Which of the following is true for the solid angle?

The correct answer is δω = δA cos θ / r².

Step-by-step Explanation:

1. Definition of Solid Angle:
The solid angle, denoted by $d\omega$ or $\delta \omega$, is a two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point.

2. Basic Formula for a Spherical Surface:
For a portion of a spherical surface of area $\delta A$ at a distance $r$ from the center, where the area element is perpendicular to the radial vector (normal vector $\hat{n}$ is aligned with the radial direction), the solid angle is defined as:
$\delta \omega =\frac{\delta A}{r^2}$

3. General Case (Inclined Surface):
If the surface element $\delta A$ is not perpendicular to the line of sight (radial direction), we must project the area element onto the plane perpendicular to the radial vector. Let $\theta$ be the angle between the normal to the area element $\delta A$ and the radial vector (direction of $r$). The projected area perpendicular to the radial direction is:
$\delta A_{\text{projected}}=\delta Acos\theta$

4. Substituting the Projected Area:
Using the projected area in the solid angle definition gives the general formula for the solid angle subtended by an area element:
$\delta \omega =\frac{\delta Acos\theta }{r^2}$

Therefore, the relation δω = δA cos θ / r² is the correct representation.