Solid angle in the spherical polar co-ordinate system:

In spherical polar co-ordinate system, there are three area elements \( {dA}_r,\ {dA}_{\theta} \) and \( {dA}_{\phi} \), out of which only the area element \( {dA}_r \) subtends a solid angle at the centre \( O \).

[To know about the three types of area elements (CLICK HERE)]

The area element \( {dA}_r \) is perpendicular to the direction of the radial vector \vec{r}, Hence only the area \( {dA}_r \) subtends a solid angle at the centre O of the sphere.

[To know in detail, why only the area element \( {dA}_r \) subtends a solid angle at the centre (CLICK HERE)]

We know that the value of this area element \( {dA}_r \) is given by,

\( {dA}_r=r^2\ \sin\theta\ d\theta\ d\phi \).

So the solid angle subtended by the area \( {dA}_r \) is given by,

\( d\omega=\displaystyle{\frac{ r^2\ \sin\theta\ d\theta\ d\phi }{r^2}}= \sin\theta\ d\theta\ d\phi \)