in Watts but I need to convert them into radiance values (W/m2/sr) for my application. I would like to know how to...
10. B Solid angle by placing one disk on a cylinder

Hello, I am trying to find an analytical expression to determine the solid angle subtended by a disk source onto the face of the cylinder. I will appreciate if someone can provide me directions. I am aware how to calculate solid angle by a point source to cylinder's face ( omega =...
11. I Prove that solid angle of any closed surface is 4pi

I googled a lot on proof of Gauss theorem and nearly every other proof (on web and so on books) state that solid angle of closed surface is 4pi but I can't find the proof of this nowhere ! I tried setting up the integral but don't know how to proceed furthur : Ω=∫(cosθ/r^2)*dA Also The one...
12. I Visualizing Solid Angle of a 3d Object (say a Sphere)

Hello Everybody! Concept of Solid Angle was pretty much straight forward until they were on surface patches were taken into account which were visualized as base of cone. I am having difficult when 3d Objects like Sphere/Cylinder . We can very easily calculate the respective area and plugin the...
13. Correcting results to fit Rutherford's Scattering Formula

I'm a Physics undergraduate at University and in my labs module I had to recreate the Rutherford Scattering experiment. The basic setup was similar to this: My problem is that this setup only records data in one plane and not 3D. In reality the particles are scattered in a sort of cone...
14. Integral of angular functions over d-dim solid angle

Hello everyone! I have a question about angular integration in arbitrary d dimensions. The interest comes from the need to use dimensional regularization. Suppose I start with a 2-dimensional integral and then I have to move to d=2-\epsilon dimension to regularize my integral. Now, suppose...
15. Calculating the Solid Angle Subtended by a Disc: How to Find ω?

The problem is to find out the solid angle ω subtended by a disc of radius a at a point P distant z from its centre along its axis. α is the semi-vertical angle of the disc at the point P in question. The answer is supposed to be ω = 2π (1 - cos α), according to an online text. However, I find...
16. Light through a tube - Solid Angle (Oblique Cone)

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...
17. I can't accept the solution manual's explanation

Homework Statement It's a long winded problem, I'll post a picture and an imgur link Imgur link: http://i.imgur.com/5wvbqO2.jpg Homework Equations Divergence Theorem \iint\limits_S \vec{F}\cdot d\vec{S} = \iiint\limits_E \nabla \cdot \vec{F} \ dV The Attempt at a Solution I'll follow...
18. Unit solid angle and finite angle

Homework Statement A point source emits visible light isotropically. Its luminous flux is 0.11 lumen. Find the flux whithin the cone that has half angle of 30 degree from the light source. Homework Equations luminous flux = luminous intensity * solid anlge The Attempt at a Solution I tried...
19. Transformation of solid angle

I'm trying to figure out how the element of solid angle transforms under a transformation between two inertial frames moving with velocity v w.r.t. each other under an arbitrary direction. But I should say I disappointed myself! Anyway, some books which contain a brief discussion on this(which...
20. Verifying Duderstadt & Hamilton's Eqs. (4-134) & (4-137) in Nuclear Transport

Show by direct calculation that Eqs. (4-134) and (4-137) in the textbook by Duderstadt and Hamilton hold, i.e.:(a) ∫ dΩΩiΩj= 4π/3 δij; i,j = x,y,z; 4π(b) ∫ dΩΩxΩyΩz = 0, if l, m, or n is odd. 4π The integrals are over 4π. This is part of the derivation of the diffusion equation...
21. Transformation of Solid Angle in Gravitational Lenses by P.Schneider et al.

When considering a small beam of null-geodesics in spacetime it is possible to define the solid angle spanned by two of the rays at the observer. At page 111 in "Gravitational Lenses" by P.Schneider et. al. they state with reference to Figure (b) that: "The dependence of this distance on the...
22. Solid Angle Rutherford Scattering

Homework Statement derive an equation for the solid angle for a Rutherford scattering detector given a detcor window area of A and a distance to the detector of D for some scattering angle \phi given that d\Omega =2\pi sin\phi d\phi Homework Equations d\Omega =2\pi sin\phi d\phi A=Dd\phi The...
23. The solid angle equation for the magnetic field of a loop

Background: Using Biot-Savart law we proved that for a closed loop with current ##I##, the magnetic field at a point P was equal to ##\vec{B}=-\frac{\mu_{0} I \nabla{\Omega}}{4 \pi}## where ##\Omega(x,y,z)## is a function of the position of P that represents the solid angle at which the loop is...
24. Great! Glad to hear you were able to figure it out. Keep up the good work!

