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Check that the ratio ke^{2}/G m_{e}m_{p}_{ }is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

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#### Solution

The given ratio is

`("ke"^2)/("Gm"_"e""m"_"p")`

Where,

G = Gravitational constant

Its unit is N m^{2 }kg^{−2}.

m_{e} and m_{p} = Masses of electron and proton.

Their unit is kg.

e = Electric charge.

Its unit is C.

k = A constant

= `1/(4piin_0)`

∈_{0} = Permittivity of free space

Its unit is N m^{2 }C^{−2}.

Therefore, unit of the given ratio `("ke"^2)/("Gm"_"e""m"_"p") = (["Nm"^2"C"^-2]["C"^-2])/(["Nm"^2"kg"^-2]["kg"]["kg"]) = "m"^0"L"^0"T"^0`

Hence, the given ratio is dimensionless.

e = 1.6 × 10^{−19 }C

G = 6.67 × 10^{−11} N m^{2 }kg^{−}^{2}

m_{e }= 9.1 × 10^{−31 }kg

m_{p} = 1.66 × 10^{−27 }kg

Hence, the numerical value of the given ratio is

`("ke"^2)/("Gm"_"e""m"_"p") =(9 xx 10^9 xx (1.6 xx 10^-19)^2)/(6.67 xx 10^-11 xx 9.1 xx 10^-31 xx 1.67 xx 10^-27) ≈ 2.3 xx 10^39`

This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant.

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