A physical quantity of the dimensions of length that can

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A physical quantity of the dimensions of length that can be formed out of c, G and e²
4 πε₀

is [c is velocity of light, G is universal constant of gravitation and e is charge]

1. c² G e² ^{1 / 2}
  4 π ε₀
2. 1 e² ^{1 / 2}
  c² G 4 π ε₀
3. 1 G e²
  c 4 π ε₀
4. 1 G e² ^{1 / 2}
  c² 4 π ε₀

Correct Option: D

Let dimensions of length is related as,

L = [c]^x [G]^y			e²		^z
			4 π ε₀

	e²	= ML³T^-2
	4 π ε₀

L = [LT^{– 1}]^x [M^{– 1}L³T^{– 2}]^y[ML³T^{– 2}]^z
[L] = [L^{x + 3y + 3z}M^{– y + z} T^{– x – 2y – 2z}]
Comparing both sides – y + z = 0 ⇒ y = z...(i)
x + 3y + 3z = 1...(ii)
– x – 4z = 0 (∵ y = z)...(iii)
From (i), (ii) & (iii)

z = y =	1	, x = – 2
	2

Hence , L = c^{– 2}		G .	e²		^{1 / 2}
			4 π ε₀

A physical quantity of the dimensions of length that can be formed out of c, G and	e²
	4 πε₀