In deriving the pressure of a gas, my book states that 'if all molecules are equally likely to be traveling in any direction, the fraction whose trajectories lie in an elemental solid angle dΩ is dΩ/4π'. This initally made sense to me, but then thinking about it, I wrote dΩ=sinθdθdφ and this...

Homework Statement For a sphere of radius r, find the solid angle Ω in steradians defined by spherical angles of: a.) 0°≤θ≤ 20°, 0°≤ø≤360°; Homework Equations dA = r2 sin dθ dø (m2) dΩ = dA / r2 = sin dθ dø (sr) The Attempt at a Solution I think I understand what a steradian (sr)...
26. Discover the Derivation of Solid Angle: Short and Simple Method

I want to know the derivation of solid angle. I have found on net. But the derivation is to long, is there any short method?
27. Laser Beam - Divergence and Solid Angle

I am somewhat confused about the connection between divergence and solid angle for a beam. I know individually what each term means... but I'm confused as to how (or even if) one can calculate the solid angle of a beam, given the divergence. I have some notes from a previous lecture series I...
28. Calculating Solid Angle Between Three Vectors: A Scientific Guide

How I do for calculate the solid angle formed by 3 vectors like: http://s30.postimg.org/61lpvmi4h/solid.jpg ?
29. How to calculate the solid angle subtended by an off axis disk

Hi, It's surprising how little information is available on this topic, considering it seems like such a fundamental problem. The only tutorial I have found is http://www.umich.edu/~ners312/Course%20Library/SolidAngleOfADiskOffAxis.pdf, and my university does not have access to the other papers...
30. Understanding the Relationship Between Solid Angle and Plane Angles

Hellow! I known an infinitesimal relation between the solid angle Î© with the azimutal angle Î¸ and zenital Ï†, that's given by dÂ²Î© = sin(Ï†) dÏ† dÎ¸. But this is infinitesimal relation, exist other relation non infinitesimal between the solind angle with the plane angles? Thanks!
31. Probability of finding a particle in a solid angle

I have an interesting question that I'm not sure how to go about solving. This question has a little general relativity and (maybe) a little QM, but I wasn't sure where to post it. Question: Imagine that a \pi0 meson traveling along the z-axis (velocity v=0.99c, rest mass M) decays into two...
32. Derivative of Solid Angle: Lightmann & al. Explained

hello please see attached snapshot from the book of Ligthmann & al. (problem book in relativity and gravitation). Can somebody please explain the expression for the derivative of the solid angle ? here it is given as dvxdvy where the v's are speeds ! How come this is so ? Thanks,
33. Why/how does integral of solid angle = pi?

Hi folks, can someone help explain this in words of one syllable or less? I am looking at a text that compares flux and intensity of a distant source, and it states that ∫∫dΩ = ∏ I know that dΩ = sinθ dθ d∅ but I don't understand where the given result comes from. What are the...
34. Inverse Square Law - Calculating the solid angle

Hi Everyone, For an individual inquiry and formal lab report task at school I have chosen to conduct an experiment to find out whether hot shoe mounted flash units follow the inverse square law and how the flash zoom is affected by the inverse square law. My first question is that In order...
35. How to calculate the solid angle of a cone with cone axis arbitrary?

I read that if the cone with apex angle 2α whose central axis is vertical, apex at the origin, then one can use spherical coordinate to calculate the solid angle of the cone ∫02∏∫0αsin\varphid\thetad\varphi However, what if the central axis is align to y-axis horizontally, instead of...

