Question

What is the solid angle subtended by a hemisphere at its center?
(a) \[\pi \] steradian
(b) \[2\pi \] steradian
(c) \[3\pi \] steradian
(d) \[4\pi \] steradian

Answer 1

Hint: In order to determine the solid angle subtended by a sphere at its center, we can easily find the answer for our question. Because the hemisphere is half a sphere.
The solid angle is the angle created in three-dimensional space that an object subtends at a point.
A solid angle can be defined in terms of an area on the surface of a sphere which is centred at the vertex of the angle. In our problem we need to find the solid angle subtended by a hemisphere at its center.

Complete step by step solution:
Solid angle is a measure of how big an object appears to an observer when looking from that point. The symbol for solid angle is \[\Omega \] (Greek letter omega). 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 this problem,
Let the SI unit of solid angle is the steradian (sr).
The solid angle is related to the area it cuts out of a sphere:
\[\Omega = \dfrac{A}{{{r^2}}}sr.\] …………………………………………………….. (1)
where,
\[\Omega = \] solid angle
\[A = \] surface area of sphere
 \[r = \] radius of the sphere
\[sr = \] SI unit steradian.
The value of the surface area of the sphere, \[A = 4\pi {r^2}\] ……………………………………………………. (2)
By substituting this value of equation (2) in equation (1), we will get,
\[\Omega = \dfrac{A}{{{r^2}}}sr.\]
 \[ = \dfrac{{4\pi {r^2}}}{{{r^{^2}}}}\]
Now, \[{r^2}\] will get cancel,
\[\Omega = 4\pi sr\] is the solid angle subtended by a sphere at its center.
Therefore, the solid angle subtended by a hemisphere at its center is ‘half’ the solid angle subtended by a sphere at its center.
\[\Omega = \dfrac{{4\pi }}{2}\]
\[\Omega = 2\pi sr\].
Therefore, the solid angle subtended by a hemisphere at its center is \[2\pi \] steradian.
Hence option (b) \[2\pi \] steradian is correct.
So, the correct answer is “Option B”.

Note: We can also find the solid angle subtended by a hemisphere at its center by directly substituting the value of surface area of the hemisphere in equation (1).
The value of surface area of hemisphere, \[A = 2\pi {r^2}\] ………………………………………… (3)
Now substituting equation (3) in (1) we will easily obtain the solid angle subtended by a hemisphere at its center.
\[\Omega = \dfrac{A}{{{r^2}}}sr.\]
\[ = \dfrac{{2\pi {r^2}}}{{{r^2}}}\]
Now, \[{r^2}\] will get cancel,
\[\Omega = 2\pi sr.\] is the solid angle subtended by a hemisphere at its center.