Question

If energy (E), velocity (V) and time (T) are chosen as the fundamental quantities, the dimensional formula of surface tension will be:
$\text{A.}\quad [EV^{-2}T^{-2}]$
$\text{B.}\quad [E^{-2}V^{-2}T^{-3}]$
$\text{C.}\quad [EV^{-2}T^{-1}]$
$\text{D.}\quad [EV^{-2}T^{-2}]$

Answer 1

Hint: Dimensional formula is a set or combination of different units which indicate a particular physical quantity. This formula tells us the dependency of a particular physical quantity on another. This formula is not unique which means more than one physical quantity can have the same dimensional formula.

Complete step-by-step answer:
In order to get the dimensional formula for viscosity, we shall know basic formulas of quantities regarding it.
Now, surface tension is force acting per unit length. Thus $T = \dfrac Fl$
Also, Work done = F.s and we know that energy has the same dimensions as that of work. So units of energy are F.s.
Or $F = \dfrac{Energy}{length}$
$\implies T = \dfrac{Energy}{l^2}$
Now, velocity is the rate of change of displacement (length). Hence,
$v = \dfrac{l}{t}$
$\implies l = vt$
And hence $T = \dfrac{Energy}{ (vt)^2}$
Denoting energy, velocity and time with $E, V\ and\ T$ respectively, we get;
$T = E^1 V^{-2} T^{-2}$
Hence $T = [E V^{-2} T^{-2}]$, option A. is correct.

So, the correct answer is “Option A”.

Additional Information: It is to be noted that many physical quantities have same physical dimensions by they can be completely different in every aspect like work and torque, both have same dimensions $[M^1 L^{2}T^{-2}]$ which are also the dimensions of heat given.

Note: Students are advised to learn basic quantities’ dimensions only (for example displacement, angle, intensity, mass, etc.) and complex quantities’ dimensions (for example torque, pressure, force, etc.) can be easily derived by knowing the formulae and proceeding as shown in the above example. Also, the above question could be solved by approaching in the way of writing surface energy which is defined as the work done per unit surface area. Hence it is stored as surface energy which also means energy stored per unit area.