Question

(The solid angle) Let \(S\) be a plane surface, oriented in accordance with a unit normal \(\mathbf{n}\). The solid angle \(\Omega\) of \(S\) with respect to a point \(O\) not in \(S\) is defined as $$ \Omega(O, S)=\pm \text { area of projection of } S \text { on } S_{1}, $$ where \(S_{1}\) is the sphere of radius 1 about \(O\) and the \(+\) or - sign is chosen according to whether \(\mathbf{n}\) points away from or toward the side of \(S\) on which \(O\) lies. This is suggested in Fig. 5.37. a) Show that if \(O\) lies in the plane of \(S\) but not in \(S\), then \(\Omega(O, S)=0\). it b) Show that if \(S\) is a complete (that is, infinite) plane, then \(\Omega(O, S)=\pm 2 \pi\). c) For a general oriented surface \(S\) the surface can be thought of as made up of small elements, each of which is approximately planar and has a normal \(\mathbf{n}\). Justify the following definition of element of solid angle for such a surface element: $$ d \Omega=\frac{\mathbf{r} \cdot \mathbf{n}}{r^{3}} d \sigma $$ where \(\mathbf{r}\) is the vector from \(O\) to the element. d) On the basis of the formula of (c), one obtains as solid angle for a general oriented surface \(S\) the integral $$ \Omega(O, S)=\iint_{S} \frac{\mathbf{r} \cdot \mathbf{n}}{r^{3}} d \sigma . $$ Show that for surfaces in parametric form, if \(O\) is the origin, $$ \Omega(O, S)=\iint_{R_{x r}}\left|\begin{array}{ccc} x & y & z \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \end{array}\right| \frac{1}{\left(x^{2}+y^{2}+z^{2}\right)^{\frac{1}{2}}} d u d v . $$ This formula permits one to define a solid angle for complicated surfaces that interac themselves. e) Show that if the normal of \(S_{1}\) is the outer one, then \(\Omega(O, S)=4 \pi\). f) Show that if \(S\) forms the boundary of a bounded, closed, simply connected region \(R\). then \(\Omega(O, S), \pm 4 \pi\), when \(O\) is inside \(S\) and \(\Omega(O, S)=0\) when \(O\) is outside \(S\). g) If \(S\) is a fixed circular disk and \(O\) is variable, show that \(-2 \pi \leq \Omega(O, S) \leq 2 \pi\) and that \(\Omega(O, S)\) jumps by \(4 \pi\) as \(O\) crosses \(S\).

Short answer

Answer: The maximum solid angle subtended by a fixed circular disk at a variable point O when O is directly above or below the center of the disk is ±2π, depending on the orientation of the normal vector 𝐧.

Step-by-step solution

a) Solid angle for an \(O\) in the plane of \(S\)

If \(O\) lies in the plane of \(S\) but not in \(S\), then every point on the surface \(S\) is equidistant from \(O\) in the plane, and the projection of \(S\) on the unit sphere \(S_1\) will cover the same area on opposite sides of \(S\) since \(\mathbf{n}\) points away from one side and towards the other. As a result, the contributions to the solid angle cancel out, and we have \(\Omega(O, S) = 0\).

b) Solid angle for a complete plane surface

If the surface \(S\) is a complete (infinite) plane, the projection of \(S\) on the unit sphere \(S_1\) will completely cover either the upper or lower hemisphere of the sphere, depending on the orientation of the unit normal \(\mathbf{n}\). Each hemisphere has an area of \(2\pi\); therefore, \(\Omega(O, S) = \pm 2 \pi\) depends on the orientation of \(\mathbf{n}\).

c) Justification for the element of the solid angle definition

For a general oriented surface \(S\), we can consider the surface as divided into small elements, each of which is approximately planar with a normal \(\mathbf{n}\). The solid angle element \(d\Omega\) should be proportional to the area of the projection of \(S\) on the unit sphere \(S_1\). Since \(S\) is approximately planar for each small element, these elements will project with an area given by the dot product of the normal vector \(\mathbf{n}\) and the position vector \(\mathbf{r}\). This is scaled by \(\frac{1}{r^3}\), as it is proportional to the area of the projection in the unit sphere. Therefore, we have \(d \Omega = \frac{\mathbf{r} \cdot \mathbf{n}}{r^{3}} d \sigma\).

d) Solid angle formula for an oriented surface in parametric form

For a surface, \(S\), in parametric form \(S(u, v) = (x(u, v), y(u, v), z(u, v))\), we want to compute the solid angle \(\Omega(O, S)\). To do so, we integrate the definition of the elemental solid angle d\(\Omega\) from part (c) over the surface: $$ \Omega(O, S)=\iint_{S} \frac{\mathbf{r} \cdot \mathbf{n}}{r^{3}} d \sigma. $$ To bring this formula to match the desired form, we have to express \(\mathbf{n}\), the unit normal vector, in terms of the partial derivatives of \(S\) w.r.t. \(u\) and \(v\): $$ \mathbf{n}(u, v) = \frac{\partial S}{\partial u} \times \frac{\partial S}{\partial v}, $$ which implies: $$ d\sigma = \left| \frac{\partial S}{\partial u} \times \frac{\partial S}{\partial v} \right| dudv. $$ Then by integrating with respect to \(u\) and \(v\) over the region \(R_{x r}\), we obtain: $$ \Omega(O, S)=\iint_{R_{x r}}\left|\begin{array}{ccc} x & y & z \\\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \end{array}\right| \frac{1}{\left(x^{2}+y^{2}+z^{2}\right)^{\frac{1}{2}}} d u d v. $$

e) Solid angle over an entire sphere \(S_1\)

