Question

If G (universal gravitational constant), h (Planck’s constant) and C (Velocity of light) were taken as the fundamental units, the dimensional formula, for pressure (p), would be

(1) [p] = [h^-2 G^-1 C⁷]

(2) [p] = [h² GC^-7]

(3) [p] = [hG² C^-7]

(4) [p] = [h^-1G^-2 C⁷]

Answer 1

(4) [p] = [h^-1G^-2 C⁷]

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

[h] = [ML²T^-1] 

[C] = [LT^-1]

From these, we get

[G h] = L⁵T^-3

Also, [C⁵] = L⁵T^-5

∴ [GH/C⁵] = T² or [T] = G^1/2 H^1/2 C^-5/2

∴ [L] = [CT] = [G^1/2 H^1/2 C^1/2]

and [M] = [hL^-2 T⁺¹] = [h(G^-1h^-1C³)(G^1/2 H^1/2 C^-5/2)]

= [h^1/2 G^-1/2 C^1/2]

Now pressure, p = [ML^-1T^-2]

∴ [P] = [h^-1G^-2C⁷]