Problem 30 - Master Resource Book in JEE Main Physics

Question

If Planck's constant ( \(h\) ) and speed of light in vacuum (c) are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities? \(\quad\) [NCERT Exemplar] (a) Mass of electron \(\left(m_{e}\right)\) (b) Universal gravitational constant \((G)\) (c) Charge of clectron (e) (d) Mass of proton \(\left(m_{p}\right)\)

Step-by-Step Solution

Answer

(b) Universal gravitational constant (G)

Step 1: Understand the Problem

Given fundamental quantities are Planck's constant \( (h) \) and speed of light in vacuum \( (c) \). We need to find which additional quantity allows us to define length, mass, and time using these as base units.

Step 2: Key Concept

Planck's constant \( (h) \) has units of \([ ext{Energy}] imes [ ext{Time}] = ([ML^2T^{-1}]) \). Speed of light \( (c) \) has units \([LT^{-1}]\). We need a third quantity that covers all necessary dimensions of length, mass, and time.

Step 3: Check Each Option for Dimensional Analysis

Evaluate each option for potential as a base unit.- **Mass of electron,** \( m_e: [M] \)- **Gravitational constant,** \( G: [M^{-1}L^3T^{-2}] \)- **Charge of electron,** \( e: [I] \)- **Mass of proton,** \( m_p: [M] \)

Step 4: Select the Correct Additional Quantity

Choosing \( m_e \) or \( m_p \) will help in defining mass but leave us unable to completely define length and time when combined with \( h \) and \( c \). The charge \( e \) introduces a different dimension (current), which is unnecessary. Using the universal gravitational constant \( G \) with \( c \) and \( h \) allows calculation of length, mass, time dimensions (due to \( G \) involving all these dimensions).

Step 5: Final Conclusion

Out of the given options, the Universal gravitational constant \( G \) is the only additional fundamental quantity that when combined with Planck’s constant \( h \) and the speed of light \( c \), can comprehensively describe length, mass, and time dimensions.

Key Concepts

Planck's Constant

Planck's constant, denoted as \( h \), is a fundamental physical constant that plays a crucial role in the field of quantum mechanics. It is primarily used to describe the sizes of quanta, or the smallest possible discrete units of energy. Planck's constant is pivotal because it establishes the scale of quantum effects. Without it, we wouldn't be able to describe phenomena that occur at the atomic and subatomic levels effectively.

The dimensions of Planck’s constant can be expressed as energy multiplied by time, or \([\text{ML}^2\text{T}^{-1}]\). Here, \( M \) represents mass, \( L \) stands for length, and \( T \) is time. This dimension suggests that Planck’s constant can serve as a key component in the dimensional analysis to define other physical quantities.

When combined with other fundamental constants such as the speed of light, \( h \) helps to bridge different realms of physics, providing a coherent framework for understanding both the vast cosmos and the minute quantum world. By understanding Planck's constant, students can appreciate its impact on modern physics and its essential role in expressing fundamental units.

Speed of Light

The speed of light, commonly denoted by \( c \), is another fundamental constant in physics, with a value of approximately \( 3 \times 10^8 \) meters per second in a vacuum. It signifies the fastest speed at which information and matter can propagate through the universe.

In dimensional terms, the speed of light is expressed as \([\text{LT}^{-1}]\), where \( L \) represents length and \( T \) stands for time. This showcases its prominence in linking these two basic aspects of the physical world. Light plays a key role in our understanding of not just optics, but also in a wider context, helping us unravel the fabric of space-time in theories like General Relativity.

The speed of light, when used alongside Planck's constant, forms a basis for establishing dimensionless quantities and converting between different physical units. In many physical theories and equations, \( c \) is used as a conversion factor between mass-energy equivalence, time, and space, reinforcing its place as a cornerstone in modern physics.

Universal Gravitational Constant

The universal gravitational constant is symbolized as \( G \) and is key to the law of universal gravitation formulated by Isaac Newton. It characterizes the intensity of the gravitational force that acts between two bodies with mass, defining how mass exerts an attraction over a distance. This constant is crucial for calculations related to gravitational forces and helps in explaining the orbits of planets and the motion of celestial bodies.

From a dimensional standpoint, \( G \) has the dimensions of \([\text{M}^{-1}\text{L}^3\text{T}^{-2}]\). Here, \( M \), \( L \), and \( T \) again stand for mass, length, and time, respectively. These dimensions make \( G \) a valuable component when combining it with other fundamental constants such as Planck's constant and the speed of light. This allows it to bridge the interrelation between mass, distance, and time in a broad spectrum of physical phenomena.

In particular, in the scenario of dimensional analysis, \( G \), when allied with \( h \) and \( c \), offers a powerful framework to define a coherent system that can express fundamental units like length, mass, and time. This blend of constants forms a triad that is especially significant in advanced and theoretical physics, serving as a linchpin for synthesizing various concepts into a unified understanding of natural laws.