p = E/c = mc = hf/c = hλ
p represents momentum.
E represents energy.
c represents the speed of light in vacuum.
m represents mass.
h represents the Planck constant.
f represents frequency.
λ represents wavelength.
Each of these quantities has specific meanings and units:
Momentum (p) is the product of an object's mass (m) and its velocity (v). In the equation p = E/c = mc, the first term represents momentum, and it is equal to the energy (E) divided by the speed of light (c).
The equation E = mc^2 represents the famous mass-energy equivalence relationship proposed by Einstein in his theory of special relativity. It states that energy (E) is equal to mass (m) multiplied by the speed of light squared (c^2).
In the equation hf/c, h represents the Planck constant, f represents frequency, and c is the speed of light. This equation relates the energy of a photon (hf) to its momentum (p) through the speed of light (c).
Finally, the equation hλ relates the Planck constant (h) to the wavelength (λ) of a wave or particle.
It's important to note that these equations are derived and applicable within specific physical theories, such as special relativity and quantum mechanics. They have been extensively tested and confirmed by experimental observations. Each equation represents a different aspect of physical phenomena and provides a mathematical description of the relationships between various quantities.
p = E/c: This equation relates momentum (p) to energy (E) through the speed of light (c). It is derived from special relativity and indicates that the momentum of a particle is equal to its energy divided by the speed of light.
mc: This equation represents the relativistic mass (m) of an object multiplied by the speed of light (c). It is another formulation derived from special relativity, which relates mass and energy.
hf/c: This equation relates the momentum (p) of a photon to its frequency (f) and the speed of light (c). It is derived from the equation for the momentum of a photon, which is given by p = hf/c, where h is the Planck constant.
hλ: This equation relates the momentum (p) of a photon to its wavelength (λ) through the Planck constant (h). It is another formulation of the equation for the momentum of a photon.
Additional: Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants. Originally proposed by the German physicist Max Planck in 1899, these units are a system of natural units because their definition is based on properties of nature. It may be mentioned here that Einstein first published his special theory of relativity in 1905, which describes his revolutionary ideas about light, time and energy.
The four universal constants, by definition, have a numerical value of 1 when expressed in these units:
1. Speed of light in vacuum, c,
2. Gravitational constant, G,
3. Reduced Planck constant, ħ, and
4. Boltzmann constant, kB.
• Planck length = ℓP = L ≈ 1.61626 × 10^−35 m;
• Planck time = tP = T ≈ 5.391247 × 10^−44 s;
• ℓP/tP is the ratio of the Planck length to the Planck time;
Since, ℓP/tP = (1.61626 × 10^−35 m) / (5.391247 × 10^−44 s);
1. To divide two numbers in scientific notation, we subtract the exponents of the 10 and divide the coefficients:
2. Coefficient: (1.61626) / (5.391247) ≈ 0.299792458
3. Exponent: (10^(-35)) / (10^(-44)) = 10^(-35 - (-44)) = 10^9
4. So the simplified value is approximately:
5. 0.299792458 × 10^9 m/s
6. Now, we recognize that this is the speed of light in a vacuum, which is denoted by 'c':
7. c ≈ 2.99792458 × 10^8 m/s
8. So, the simplified expression is:
9. (1.61626 × 10^−35 m) / (5.391247 × 10^−44 s) ≈ 2.99792458 × 10^8 m/s;
The ratio of the Planck length to the Planck time (ℓP/tP) yields a value to the speed of light in a vacuum, c;
This is a fundamental constant in physics and is denoted by 'c'.
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