Question

The solid angle subtended by the periphery of an area $1c{m^2}$ at a point situated symmetrically at a distance of $5cm$ from the area is
A) $2 \times {10^{ - 2}}steredian$
B) $4 \times {10^{ - 2}}steredian$
C) $6 \times {10^{ - 2}}steredian$
D) $8 \times {10^{ - 2}}steredian$

Answer 1

Solid angle is the three-dimensional analogue of an angle subtended by planes meeting at a certain point. It is measured as the ratio of the area and the square of the distance of area from the point.

Formula Used:
If a solid angle is subtended by the periphery of an area $A$ at a point situated at a distance of $d$ from the from the area, then the solid angle $\Omega$ is defined by
$\Omega = \dfrac{A}{{{{\left( d \right)}^2}}}$
where the solid angle $\Omega$ has the unit of steradian.

Complete step by step answer:
Given:
The area of the subtended region, $A$ is $1cm$.
The distance from the area and the point those subtends the solid angle, $d$ is $5cm$
To get: The solid angle $\Omega$
Step 1:
Calculate the solid angle from the eq (1).
$\Omega = \dfrac{A}{d} \\\ = \dfrac{1}{{{{\left( 5 \right)}^2}}}steradian \\\ = \dfrac{1}{{25}}steradian \\\ = 4 \times {10^{ - 2}}steradian \\\$

Final Answer:
: The solid angle subtended by the periphery of an area $1c{m^2}$ at a point situated symmetrically at a distance of $5cm$ from the area is (B) $4 \times {10^{ - 2}}$ steradians.

Additional Information: The length of a circular arc subtended by lines form an angle, similarly a solid angle is the area of the segment of a unit sphere subtended by the planes meeting at the apex. So, likewise you can calculate the solid angle as the ratio of the area and the square of the distance of the area from the point. This has a wide application in astronomy and astrophysics, where a very far away celestial object is studied. Here, the solid angle is the ratio of the area of the body, projected along the viewpoint roughly proportional to the squared distance. Thus, several properties like luminosity are studied from this. Also, solid angles are highly useful in understanding the scattering experiments.

Note: To calculate the solid angle you need to be careful about the units of the area $A$ and the distance from the point $d$ that subtends the solid angle. If the units are not the same then there should be some metric scaling. Otherwise you would get a wrong answer. It is also important to assure that the area is perpendicular to the distance while using the formula, the area in the formula basically is the area perpendicular to the distance.