Problem 45 - Master Resource Book in JEE Main Physics

Question

Assertion Pressure has the dimensions of energy density. Reason Energy density \(=\frac{\text { energy }}{\text { volume }}=\frac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}\) \(\left[\mathrm{ML}^{-1} \mathbf{T}^{-2}\right]=\) pressure

Step-by-Step Solution

Answer

The assertion is correct; pressure has the dimensions of energy density.

Step 1: Understand Dimensions

To solve this problem, first analyze the physical quantities involved. Pressure is defined as force per unit area, and energy density is energy per unit volume. We'll express these quantities in terms of fundamental dimensions (Mass \([M]\), Length \([L]\), and Time \([T]\)).

Step 2: Dimension of Pressure

Pressure is force per unit area. The force has the dimension \(MLT^{-2}\), and area has the dimension \(L^2\). Thus, the dimension of pressure is \([Pressure] = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}\).

Step 3: Dimension of Energy Density

Energy density is defined as energy per unit volume. Energy (or work) has the dimension \(ML^2T^{-2}\), and volume has the dimension \(L^3\). Therefore, the dimension of energy density is \([Energy Density] = \frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2}\).

Step 4: Compare Dimensions

Now, compare the dimensions of pressure and energy density. Both have the dimension \(ML^{-1}T^{-2}\). Thus, this confirms that pressure indeed has the same dimensions as energy density.

Key Concepts

Pressure

Pressure is a fundamental concept in physics, often defined as the force exerted per unit area. This means that when a force is applied to a surface, the pressure is determined by dividing the force by the area over which it acts.
In terms of fundamental dimensions, force has the dimension of mass times acceleration, which is represented as

• Mass \([M]\)
• Length \([L]\) measured within distance
• Time \([T]\) as the unit for duration

Acceleration is velocity change over time, combining length per time squared: \(LT^{-2}\).
Therefore, the full dimension of force becomes \(MLT^{-2}\).
Area, being a two-dimensional space, results in dimensions of \(L^2\). When we divide force by area for pressure, it results in the dimensional formula:

• Pressure: \[\text{Pressure} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}\]

This equation tells us how pressure exists as force spread across a surface.

Energy Density

Energy density describes how much energy is stored in a given volume. It's a way to express energy concentration in a particular region. Consider energy as a form of work, quantified by the ability to exert force over a distance.
This energy thus shares the same fundamental dimension as work: \(ML^2T^{-2}\).
Volume represents a three-dimensional space, bearing the dimensions \(L^3\). Examining these terms together, energy density's dimension is rendered as follows:

• Energy Density: \[\text{Energy Density} = \frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2}\]

These simplified dimensions reveal that energy density inherently matches the dimensional framework of pressure, embodying energy allocation within a volume.

Fundamental Dimensions

Fundamental dimensions are the cornerstones of dimensional analysis, allowing physical quantities to be expressed in consistent terms. By breaking down complex units into basic units of mass, length, and time, we derive understanding about how different quantities relate.

• Mass is represented as \(M\) and includes weight-related properties.
• Length is \(L\), covering distances or extents in space.
• Time, \(T\), is essential for measuring motion or dynamics.

Using these dimensions, we gain insight into the relationships among physical quantities. For instance, relating pressure and energy density becomes an exercise in recognizing that fundamentally, they share the same dimensional identity: \(ML^{-1}T^{-2}\).
This understanding, facilitated by fundamental dimensions, validates reasons for such comparisons and strengthens our comprehension of physically abstract concepts like pressure or energy, ultimately streamlining comparisons between them.