If ε0 is permittivity of free space, e is charge of proton, G is universal gravitational constant and mp is mass of a proton then the dimensional formula for (e2/4 π ε0 G m p 2) is

Question

If ε0 is permittivity of free space, e is charge of proton, G is universal gravitational constant and mp is mass of a proton then the dimensional formula for (e2/4 π ε0 G m p 2) is (A) [ M 1 L 1 T -3 A -1] (B) [ M 0 L 0 T 0 A 0] (C) [ M 1 L 3 T -3 A -1] (D) [ M -1 L -3 T 4 A 2]

Tardigrade Admin · Accepted Answer

Gravitational force F1=(G Mp2/r2) Electrostatic force F2=(1/4 π ε0) (e2/r2) (F2/F1)=(e2/4 π ε0 G MP2) ∴ Dimension less [ M ° L ° T ° A °]

Q. If $ε_{0}$ is permittivity of free space, $e$ is charge of proton, $G$ is universal gravitational constant and $m_{p}$ is mass of a proton then the dimensional formula for $\frac{e ^{2}}{4 π ε _{0} G m _{p}^{2}}$ is

$[M^{1} L^{1} T^{- 3} A^{- 1}]$

$[M^{0} L^{0} T^{0} A^{0}]$

$[M^{1} L^{3} T^{- 3} A^{- 1}]$

$[M^{- 1} L^{- 3} T^{4} A^{2}]$

Gravitational force $F_{1} = \frac{G M _{p}^{2}}{r ^{2}}$
Electrostatic force $F_{2} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}}$
$\frac{F _{2}}{F _{1}} = \frac{e ^{2}}{4 π ε _{0} G M _{P}^{2}}$
$∴$ Dimension less $[M^{\circ} L^{\circ} T^{\circ} A^{\circ}]$

Q. If ε0​ is permittivity of free space, e is charge of proton, G is universal gravitational constant and mp​ is mass of a proton then the dimensional formula for 4πε0​Gmp2​e2​ is

[M1L1T−3A−1]

[M0L0T0A0]

[M1L3T−3A−1]

[M−1L−3T4A2]