If \(S_1\) is the entire sphere with the outer normal, we can use the formula from part (d) to calculate the solid angle subtended by \(S_1\) to the point \(O\). We can parametrize the sphere \(S_1\) in spherical coordinates \((\theta, \phi)\), where \(x = \sin\theta\cos\phi\), \(y = \sin\theta\sin\phi\), and \(z = \cos\theta\). Then, we compute the solid angle using the integration formula from part (d): $$ \Omega(O, S_1)=\iint_{0 \leq \theta \leq \pi, 0 \leq \phi \leq 2\pi}\left|\begin{array}{ccc} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\theta\sin\phi & \sin\theta\cos\phi & 0 \end{array}\right| d\theta d\phi \\ = \int_{0}^{\pi}\int_{0}^{2\pi}\sin^2\theta d\theta d\phi \\ = 4\pi. $$ In conclusion, we find that if \(S_1\) is an entire sphere with the outer normal, \(\Omega(O, S_1)=4\pi\).

f) Solid angle inside and outside a bounded, closed surface

If the surface \(S\) forms the boundary of a bounded, closed, and simply connected region \(R\), then the solid angle \(\Omega(O, S)\) depends on whether the point \(O\) is inside or outside the surface \(S\). The solid angle is defined in terms of the projection of the surface on the unit sphere. As a result: 1. If \(O\) is inside \(S\), the entire sphere's surface will be covered by the projection of the surface, implying that \(\Omega(O, S) = \pm4\pi\), depending on the orientation of the normal vector \(\mathbf{n}\). 2. If \(O\) is outside \(S\), the projection of the surface will not cover any part of the sphere, implying that \(\Omega(O, S) = 0\).

g) Solid angle for variations of a fixed circular disk

If \(S\) is a fixed circular disk and the point \(O\) is variable, then the solid angle \(\Omega(O, S)\) depends on the position of \(O\) relative to \(S\). When \(O\) is outside the disk, the solid angle is zero since there is no projection of \(S\) on the unit sphere \(S_1\). When \(O\) comes closer to the disk surface, the solid angle increases, reaching a maximum of \(\pm2\pi\) when \(O\) is directly above or below the center of the disk. Note that the sign depends on the orientation of the normal vector \(\mathbf{n}\). As \(O\) crosses the disk, the solid angle changes abruptly by \(4\pi\), meaning it jumps from \(-2\pi\) to \(2\pi\) or vice versa. Therefore, for a variable point \(O\) and a fixed circular disk \(S\), the solid angle \(\Omega(O, S)\) satisfies \(-2\pi \leq \Omega(O, S) \leq 2\pi\), and it jumps by \(4\pi\) as \(O\) crosses the disk.

Key concepts

plane surface

A plane surface is a flat, two-dimensional surface that extends infinitely in its defined directions, much like a piece of paper lying on a table. It is characterized by its flatness and lacks any curvature. This kind of surface can be oriented in a three-dimensional space, allowing us to define its position and behavior in relation to other geometric entities.
Plane surfaces are crucial in various mathematical contexts, especially when dealing with projections and solid angles, as their simplicity provides ease in calculations. Understanding plane surfaces serves as a foundation for exploring more complex geometrical concepts.

oriented surface

An oriented surface has a designated direction, decided by its unit normal vector. This unit normal is a perpendicular vector that signifies which side of the surface is considered positive and which side negative.

This orientation is essential when determining the solid angle, as it decides whether the solid angle is considered positive or negative depending on whether the normal vector points toward or away from the reference point. Thus, understanding how surfaces are oriented is vital in multiple physics and engineering applications, where the direction of vectors greatly affects the outcomes.

parametric form

In geometry, a surface's parametric form describes its points in terms of parameters. Instead of traditional Cartesian coordinates (x, y, z), the surface is presented as a function of two parameters, such as \(u\) and \(v\). Here, the surface \(S(u, v)\) is represented as \((x(u,v), y(u,v), z(u,v))\).

This form is particularly beneficial in dealing with complex surfaces or when performing integrations over surfaces, as it simplifies many operations. By using a parametric form, we can better handle the calculation of areas, gradients, normals, or projections on different surfaces. It turns complex mathematical operations into chain of simple algebraic rules.

unit normal

A unit normal vector is a vector perpendicular to a surface that has a length of one. It's denoted by \(\mathbf{n}\) and is used to define the orientation of a surface within three-dimensional space.

For a surface described in parametric form, the unit normal can be found by taking the cross product of two tangent vectors obtained by partial differentiating the parametric equations. It is common to see this in the form \[ \mathbf{n}(u, v) = \frac{\partial S}{\partial u} \times \frac{\partial S}{\partial v} \].

This concept is crucial when dealing with projections and solid angles, as it impacts whether quantities should be considered with positive or negative values based on the vector's direction.

projection area

The projection area refers to the "shadow" that one surface casts onto another plane or sphere. When a surface is "projected," it is effectively translated or transformed into a different plane or curved surface, like a sphere. This is akin to how the shadow of an object is cast onto a wall or floor by a light source.

In this context, projecting a plane surface onto a unit sphere results in the area of the spherical cap that accommodates the plane's outline. This projection is crucial in defining the solid angle for the surface because it provides a geometrical interpretation of how much of the sphere the surface covers. The projection area depends heavily on the surface's orientation and distance to the reference point, which affects the solid angle's computation.