This is only an example from Kraus Antenna 3rd edition page 404. The question is really a math problem involves calculation of ratio of solid angles. Just ignore the antenna part. this is directly from the book: Example 12-1.1 Mars temperature The incremental antenna temperature for the...
37. Integral of differential cross section over solid angle

Homework Statement Find σ , the differential cross section, starting from the expression below and integrating over solid angle Ω Homework Equations dσ/dΩ = r2sin2θ The Attempt at a Solution dσ = r2sin2θ dΩ I remember that dΩ = sinθ dθ dμ and doing the μ integral from 0...
38. Stephan-Boltzmann law, solid angle integral, and an extra pi factor

Homework Statement This isn't a homework question, I am writing up my scribbled notes from todays lecture and have got stuck on some calculus, and lost the thread of the argument. Last week, we integrated Plancks law to find B(T) = ∫ Bv(T) dv = 2∏4(kT)4 / 15c2h3 Then...
39. Total Solid Angle of 6 Squares on Unit Sphere

Homework Statement Hi I am looking at a unit sphere. Two squares are projected onto the sphere on opposite ends, as shown in figure 1 (the figure only shows one square, the other one is at the opposite end). There are two more sets of these squares, each set in its own dimension, so there...
40. How Is the Solid Angle for a Circular Detector Calculated?

[b]1. The solid angle subtended by a 100 cm^2 circular detector at a distance of 1 m is ______steradians. [b]2. Ω = A/r^2 and A =∏r^2 (area of a circle) [b]3. I originally tried to find r by solving 100 = ∏r^2 and I got r = 5.6. I then tried to plug into the first equation for Ω only to...
41. Modeling the Solid Angle with Vpython

A radiation source at some point through the collimator of a detector. Modeled as 100 spheres tracing out the image of a circle on a plane some distance above the original position of the source. What I want to do is collect all the spheres that go through the top of a cylinder and then...
42. Solid angle & flux out of cone

Hi, I was reading my astrophysics textbook and came across solid angles. I'm not sure I fully understand, for example there was a problem in the book that went as follows. The attached "math.jpg" shows a light source (yellow) in the centre of an arc. The problem is 2D, but the arc is...
43. What is the Beam Solid Angle of an Antenna?

The definition of Beam solid angle: For an antenna with single main lobe, the Beam solid angle defines as: The solid angle \; \Omega_A\; where all the radiated power would flow with radiation intensity equal to maximum and constant inside the Beam solid angle \; \Omega_A\;. The book gave...
44. Question on solid angle of sphere.

I understand the surface of the sphere is 4\pi sr. where the area of one sr is r^2. My question is why d\Omega = sin \theta \;d \theta \;d\phi? Can anyone show me how to derive this. Is it because surface area dS = (Rd\theta)(R\; sin\;\theta\;d\phi)\;\hbox { so if }\; \Omega = \frac S {R^2}...
45. Gauss's law derivation using solid angle

So, I was trying to find a rigorous mathematical derivation of gauss's law(please I don't want to hear again any field lines nonsense) and I stumbled upon jackson's proof which uses the solid angle concept and seems a solid enough proof(stupid joke:smile:).The problem is that it's the first time...
46. Solid angle , never seen derivative

hello, Please attached snapshot of an answer to an Ex. I was stunned at the formula for the derivative of the solid angle which is : d(solid angle) = dvx * dvy I would appreciate if somebody can provide some hints on how one can find it ? Thank you,
47. Solid angle (and integral of a sine function)

Hello, I was following the derivation of the solid angle of right rectangular pyramid that I found at http://www.slac.stanford.edu/~bgerke/notes/solid_angle.pdf" . I don't quite understand the step between the 3rd to the 4th equation. In particular how the integral...
48. Determine the solid angle subtended by a detector

[b]1. . A cylindrical detector, similar to the NaI detectors you use in lab, has a diameter of 5 inches and a length of 5 inches. A 60Co source is placed on the cylindrical axis, 20 cm away from the front face of the detector. a. Determine the solid angle subtended by the detector...
49. Deriving Solid Angle Formulation from Definition

Hello, it is often written in books that the solid angle \Omega subtended by an oriented surface patch can be computed with a surface integral: \Omega = \int\int_S \frac{\mathbf{r}\cdot \mathbf{\hat{n}} }{|\mathbf{r}|^3}dS where r is the position vector for the patch dS and n its normal (see...
50. Calculate the hieght of the cones by using Solid angle?

Calculate the hieght of the cones by using Solid angle? pleasez help Homework Statement Calculate the hieght of the cones by using Solid angle? Homework Equations d(omega) The Attempt at a Solution what can i do in that question?