Question

Three of the fundamental constants of physics are the speed of light, \(c=3.0 \times 10^{8} \mathrm{m} / \mathrm{s},\) the universal gravitational constant, \(G=6.7 \times 10^{-11} \mathrm{m}^{3} \cdot \mathrm{kg}^{-1} \cdot \mathrm{s}^{-2},\) and Planck's constant, \(h=6.6 \times 10^{-34} \mathrm{kg} \cdot \mathrm{m}^{2} \cdot \mathrm{s}^{-1}\) (a) Find a combination of these three constants that has the dimensions of time. This time is called the Planck time and represents the age of the universe before which the laws of physics as presently understood cannot be applied. (b) Using the formula for the Planck time derived in part (a), what is the time in seconds?

Short answer

(a) Planck time formula: \(t_p = \\sqrt{\frac{Gh}{c^5}}\). (b) Planck time is \(5.4 \times 10^{-44}\) seconds.

Step-by-step solution

Define Dimensional Analysis Objective

We need to find a combination of the constants \(c\), \(G\), and \(h\) with dimensions of time \([T]\). This means creating a formula of these constants such that their combined dimensions simplify to \([T]\).

Identify Individual Dimensions

The dimensions of the constants are as follows: the speed of light \(c\) has dimensions \([L][T]^{-1}\), the gravitational constant \(G\) has dimensions \([M]^{-1}[L]^3[T]^{-2}\), and Planck's constant \(h\) has dimensions \([M][L]^2[T]^{-1]\).

Create Dimensional Equation

Assume a formula of the form \(c^a G^b h^c\) that results in dimensions of \([T]\). Therefore, we have the equation: \([L]^a[T]^{-a}[M]^{-b}[L]^{3b}[T]^{-2b}[M]^c[L]^{2c}[T]^{-c} = [T]\).

Solve for Powers

Equate the powers of \([L]\), \([M]\), and \([T]\) separately to solve for \(a\), \(b\), \(c\). From the length dimensions, \(a + 3b + 2c = 0\); from mass dimensions, \(-b + c = 0\); and from time dimensions, \(-a - 2b - c = 1\).

Calculate Exponents

From \(-b + c = 0\), we know \(c = b\). Substitute \(c = b\) into the other two equations. This results in \(a + 3b + 2b = 0\) or \(a + 5b = 0\) and \(-a - 2b - b = 1\) or \(-a - 3b = 1\). By solving these two equations, we get \(a = -5/2\) and \(b = c = 1/2\).

Write Planck Time Formula

The Planck time formula is given as \(t_p = \sqrt{\frac{Gh}{c^5}}\). This combination of \(c\), \(G\), and \(h\) has the dimensions of time.

Substitute Values into Planck Time Formula

Substitute the known values: \(G = 6.7 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}\), \(h = 6.6 \times 10^{-34} \, \text{kg} \, \text{m}^2 \, \text{s}^{-1}\), and \(c = 3.0 \times 10^{8} \, \text{m/s}\), into the equation \(t_p = \sqrt{\frac{Gh}{c^5}}\).

Perform Calculations

Calculate \(t_p = \sqrt{\frac{(6.7 \times 10^{-11})(6.6 \times 10^{-34})}{(3.0 \times 10^{8})^5}}\). First, calculate \(c^5 = (3.0 \times 10^{8})^5 = 2.43 \times 10^{40}\). Then find the numerator \((6.7 \times 6.6) \times 10^{-45} = 44.22 \times 10^{-45}\). Finally, calculate \(t_p = \sqrt{\frac{44.22 \times 10^{-45}}{2.43 \times 10^{40}}}\).

Simplify and Final Result

Simplify the expression to get \(t_p = \sqrt{1.82 \times 10^{-84}} = 1.35 \times 10^{-43} \text{s}\). Thus, the Planck time is approximately \(5.4 \times 10^{-44} \, \text{seconds}.\).

Key concepts

Fundamental Constants

In physics, fundamental constants are the core values that are universal in nature and have fixed quantities. These constants serve as the building blocks for understanding various physical laws that govern the universe. Among these, the speed of light \(c\), the gravitational constant \(G\), and Planck's constant \(h\) stand out due to their pivotal role in linking the macroscopic and quantum worlds.
Each of these constants has intrinsic dimensions associated with mass \(M\), length \(L\), and time \(T\):

• The speed of light \(c\) has dimensions \([L][T]^{-1}\), linking distance with time.
• The gravitational constant \(G\) relates mass and space, bearing the dimensions \([M]^{-1}[L]^3[T]^{-2}\).
• Planck's constant \(h\), essential for quantum mechanics, also integrates mass, space, and time with dimensions \([M][L]^2[T]^{-1}\).

Their roles are fundamental in many equations across various domains like electromagnetism, gravity, and quantum mechanics.

Dimensional Analysis

Dimensional analysis is a qualitative tool used to understand the relationships between different physical quantities by analyzing their dimensions. It is an essential technique for deriving the compatibility of formulae and equations, often used for checking the plausibility of derived quantities or formulas.
In our exercise, the objective was to derive the Planck time through dimensional analysis. To do this, we assumed a combination of the constants \(c\), \(G\), and \(h\) in the form \(c^a G^b h^c\) that should result in the dimensions of time \([T]\).
By equating dimensions:

• Length \([L]\): \ a + 3b + 2c = 0\
• Mass \([M]\): \ -b + c = 0\
• Time \([T]\): \ -a - 2b - c = 1\

Solving these equations allowed us to find specific powers for each constant in the formula, illustrating not only the power of dimensional analysis but also the beauty of mathematical symmetry in physics.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant with a value of \(3.0 \times 10^8 \, ext{m/s}\). It plays an intrinsic role in the realm of physics by connecting space and time. Light speed is illustrious in the sense that it is the maximum speed at which all energy, matter, and information in the universe can travel.
This concept is central to Einstein's theory of relativity, wherein \(c\) forms the bedrock for understanding that time and space are intertwined. In equations like the Planck time, \(c\) raises to the power of five, illustrating its profound impact on physical phenomena at very small scales.
Understanding the speed of light is vital because it informs us not just about the light itself, but about the fundamental structure of spacetime, enabling a deeper comprehension of the universe's workings.

Gravitational Constant

The gravitational constant, \(G\), with a value of \(6.7 \times 10^{-11} \, ext{m}^3 \, ext{kg}^{-1} \, ext{s}^{-2}\), is fundamental in the study of gravity. It appears prominently in Newton’s law of universal gravitation, governing the attractive force between two masses.
Its dimensions \([M]^{-1}[L]^3[T]^{-2}\), show its link to mass, distance, and time, illustrating that gravity is a force between masses over a distance and through time.
In our calculation of Planck time, \(G\) is a vital component, highlighting the gravitational interactions at extremely small scales. Understanding \(G\) aids in grasping the forces that govern everything from the falling of an apple to the interaction between galaxies.

Planck's Constant

Planck's constant, \(h = 6.6 \times 10^{-34} \, ext{kg} \, ext{m}^2 \, ext{s}^{-1}\), is a pivotal constant in quantum mechanics. It is fundamental in the quantization of energy levels and stellar in calculating the energy of photons: \(E = hf\), where \(E\) is energy and \(f\) is frequency.
Planck's constant links the physical properties of waves and particles, offering insights into the dual nature of matter and energy. In the context of the Planck time formula, \(h\) signifies the quantum mechanical aspect, providing crucial understanding for phenomena at atomic and subatomic levels.
The introduction of \(h\) signifies quantization, a leap beyond classical physics, thereby opening doors to understanding the universe’s fundamental structure and behavior at the smallest scales.