Question

A physical quantity of the dimensions of length that can be formed out of c, G and `e^2/(4piε_0)` is (c is velocity of light, G is universal constant of gravitation and e is charge):

पर्याय

• `1/c^2 [G e^2/(4piε_0)]^(1/2)`

• `c^2 [G e^2/(4piε_0)]^(1/2)`

• `1/c^2 [e^2/(G4piε_0)]^(1/2)`

• `1/c G e^2/(4piε_0)`

Answer 1

`1/c^2 [G e^2/(4piε_0)]^(1/2)`

Explanation:

Let the physical quantity formed of the dimensions of length be given as,

[L] = `[c]^x [G]^y [e^2/(4piε_0)]^z`  ......(i)

Now,

Dimensions of velocity of light `[c]^x = [LT^-1]^x`

Dimensions of universal gravitational constant `[G]^y = [M^-1L^3T^-2]^y`

Dimensions of `[e^2/(4piε_0)]^z` = [ML³T^–2]^z 

Substituting these in equation (i)

[L] = [LT^–1]^x [M^–1L³T^–2]^y [ML³T^–2]^z

= `[L^(x + 3y + 3z) M^(-y + z) T^(-x - 2y - 2z)]`

Solving for x, y, z

x + 3y + 3z = 1

– y + z = 0

x + 2y + 2z = 0

Solving the above equations,

x = – 2, y = `1/2`, z = `1/2`

∴ L = `1/c^2 [G e^2/(4piε_0)]^(1/2)`