Soumendra Nath Thakur
09-07-2024
Max Planck formulated his expressions for the Planck scale in 1899, which was before the development of both relativity and quantum mechanics. The relationship between the Planck length (ℓₚ) and the Planck time (tₚ) to find the maximum speed of electromagnetic waves.
Given:
ℓₚ is the Planck length, which is the smallest possible perceptible length.
tₚ is the Planck time, which is the shortest possible meaningful time.
The speed of propagation vᵥᵥₐᵥₑ is the distance the wave travels in a given time, which is one wavelength in a time of one period. In equation form, it is written as: vᵥᵥₐᵥₑ = λ/T.
Since, vᵥᵥₐᵥₑ = λ/T = fλ, the maximum speed of electromagnetic waves, vᵥᵥₐᵥₑ(ₘₐₓ), can be given by the ratio of the Planck length to the Planck time:
vᵥᵥₐᵥₑ(ₘₐₓ) = ℓₚ/tₚ
The Planck length (ℓₚ) is defined as:
ℓₚ = √ℏG/c³
The Planck time (tₚ) is defined as:
tₚ = √ℏG/c⁵
To find vᵥᵥₐᵥₑ(ₘₐₓ), substitute these definitions into the ratio:
vᵥᵥₐᵥₑ(ₘₐₓ) = √(ℏG/c³)/√(ℏG/c⁵)
Simplify the expression:
vᵥᵥₐᵥₑ(ₘₐₓ) = √(ℏG/c³)/√(ℏG/c⁵) = √c²
vᵥᵥₐᵥₑ(ₘₐₓ) = c
Thus, the maximum speed of electromagnetic waves is:
vᵥᵥₐᵥₑ(ₘₐₓ) = c
This confirms that the maximum speed of electromagnetic waves is the speed of light (c), which aligns with our current understanding of physics.
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