Finding Planck time in terms of [G], [h], and [c]

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SUMMARY

The discussion focuses on deriving the Planck time using the fundamental constants: gravitational constant [G], Planck's constant [h], and the speed of light [c]. The correct exponents for the equation [G]^a [h]^b [c]^g to yield dimensions of time are a = 1/2, b = 1/2, and g = -5/2. The calculated value for Planck time, tPlanck = G^(1/2) * h^(1/2) * c^(-5/2), results in approximately 1.3512189 x 10^-43 seconds, which is incorrect due to the use of [h] instead of the reduced Planck constant [ħ].

PREREQUISITES
  • Understanding of dimensional analysis in physics
  • Familiarity with fundamental constants: gravitational constant [G], Planck's constant [h], and speed of light [c]
  • Basic algebra for solving equations with exponents
  • Knowledge of the reduced Planck constant [ħ] and its significance
NEXT STEPS
  • Research the significance and calculation of the reduced Planck constant [ħ]
  • Study dimensional analysis techniques in physics
  • Explore the derivation of other Planck units, such as Planck length and Planck mass
  • Learn about the implications of Planck time in quantum mechanics and cosmology
USEFUL FOR

Students and professionals in physics, particularly those studying quantum mechanics, general relativity, and dimensional analysis. This discussion is beneficial for anyone looking to deepen their understanding of fundamental constants and their applications in theoretical physics.

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Homework Statement



Find a set of 3 exponents a, b, g such that the quantity [G]^a [h]^b [c]^g has dimensions of time. In other words:
[T] = [G]^a [h]^b [c]^g
(hint: you should get 3 algebraic equations in the 3 unknowns a, b, g which are not that hard to solve. Now determine the socalled
Planck time from tPlanck = G^a h^b c^g. [3 Points].

Homework Equations



G = 6.67266 x 10^-11 m^3 kg^-1 s^-2
h = 6.62606896x10^-34 kg m^2 s^-1
c = 299,792,458 m s^-1

The Attempt at a Solution



I solved a = 1/2, b = 1/2, g = -5/2, which cancel out the dimensions other than [T],
and then I tried to calculate the value for Placnk time:

(6.67266 x 10^-11)^(1/2) (6.62606896x10^-34)^(1/2) (299,792,458)^(-5/2)
= 1.3512189 x10^-43

Which is not what it should be.. where did I make a mistake..?
 
Physics news on Phys.org
You used h rather than \hbar.
 
How did you find out those exact values of the exponents?
 

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