The ratio of electrostatic and gravitational forces acting between electron and proton separated by a distance $5\times {10}^{-11}m$, will be (Charge on electron $=1.6\times {10}^{-19}C$, mass of electron $=9.1\times {10}^{–31}$ kg, mass of proton $=1.6\times {10}^{-27}kg,G=6.7\times {10}^{-11}N{m}^2/k{g}^2)$

Step 1: Given that:

The distance(r) between electron and proton = $5\times {10}^{-11}m$

Charge on electron = charge on proton = e = $1.6\times {10}^{-19}C$

Mass of electron($m_e$) = $9.1\times {10}^{-31}kg$

Mass of proton($m_p$) = $1.6\times {10}^{-27}kg$

G = $6.7\times {10}^{-11}N{m}^{2}k{g}^{-2}$

Step 2: Calculation of Electrostatic force:

The electrostatic force between two charged bodies with charges $q_1$ and $q_2$ separated by a distance r between them is given as;

$F_e=k_e\frac{q_1{q}_2}{r^2}=9\times {10}^9\frac{q_1{q}_2}{r^2}$

Putting the values, we get;

$F_e=9\times {10}^9\frac{1.6\times {10}^{-19}C\times 1.6\times {10}^{-19}C}{{(5\times {10}^{-11}m)}^2}$

$F_e=\frac{14.4\times 1.6\times {10}^{-30}}{25\times {10}^{-22}}$

$F_e=\frac{23.04}{25}\times {10}^{-8}$

$F_e=0.9216\times {10}^{-8}$

$F_e=9.22\times {10}^{-9}N$

Step 3: Calculation of gravitational force:

The gravitational force between two bodies of masses $m_1$ and $m_2$ placed at a distance r from each other is given by;

$F_g=G\frac{m_1{m}_2}{r^2}=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}\frac{m_1{m}_2}{r^2}$

Putting all the values, we get;

$F_g=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}\frac{9.1\times {10}^{-31}kg\times 1.6\times {10}^{-27}kg}{{(5\times {10}^{-11}m)}^2}$

$F_g=\frac{97.1152\times {10}^{-69}}{25\times {10}^{-22}}$

$F_g=3.88\times {10}^{-47}N$

Step 4: Comparision of electrostatic force to the gravitational force:

$\frac{F_e}{F_g}=\frac{3.88\times {10}^{-47}N}{9.22\times {10}^{-9}N}$

$\frac{F_e}{F_g}=0.42\times {10}^{-38}$

$\frac{F_e}{F_g}=4.2\times {10}^{-39}$

Thus,

The ratio of electrostatic force to the gravitational force between electron and proton is $4.2\times {10}^{-39}:1$.