Question

The unit of solid angle is steradian. What is the dimensional formula for steradian?
A.$\left[ {{L^1}{M^1}{T^{ - 1}}} \right]$
B. $\left[ {{L^0}{M^0}{T^0}} \right]$
C. $\left[ {{L^2}{M^{ - 1}}{T^1}} \right]$
D. $\left[ {{L^{ - 2}}{M^1}{T^0}} \right]$

Answer 1

Hint- Solid angle can be considered as a 3D analogue of a plane angle. It is given by the equation
$\omega = \dfrac{A}{{{R^2}}}$

Complete step by step answer:
Solid angle is a three-dimensional angle subtended by any object. The unit of solid angle is steradian.
Solid angle can be considered as a 3D analogue of a plane angle. It is given by the equation
$\omega = \dfrac{A}{{{R^2}}}$
Where $\omega$ is the solid angle, $A$ is the area and $R$ is the radius.
Therefore, dimension of solid angle, $\left[ \omega \right]$ can be written as
$\left[ \omega \right] = \dfrac{{\left[ A \right]}}{{{{\left[ R \right]}^2}}}$
Dimension of area $\left[ A \right]$ is $\left[ {{L^2}{M^0}{T^0}} \right]$
Dimension of square of radius ${\left[ R \right]^2}$ is ${\left[ {{L^1}{M^0}{T^0}} \right]^2} = \left[ {{L^2}{M^0}{T^0}} \right]$
Therefore, dimension of solid angle, $\left[ \omega \right]$ is
$\left[ \omega \right] = \dfrac{{\left[ A \right]}}{{{{\left[ R \right]}^2}}} \\\ = \dfrac{{\left[ {{L^2}{M^0}{T^0}} \right]}}{{\left[ {{L^2}{M^0}{T^0}} \right]}} \\\ = \left[ {{L^0}{M^0}{T^0}} \right] \\\$
Which means solid angle is a dimensionless quantity. Since steradian is the unit of solid angle. It’s dimension is same as $\left[ {{L^0}{M^0}{T^0}} \right]$
Thus, the answer is option B

Note: Even when a quantity is dimensionless it can still have a unit. For example, plane angle is a dimensionless quantity but it has a unit which is angle. Similarly, solid angle is a dimensionless quantity but it has a unit which is steradian represented by the symbol $sr$