Question

The dimensions of potential gradient are:-
A. ${\text{ML}}{{\text{T}}^{ - 3}}{{\text{A}}^{ - 1}}$
B. ${\text{ML}}{{\text{T}}^{ - 2}}{{\text{A}}^{ - 2}}$
C. ${{\text{M}}^{ - 1}}{{\text{L}}^{ - 1}}{{\text{A}}^{ - 1}}$
D. ${\text{ML}}{{\text{T}}^{ - 2}}$

Answer 1

Hint: A potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. So, we are to deduce the formula of potential gradient on the basis of the quantities whose dimensions are known to us. Then apply their dimensional analysis to get the dimensions of the potential gradient.

Complete step by step answer:
Potential gradient is the rate of change of potential (energy) with position.
For example, if ${\text{V}}\left( x \right)$ is potential then on a ${\text{V}}\left( x \right)$ versus $x$ graph, the gradient is the slope of the curve at any point $x$.
So,${\text{Gradient = }}\dfrac{{{\text{Change in potential}}}}{{{\text{Change in position point}}}}$
So, Dimension $\left[ {{\text{Gradient}}} \right] = \dfrac{{\left[ {{\text{Potential/Energy}}} \right]}}{{{\text{[Length]}}}}$

Now, the dimension of potential/energy, we know,
$\left[ {{\text{Potential}}} \right] = {\text{M}}{{\text{L}}^2}{{\text{T}}^{ - 2}}$
And, the dimension of length, we know,
$\left[ {{\text{Length}}} \right] = {\text{L}}$
So, applying the dimensions in the dimensional formula of gradient, we get,
$\left[ {{\text{Gradient}}} \right] = \dfrac{{{\text{M}}{{\text{L}}^2}{{\text{T}}^{ - 2}}}}{{\text{L}}}$
$\therefore \left[ {{\text{Gradient}}} \right] = {\text{ML}}{{\text{T}}^{ - 2}}$

Therefore, the correct option is D.

Note: -The dimensional analysis is required to check the correctness and validity of any relation. If the dimensions of the right hand side and left hand side of any relation doesn’t match, then they will not be correct, dimensional analysis plays a vital role in validity. We can also use dimensional analysis to derive the relationship between various physical quantities. We can determine the dimensions of unknown quantities if we know the formula of the quantity. We can convert one system of units to another system of units with the help of dimensional analysis.