Question

The dimensions of intensity of energy are:
A. $ML^2T^{-1}$
B. $ML^2T^{-2}$
C. $MT^{-3}$
D. $ML^{-2}T^{3}$

Answer 1

Hint: The intensity of energy is the power transferred per unit time. To deduce a quantity to its dimensional form means expressing the units of the quantity in terms of just powers of mass, length and time. In other words, reduce the units to SI base units from the SI units of each of the involved quantities and you will be able to express them in terms of the fundamental quantities mentioned before.

Formula used:
Intensity of energy $I = \dfrac{E}{At}$ where E is the energy transferred, A is the perpendicular area and t is the time.

Complete answer:
Let us try and understand what intensity of energy means.
The intensity of radiant energy can be described as the amount of power that is transferred per unit area. Recall that power is just the amount of energy that is transferred per unit time. Also, keep in mind that the area is measured on the plane that is in a direction perpendicular to the propagation of energy.
Now, a dimensional formula for any physical quantity looks like $\left[ M^x L^y T^z\right]$, where M, L, and T, are the mass, length and time respectively, and x,y,and z are any integers representing dimensions.
Let us now try to dimensionally analyse what our definition of intensity of energy looks like.
If the intensity of energy $I$ is the power $P$ transferred per unit area $A$ and if power is the energy $E$ transferred per unit time $t$ then the intensity of energy can be redefined as the energy transferred per unit area per unit time, i.e.,
$I = \dfrac{P}{A}$ and $P=\dfrac{E}{t} $, then, $I = \dfrac{E}{At}$
Now, in order to obtain the dimensional formula for intensity of energy, let us express the above quantities in their SI base units first. Therefore, we have:
Energy E :
SI units: joule (J)
SI base units: $m^2.kg.s^{-2}$
Dimensional formula: $\left[ M^1 L^2 T^{-2}\right]$
Area A:
SI units: square meter ($m^2$), already in base units
Dimensional formula: $\left[ M^0 L^2 T^0\right] =\left[L^2 \right]$
Time t:
SI units: seconds (s), already in base units.
Dimensional formula: $\left[ M^0 L^0 T^1\right] =\left[T^1 \right]$
Putting all this together into the intensity of energy relation we get something like:
$I = \dfrac{E}{At} = \dfrac{\left[ M^1 L^2 T^{-2}\right]}{ \left[L^2 \right] \left[T^1 \right]} = \left[M^1 L^0 T^-3\right] \Rightarrow I = \left[MT^{-3}\right]$

So, the correct answer is “Option C”.

Note:
Additionally, we can look at Power P as well:
Power (P) =$\dfrac{Work}{Time} = watts = joule \times second^-1$
SI units: watt (W)
SI base units: $kg.m^2.s^{-3}$
Dimensional formula: $\left[ M^1 L^2 T^{-3}\right]$
Therefore, the dimensional formula allows us to express the unit of a physical quantity in terms of the fundamental quantities mass (M), length (L) and time (T).
Also, while deducing the dimensional formula from the base units be careful in assigning the right power to the right fundamental quantity each time as any discrepancy in this will yield absurd results.