Problem 18 - Master Resource Book in JEE Main Physics

Question

An important milestone in the evolution of the universe just after the Big Bang is the Planck time \(t_{p}\), the value of which depends on three fundamental constants speed \(c\) of light in vacuum, gravitational constant \(G\) and Planck's constant \(h\). Then, \(t_{p} \propto\) (a) \(\mathrm{Ghc}^{5}\) (b) \(\frac{c^{5}}{G h}\) (c) \(\frac{G h}{c^{5}}\) (d) \(\left(\frac{G h}{c^{5}}\right)^{1 / 2}\)

Step-by-Step Solution

Answer

(d) \( \left(\frac{Gh}{c^5}\right)^{1/2} \)

Step 1: Understanding Planck Time

Planck time \( t_{p} \) is a unit of time in the system of Planck units, representing the time it takes for light to travel one Planck length in a vacuum. It depends on the gravitational constant \( G \), Planck's constant \( h \), and the speed of light \( c \).

Step 2: Proportionality Relation for Planck Time

The proportionality for Planck time \( t_{p} \) can be derived from its formula. \( t_{p} \) is given by the square root of the ratio of the product of gravitational constant \( G \) and Planck's constant \( h \) to the fifth power of the speed of light \( c \): \[ t_{p} \propto \sqrt{\frac{Gh}{c^5}} \].

Step 3: Understanding the Options

Given the options, we need to match the derived proportionality expression for \( t_{p} \) to one of the options. The correct relation from our derivation is \( \left(\frac{Gh}{c^5}\right)^{1/2} \).

Step 4: Selecting the Correct Answer

Compare our derived expression \( \left(\frac{Gh}{c^5}\right)^{1/2} \) with the given options. Option (d) matches our derived expression. Therefore, option (d) is the correct choice.

Key Concepts

Fundamental Constants

Fundamental constants are the building blocks of physics. They are special numerical values that are universally constant and do not change with time or space. These constants are essential because they form the foundation for the laws of physics that govern our universe.
For example, the speed of light in a vacuum, denoted as \(c\), is a fundamental constant with a value of approximately \(3 \, \times \, 10^8\) meters per second. This constant is crucial in the equations of relativity and describes the maximum speed that anything in the universe can travel.

• The gravitational constant \(G\) is another key constant, and it's involved in the formula for gravitational force. It signifies the strength of gravity between two masses in the universe.
• Planck's constant \(h\) is vital in quantum mechanics and helps to relate the energy of photons to their frequency.

Understanding these constants allows us to create units like the Planck units and explore phenomena from across the universe, starting from the smallest particles to vast cosmic structures.

Big Bang Cosmology

Big Bang Cosmology is a branch of science that studies the origins and development of the universe. According to this theory, the universe began as a hot, dense singularity approximately 13.8 billion years ago and has been expanding ever since.
The Big Bang marked the birth of the universe, leading to the formation of all matter and fundamental forces.

• This model helps explain phenomena such as the cosmic microwave background radiation, which is the thermal radiation left over from the early stages of the universe after the Big Bang.
• It also accounts for the distribution of galaxies and the expansion of space, confirmed by observations of redshifts in galaxy light.

By studying the Big Bang, scientists gain insights into the very first moments of time, including important milestones like Planck Time— one of the smallest timescales conceivable in physics where quantum gravitational effects are significant.

Planck Units

Planck Units offer a natural system of measurement designed to provide insight into the universe using fundamental constants. Introduced by Max Planck, these units are based on constants like the speed of light \(c\), the gravitational constant \(G\), and Planck's constant \(h\).
Unlike conventional measurements such as meters or seconds, Planck units aim to simplify the equations of physics. They express quantities such as mass, length, and time, using relationships between basic constants.

• Planck time \(t_{p}\) signifies the smallest measurable time interval in quantum physics. It derives from these constants, showing how deep these constants are tied to our understanding of the universe.
• Other examples include Planck length, which provides a lower limit on the smallest possible size of a particle, and Planck mass, which is the mass of a particle whose Compton wavelength is equal to the Planck length.

Planck units help create a framework where the equations of physics appear simpler, offering a more profound insight into universal laws.

Gravitational Constant

The gravitational constant, symbolized as \(G\), is fundamental in understanding the force of gravity in our universe. Its value is approximately \(6.674 \, \times \, 10^{-11} \, \text{Nm}^2/\text{kg}^2\).
Gravity is one of the four fundamental forces but is unique as it affects all matter in the universe and operates over vast distances.

• \(G\) appears in Newton's law of universal gravitation and is key in Einstein's general theory of relativity.
• It aids in calculating the force of gravity between two masses, crucial for understanding celestial mechanics and phenomena such as the orbits of planets and the behavior of stars.

Since gravity is consistent and ever-present, \(G\) also plays a central role in various physical theories, including those aiming to unify all natural forces. Its constancy allows scientists to explore gravitational interactions not only on Earth but also across the cosmic landscape.