Question

If Planck’s constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities?

1. Mass of electron (m_e)
2. Universal gravitational constant (G)
3. Charge of electron (e)
4. Mass of proton (m_p)

Answer 1

a. Mass of electron (m_e)

b. Universal gravitational constant (G)

d. Mass of proton (m_p)

Explanation:

We know that dimension of h

`[h] = ([ML^2T-2])/([T^-1]) = [ML^2T^-1]`

`[G] = [M^-1L^3T^-2]`

`[c] = [LT^-1]`

For mass, let m = c^ah^bG^c

Where a, b and c are exponents of c, h and G respectively.

⇒ `[ML^0T^0] = [LT^-1]^a [ML^2T^-1]^b [M^-1L^3T^-2]^c`

Using the principle of homogeneity of dimensions,

`b - c` = 1  .....(i)

`a + 2b + 3c` = 0  ......(ii)

`-a + b - 2c` = 0  ......(iii)

Adding equations (i), (ii) and (iii),

`2b` = 1 ⇒ `b = `1/2`

From equation (i), `a = b - 1 = 1/2 - 1 = - 1/2`

From equation (iii), `a = - (b + 2c) = - (1/2 - 1) = 1/2`

Hence, `m = c^(1/2) h^(1/2) G^(-1/2)` ⇒ `m = sqrt((ch)/G)`

Similarly, for length, L = `sqrt((hG)/c^3)` and for time, T = `sqrt((hG)/c^5)` 

∴ Plank's length = `sqrt((Gh)/c^3)`; Planck's time = `sqrt((Gh)/c^5)`;

And Planck's mass = `sqrt((ch)/G)`

Mass can be expressed by m_e and m_p

Hence, a, b or d any can be used to express L, M and T in terms of three chosen fundamental quantities.