A physical quantity of the dimensions of length that can be formed out of c, G and \(\frac{e^2}{4 \pi \epsilon_o}\) [c is velocity of light, G is universal constant of gravitation and e is charge]

\(\frac{1}{c^2}\) [G \(\frac{e^2}{4 \pi \epsilon_o}\)]^1/2

G = M^-1 L³ T^-2

c = L¹ T^-1

\(\frac{e^2}{4πε_o}\) = M¹ L³ T^-2

l ∝ c^P G^Q [\(\frac{e^2}{4πε_o}\)]^R 

l ∝ L^P T^-P (M^-1 L³ T^-2)^Q (M¹ L³ T^-2)^R

M⁰ L¹ T⁰ = M^-Q+R L^P+3Q+3R T^-P-2Q-2R

-Q + R = 0

Q = R

P+3Q+3R = 1

-P-2Q-2R = 0

On solving, Q = R = \(\frac{1}{2}\), P = -2

l ∝ \(\frac{1}{c^2}\) [G \(\frac{e^2}{4 \pi \epsilon_o}\)]^1/2