Question

The solar constant is defined as the energy incident per unit area per second. The dimensional formula for solar constant is:
${\text{(A) [}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{0}}}{{\text{T}}^{\text{0}}}{\text{]}} \\\ {\text{(B) [ML}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}} \\\ {\text{(C) [}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{2}}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}} \\\ {\text{(D) [M}}{{\text{L}}^{\text{0}}}{{\text{T}}^{{\text{ - 3}}}}{\text{]}} \\\$

Answer 1

For finding the dimensional formula of any quantity first of all write the formula related to that quantity. Here the solar constant is defined as the energy incident per unit area per second. Write the dimensional formula of the power and the dimensional formula of area and then simplify to get the dimensional formula of solar constant.

Complete solution:
Solar constant is defined as the total radiation energy received from the Sun per unit of time per unit of area.
Units of solar constant ${\text{ = }}\dfrac{{{\text{power}}}}{{{\text{area}}}}$
The S.I. unit of power is watt (represented by W)
S.I. unit of area is metre square (represented by ${{\text{m}}^{\text{2}}}$)
Thus, the S.I. units of solar constant ${\text{ = }}\dfrac{{\text{W}}}{{{{\text{m}}^{\text{2}}}}}$
A body is said to have power of ${\text{1 Watt}}$ if the body does work of ${\text{1 Joule}}$ in ${\text{1 second}}$.
So, ${\text{1 Watt = }}\dfrac{{{\text{1 joule}}}}{{{\text{1 sec}}}}$
Also, One joule of work is done on an object when a force of one newton (represented by ${\text{1 N}}$) is applied over a distance of one meter (represented by ${\text{1 m}}$).
So, ${\text{1 joule = }}\dfrac{{{\text{1 newton}}}}{{{\text{1 metre}}}}$
Thus, the S.I. units of solar constant is $\dfrac{{{\text{N m}}}}{{{\text{s }}{{\text{m}}^2}}}$.
Now the dimensional formula of force whose S.I. unit is newton is given by ${\text{[ML}}{{\text{T}}^{{\text{ - 2}}}}{\text{]}}$
Dimensional formula of distance whose S.I. unit is metre is given by ${\text{[}}{{\text{M}}^{\text{0}}}{{\text{L}}^{\text{1}}}{{\text{T}}^{\text{0}}}{\text{]}}$
Dimensional formula of time whose S.I. units is second is given by ${\text{[ML}}{{\text{T}}^1}{\text{]}}$
Thus, dimensional formula of solar constant is $\dfrac{{{\text{[}}{{\text{M}}^1}{{\text{L}}^1}{{\text{T}}^{{\text{ - 2}}}}{\text{][L]}}}}{{{\text{[}}{{\text{T}}^1}{\text{][}}{{\text{L}}^{\text{2}}}{\text{]}}}}{\text{ = [}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^{{\text{ - 3}}}}{\text{]}}$
The dimensional formula for solar constant is ${\text{[}}{{\text{M}}^1}{{\text{L}}^0}{{\text{T}}^{{\text{ - 3}}}}{\text{]}}$

Therefore, option (C) is the correct choice.

Note: Dimensions are denoted with square brackets. The dimensional formula of length, mass, time, electric current, thermodynamic temperature, luminous intensity and amount of substance are [L], [M], [A], [K], [Cd] and [mol] respectively. These are the quantities from which all other secondary quantities can be obtained.