Physics

A physical quantity of the dimension of length that can be formed out of c, G and $$\frac{e^{2}}{4\pi \epsilon _{0}} $$ is [c is velocity of light, G is universal constant of gravitation and e is charge]


SOLUTION
Given,

$$C,G,\dfrac{e^2}{4\pi \varepsilon_0}$$

So,

$$[L]=[C]^a,[G]^b[\dfrac{e^2}{4\pi \varepsilon_0}]^c$$

$$[L]=[LT^{-1}]^a,[ML^3T^{-2}]^b[ML^3T^{-2}]^c$$

$$[L]=L^{a+3b+3c}M^{-b+c}T^{-a-2b-2c}$$

on comparing we get,

$$a=3b+3c=1,$$

$$-b+c=0$$

$$a+2b+2c=0$$

On solving we get,

$$a=-2,b=\dfrac{1}{2},c=\dfrac{1}{2}$$

Thus,

$$L=\dfrac{1}{C^2}\sqrt{[\dfrac{Ge^2}{4\pi\varepsilon_0}]}$$
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Subjective Medium Published on 18th 08, 2020
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