Question

The dimensional formula of the pressure gradient is
A. \[M{L^{ - 1}}{T^{ - 2}}\]
B. \[{M^1}{L^{ - 2}}{T^{ - 2}}\]
C. \[{M^1}{L^2}{T^{ - 2}}\]
D. \[{M^1}{L^{ - 1}}{T^{ - 3}}\]

Answer 1

Hint: Dimension of pressure gradient can be found by using the formula
P= (change in pressure)/ (change in distance)
To find the dimension of pressure gradient first, we need to find the dimension of the pressure of that location and then it is divided by dimension of distance.

Complete step by step solution:
Pressure gradient helps to determine the direction in which pressure increases most rapidly from a location. The pressure gradient can be measured by finding the difference in pressure between two locations and dividing the difference by the distance between the two locations. The pressure gradient force or the difference in air pressure is caused by unequal heating of the earth surface.

A pressure gradient is a dimensional unit expressed as Pascal’s per meter (Pa/m). To find the dimension of the pressure gradient first find the dimension of pressure where pressure is defined as the force (F) applied perpendicular to the surface of an object whose area is given as (A).
\[P = \dfrac{F}{A} = \dfrac{{m \times a}}{A} = \dfrac{{\left[ M \right]\left[ {L{T^{ - 2}}} \right]}}{{\left[ {{L^2}} \right]}} = \left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]\]
As pressure gradient is given as the ratio of change in pressure between two locations to the distance between two locations hence the dimension of pressure gradient will be
Pressure Gradient\[ = \dfrac{P}{L} = \dfrac{{\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]}}{{\left[ L \right]}} = \left[ {{M^1}{L^{ - 2}}{T^{ - 2}}} \right]\]

Hence the dimension of Pressure Gradient is \[\left[ {{M^1}{L^{ - 2}}{T^{ - 2}}} \right]\].
Option (B) is correct.

Note: It is to be noted here that the question is asking for the dimensional formula for which we should always use the SI units and not for any other units. There are different ways of indicating the units such as CGS units and MKS units but SI units are to be used unless and until the question demands any other units.