Question

If the velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

Answer 1

Let m ∝ c^xh^yG^z 

m = Kc^xh^yG^z           ....(A) 

h = [ML²T^-1], 

c = [LT^-1], 

G = [M^-1L³T^-2]

(k = dimensionless)

Or, [ML⁰T⁰] = [LT^-1] ^x [ML²T^-1] ^y [M^-1L³T^-2] ^z

[M^y-zL^x+2y+3zT^x-y-2z]

Comparing powers –

y – z = 1  .................. (1)

x + 2y + 3z = 0  .........(2)

-x – y – 2z = 0 ............(3)

Adding above all three equations-

2y = 1 

⇒ y = \(\frac{1}{2}\)

o, z = \(-\frac{1}{2}\), x =\(\frac{1}{2}\)

Putting in eq.n (A)-

= kc^{\(\frac{1}{2}\)}h^{\(\frac{1}{2}\)}G^{\(-\frac{1}{2}\)}

m = \(k\sqrt\frac{ch}{G}\)

(ii) Let L∝c^xh^yG^z

L = kc^xh^yG^{z         ..........}(B)

Substituting in B

[M⁰LT⁰] = [LT^-1]^x × [ML²T^-1] ^y × [M^-1L³T^-2]^z

= [M^y-zL^x+2y+3zT^-x-y-2z]

Comparing powers-

y – z = 0           ....(a)

x + 2y + 3z = 1  ....(b)

-x – y – 2z = 0     ....(c)

Adding (a), (b), (c), we get-

y = \(\frac{1}{2}\) z= \(\frac{1}{2}\) , x = \(-\frac{3}{2}\)

Putting in (B)-

L = kc^{\(-\frac{3}{2}\)}h^{\(\frac{1}{2}\)}B^{\(\frac{1}{2}\)}

L = \(k\sqrt\frac{hG}{c^3}\)

(iii) Let L∝c^xh^yG^z

T = kc^xh^yG^z  ..............(C)

Substituting in B

[M⁰L⁰T] = [LT^-1]^x × [ML²T^-1]^y × [M^-1L³T ^-2]^z

= [M^y-zL^x+2y+3zT^-x-y-2z]

Comparing powers

y – z = 0 .........(1)

x + 2y + 3z = 1  ..........(2)

-x – y – 2z = 0       .........(3)

Adding (1), (2), (3), we get-

y = \(\frac{1}{2}\) , z= \(\frac{1}{2}\) , x = \(-\frac{5}{2}\)

Putting in (B)-

T = kc^{\(-\frac{5}{2}\)}h^{\(\frac{1}{2}\)}B^{\(\frac{1}{2}\)}

T = \(k\sqrt\frac{hG}{c^5}\)