Question

163. A physical quantity of the dimensions of length thatcan be formed out of c, G and e²/(4π∈₀) is [c is velocityof light, G is universal constant of gravitation and eis charge]
(1) 1/c²[G. e²/(4π∈₀)]¹/²

(2) c²[G. e²/(4π∈₀)]¹/²

(3) 1/c²[e²/(G.4π∈₀)]¹/²

(4) (1/c) .G. e²/(4π∈₀)

Answer 1

Thus the dimensions of length formed are L = 1 / c^2 [G.e^2 / 4πε0]^1 / 2

Explanation:

[L] = [c] a [G] b [e24πε0] c

[L] = [LT − 1] a [M − 1L3T − 2]b[ML3T − 2]c

[L] = La + 3b + 3cM − b + cTa − 2b − 2c

a + 3b + 3c = 1

− b + c = 0

a + 2b + 2c = 0

a + 2b + 2c=0

On solving:

a  = −2,b = 1 / 2, c = 1 / 2

L = 1 / c^2 [G.e^2 / 4πε0]^1 / 2

Thus the dimensions of length formed are L = 1 / c^2 [G.e^2 / 4πε0]^1 / 2