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Chapter 12: Problem 41

What does a solid angle represent, and how does it differ from a plane angle? What is the value of a solid angle associated with a sphere?

Short Answer

Expert verified
Answer: A plane angle is a two-dimensional angle formed by two intersecting lines, measured in degrees, radians, or grads. A solid angle is a three-dimensional angle describing the area enclosed by a cone on a spherical surface, measured in steradians. The value of a solid angle associated with a sphere is 4π steradians.

Step by step solution

01

Define plane angle

A plane angle is a two-dimensional angle measured between two intersecting lines or rays. It is usually represented using degrees, radians, or grads. The most common example is an angle between two lines that meet at a vertex, or a point where the two lines intersect.
02

Define solid angle

A solid angle is a three-dimensional angle that is used to describe the apparent size of an object or the amount of light coming from a specific area on a spherical surface. It is usually represented using steradians (sr). One great example is the angle formed by a conical region extended from the center of a sphere to a surface patch on the sphere.
03

The difference between plane angle and solid angle

While plane angles describe the angles between two intersecting lines in a two-dimensional space, solid angles describe the area that a conical region encloses on a spherical surface in a three-dimensional space. In other words, plane angles are measured in a plane, while solid angles are measured in a volume or on a curved surface. Plane angles are measured in degrees, radians, or grads, whereas solid angles are measured in steradians.
04

Calculate the value of a solid angle associated with a sphere

To find the solid angle associated with a sphere, we need to consider the formula for a solid angle. In general, the solid angle (Ω) at the apex of a cone is calculated as the ratio of the area (A) enclosed by that cone on the sphere's surface to the square of the sphere's radius (R): Ω = A / R^2 The area of a whole sphere with radius R is given by the formula: A = 4 * pi * R^2 Now, we can plug the area of the sphere into the formula for the solid angle: Ω = (4 * pi * R^2) / R^2 R^2 in the numerator and denominator cancel each other out, resulting in: Ω = 4 * pi (steradians) So, the value of a solid angle associated with a sphere is 4π steradians.

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Most popular questions from this chapter

The wavelength at which the blackbody emissive power reaches its maximum value at \(300 \mathrm{~K}\) is (a) \(5.1 \mathrm{\mu m}\) (b) \(9.7 \mu \mathrm{m}\) (c) \(15.5 \mu \mathrm{m}\) (d) \(38.0 \mu \mathrm{m}\) (e) \(73.1 \mu \mathrm{m}\)

A horizontal plate is experiencing uniform irradiation on the both upper and lower surfaces. The ambient air temperature surrounding the plate is \(290 \mathrm{~K}\) with a convection heat transfer coefficient of \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Both upper and lower surfaces of the plate have a radiosity of \(4000 \mathrm{~W} / \mathrm{m}^{2}\), and the plate temperature is maintained uniformly at \(390 \mathrm{~K}\). If the plate is not opaque and has an absorptivity of \(0.527\), determine the irradiation and emissivity of the plate.

A small circular surface of area \(A_{1}=2 \mathrm{~cm}^{2}\) located at the center of a 2-m-diameter sphere emits radiation as a blackbody at \(T_{1}=1000 \mathrm{~K}\). Determine the rate at which radiation energy is streaming through a \(D_{2}=1\)-cm-diameter hole located \((a)\) on top of the sphere directly above \(A_{1}\) and \((b)\) on the side of sphere such that the line that connects the centers of \(A_{1}\) and \(A_{2}\) makes \(45^{\circ}\) with surface \(A_{1}\).

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