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: 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 $sr$