Question

(a) Find the ratio of the electrostatic to gravitational force between two electrons. (b) What is this ratio for two protons? (c) Why is the ratio different for electrons and protons?

Short answer

(a) The ratio of the electrostatic to the gravitational force between two electrons is \(4.17 \times {10^{42}}\).

(b) The ratio of the electrostatic to the gravitational force between two protons is \(1.24 \times {10^{36}}\).

(c) The ratio of electrostatic to gravitational force for electron and proton is different because of their different mass.

Step-by-step solution

Gravitational force

All the objects in the universe attract each other with force due to the virtue of their mass. This force is known as gravitational force.

The expression for the gravitational force is,

\({F_g} = \frac{{GmM}}{{{r^2}}}\)

Here, G is the universal gravitational constant, m and M are the mass of the object separated by a distance r.

Ratio of the electrostatic to gravitational force between two electrons

(a)

The electrostatic force between two electrons is,

\({F_e} = \frac{{K{q^2}}}{{{r^2}}}\)

Here, K is the electrostatic force constant \(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), q is the charges on the electrons \(\left( {q = 1.6 \times {{10}^{ - 19}}{\rm{ }}C} \right)\), and r is the separation between the electrons.

The gravitational force between two electrons is,

\({F_g} = \frac{{Gm_e^2}}{{{r^2}}}\)

Here, G is the universal gravitational constant \(\left( {G = 6.67 \times {{10}^{ - 11}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/k}}{{\rm{g}}^{\rm{2}}}} \right)\), \({m_e}\) is the mass on an electron \(\left( {{m_e} = 9.1 \times {{10}^{ - 31}}{\rm{ kg}}} \right)\), and r is the distance of separation between the electrons.

The ratio of the electrostatic force to gravitational force between two electrons is,

\(\begin{aligned} {\left( {\frac{{{F_e}}}{{{F_g}}}} \right)_e} &= \frac{{\frac{{K{q^2}}}{{{r^2}}}}}{{\frac{{Gm_e^2}}{{{r^2}}}}}\\ &= \frac{{K{q^2}}}{{Gm_e^2}}\end{aligned}\)

Substituting all known values,

\(\begin{aligned} {\left( {\frac{{{F_e}}}{{{F_g}}}} \right)_e} &= \frac{{\left( {9 \times {{10}^9}{\rm{ }}N \cdot {m^2}/{C^2}} \right) \times {{\left( {1.6 \times {{10}^{ - 19}}{\rm{ }}C} \right)}^2}}}{{\left( {6.67 \times {{10}^{ - 11}}{\rm{ }}N \cdot {m^2}/k{g^2}} \right) \times {{\left( {9.1 \times {{10}^{ - 31}}{\rm{ }}kg} \right)}^2}}}\\ &= 4.17 \times {10^{42}}\end{aligned}\)

Hence, the ratio of the electrostatic to the gravitational force between two electrons is \(4.17 \times {10^{42}}\).

Ratio of the electrostatic to gravitational force between two protons

(b)

The electrostatic force between two protons is,

\({F_e} = \frac{{K{q^2}}}{{{r^2}}}\)

Here, K is the electrostatic force constant \(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), q is the charges on the protons \(\left( {q = 1.6 \times {{10}^{ - 19}}{\rm{ }}C} \right)\), and r is the separation between the protons.

The gravitational force between two protons is,

\({F_g} = \frac{{Gm_p^2}}{{{r^2}}}\)

Here, G is the universal gravitational constant \(\left( {G = 6.67 \times {{10}^{ - 11}}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/k}}{{\rm{g}}^{\rm{2}}}} \right)\), \({{\rm{m}}_{\rm{p}}}\) is the mass on an proton \(\left( {{m_p} = 1.67 \times {{10}^{ - 27}}{\rm{ kg}}} \right)\), and r is the distance of separation between the protons.

The ratio of the electrostatic force to gravitational force between two protons is,

\(\begin{aligned} {\left( {\frac{{{F_e}}}{{{F_g}}}} \right)_p} &= \frac{{\frac{{K{q^2}}}{{{r^2}}}}}{{\frac{{Gm_p^2}}{{{r^2}}}}}\\ &= \frac{{K{q^2}}}{{Gm_p^2}}\end{aligned}\)

Substituting all known values,

\(\begin{aligned} {\left( {\frac{{{F_e}}}{{{F_g}}}} \right)_e} &= \frac{{\left( {9 \times {{10}^9}{\rm{ }}N \cdot {m^2}/{C^2}} \right) \times {{\left( {1.6 \times {{10}^{ - 19}}{\rm{ }}C} \right)}^2}}}{{\left( {6.67 \times {{10}^{ - 11}}{\rm{ }}N \cdot {m^2}/k{g^2}} \right) \times {{\left( {1.67 \times {{10}^{ - 27}}{\rm{ }}kg} \right)}^2}}}\\ &= 1.24 \times {10^{36}}\end{aligned}\)

Hence, the ratio of the electrostatic to the gravitational force between two protons is \(1.24 \times {10^{36}}\).

Different ratios for the electron and proton

(c)

Since, the ratio of the electrostatic force to the gravitational force depends on the mass of electron and proton and the mass of electron and proton is different. Hence, the ratio of the electrostatic force to the gravitational force is different.