Dimensions of Pressure Gradient

The pressure gradient (generally of air, but more broadly of any fluid) is a physical quantity in atmospheric science that describes the direction and rate at which pressure increases most rapidly around a specific location. The pressure gradient is really a three-dimensional quantity measured in pascals per meter (Pa/m). Arithmetically, it is the gradient of pressure as a function of position. The force density has been defined as the negative gradient of pressure.

A pressure gradient is just a local characterization of the air (more generally of the fluid under investigation). The pressure gradient will only be defined at the spatial scales where the pressure (or, more broadly, fluid dynamics) is defined.

The pressure gradient within planetary atmospheres (including the Earth’s) is a vector pointing roughly downwards, because pressure changes most rapidly vertically, increasing downwards (see vertical pressure variation). The strength (or norm) of the pressure gradient in the troposphere is typically around 9 Pa/m (or 90 hPa/km).

The pressure gradient frequently includes a small but critical horizontal component that is largely responsible for wind circulation in the atmosphere. The lateral pressure gradient is a two-dimensional vector that results from the pressure gradient being projected onto a local horizontal plane. One such horizontal pressure gradient force is directed from higher to lower pressure near the Earth’s surface. Its specific orientation at any given time and location is strongly influenced by the weather conditions. At semi-latitudes, the normal horizontal pressure gradient can be in the order of 102 Pa/m (or 10 Pa/km), though much higher values can be found along meteorological fronts.

The pressure-gradient force is created when there is a pressure difference across a surface. In fact, pressure has been defined as a force per unit area applied to a surface. When there is no additional force to balance it, a difference in pressure across a surface implies a difference in force, which can result in acceleration according to Newton’s second law of motion.

Dimensional Formula of Pressure Gradient

The dimensional formula of pressure gradient can be represented as:

[M¹ L^-2 T^-2]

Here,

M = Mass

L = Length

T = Time

Derivation:

We have, Pressure Gradient = Pressure × [Distance]^-1 . . . . (1)

Then the dimensional formula of distance = [M⁰ L¹ T⁰] . . . . (2)

⇒ Pressure = Force × [Area]^-1 . . . . (3)

As because, Force = mass × acceleration

Therefore, the dimensions of force = [M] × [L¹ T^-2] = [M¹ L¹ T^-2] . . . (4)

Similarly, the dimensional formula of area = [M⁰ L² T⁰] . . . (5)

When substituting equation (4) and (5) in equation (3) we get,

Pressure (P) = [M¹ L¹ T^-2] × [M⁰ L² T⁰]^-1

Thus, the dimensional formula of pressure = [M¹ L^-1 T^-2] . . . . (6)

When substituting equation (2) and (6) in equation (1) we get,

Pressure Gradient = Pressure × [Distance]^-1

That is, P = [M¹ L^-1 T^-2] × [M⁰ L¹ T⁰]^-1 = [M¹ L^-2 T^-2]

Thus, the pressure gradient can be dimensionally represented as [M¹ L^-2 T^-2].

Significance of dimensions of pressure gradient in IIT JEE exam

It is critical to take a holistic approach to every facet of a subject’s chapter. It will not only adequately prepare you for the exam, but will also clarify your understanding of each topic. It will help you in JEE preparation and answer conceptual problems in the exam. The number of questions from the chapter unit and dimensions would be one or two, with a weightage of roughly four marks.

FAQs for Dimensions of Pressure Gradient

Q. What is called a pressure gradient?

Ans: A “pressure gradient” is indeed the change in pressure measured over a given distance. The pressure gradient produces a net force directed from high to low pressure, which is known as the “pressure gradient force.”

Q. How do you calculate pressure gradient?

Ans: The pressure gradient could be calculated mathematically by dividing the pressure difference between two locations (in Pascals) by the distance between the two locations (in meters).

Q. What is the dimensional formula of pressure gradient?

Ans: The pressure gradient can be dimensionally represented as [M¹ L^-2 T^-2].

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