Problem 11 - Master Resource Book in JEE Main Physics

Question

Given \(X=\left(G h / c^{3}\right)^{1 / 2}\), where \(G, h\) and \(c\) are gravitational constant, Planck's constant and the velocity of light respectively. Dimensions of \(X\) are the same as those of (a) mass (b) time (c) length (d) acceleration

Step-by-Step Solution

Answer

The dimensions of \(X\) match length (option c).

Step 1: Identify Given Variables and Their Dimensions

In the equation \(X = \left(Gh/c^3\right)^{1/2}\), \(G\), \(h\), and \(c\) are given as constants with known dimensions. \(G\) is the gravitational constant with dimensions \([M^{-1} L^3 T^{-2}]\), \(h\) is the Planck's constant with dimensions \([M L^2 T^{-1}]\), and \(c\) is the speed of light with dimensions \([L T^{-1}]\).

Step 2: Substitute Dimensional Formulae into X

Substitute the dimensional formulae of each constant into the expression for \(X\): \(X = \left([M^{-1} L^3 T^{-2}][M L^2 T^{-1}]/[L^3 T^{-3}]\right)^{1/2}\).

Step 3: Simplify the Dimensional Equation

Simplify \([M^{-1} L^3 T^{-2}][M L^2 T^{-1}]/[L^3 T^{-3}]\). This results in \([M^0 L^2 T^0]\), which simplifies to \([L]\). This indicates the dimensions of \(X\) are \([L]\).

Step 4: Match Dimensions with Given Options

The dimensions of \(X\) are \([L]\), which corresponds to length. Among the given options, length is represented by option (c).

Key Concepts

Gravitational Constant

The gravitational constant, denoted by \(G\), is a fundamental constant in physics that plays a critical role in the law of universal gravitation formulated by Isaac Newton. It essentially sets the scale of the gravitational force between two masses. It has a fixed value of approximately \(6.674 \times 10^{-11} \: \text{N} \, \text{(m/kg)}^2\). The dimensions of the gravitational constant are denoted as \([M^{-1} L^3 T^{-2}]\).
These dimensions help us understand how \(G\) ties mass, length, and time together in terms of gravitational interaction. When dealing with equations involving \(G\), such as the one in the exercise, these dimensions are crucial for converting the equation into a form that matches physical quantities like force or energy.

• \(M\) represents mass.
• \(L\) represents length.
• \(T\) represents time.

Planck's Constant

Planck's constant, symbolized by \(h\), is a key figure in quantum mechanics. It relates the energy of a photon to the frequency of its associated electromagnetic wave. The value of Planck's constant is approximately \(6.626 \times 10^{-34} \, \text{Js}\).
It has dimensions of \([M L^2 T^{-1}]\). These dimensions highlight that Planck's constant ties energy (in joules), distance squared, and inverse time together.
Understanding \(h\) is essential in various quantum phenomena, such as the photoelectric effect, where knowing the frequency of light helps calculate the energy of emitted electrons. When used in dimensional analysis, the properties of \(h\) can help predict outcomes in quantum systems.

• \(M L^2\) signifies energy-related aspects.
• \(T^{-1}\) indicates a component related to time or frequency.

Velocity of Light

The velocity of light, commonly represented by \(c\), is a crucial constant in physics, particularly in the theories of relativity and optics. The exact value of this constant is approximately \(3 \times 10^8 \, \text{m/s}\).
The dimensions for the velocity of light are given as \([L T^{-1}]\). These dimensions reflect how the speed of light integrates space and time. It sets a universal speed limit that cannot be surpassed by any object with mass.
The velocity of light plays a central role not only in the realm of physics but also in dimensional analysis, where it assists in deriving relationships in various equations. In the context of our exercise, \(c\) helped balance dimensional equations for determining the nature of \(X\).

• \(L\) relates to space.
• \(T^{-1}\) involves the time factor.

Units of Measurement

Units of measurement are fundamental in the field of physics and engineering. They provide a standard way to quantify and describe physical quantities, allowing communication and understanding to be clear and consistent.

• Units like meters, kilograms, and seconds are part of the SI system, which is widely adopted globally.
• Understanding the relationships between different units is key in dimensional analysis, which involves breaking down complex equations to discern underlying physical quantities.

Dimensional analysis, as demonstrated in the exercise, involves equating the units of measurement to balance equations. This technique verifies that mathematical expressions are physically meaningful by ensuring consistency between units.
For instance, harmonizing the units of \(G\), \(h\), and \(c\) provides insight into determining the dimensions of \(X\), leading us to conclude that it has the dimensions of length \([L]\).
In academics and real-world applications beyond physics, units of measurement facilitate crucial calculations, predictions, and comparisons necessary for scientific inquiry and technological advancements.