Question

(II) Compare the electric force holding the electron in orbit\(\left( {{\bf{r = 0}}{\bf{.53 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}\;{\bf{m}}} \right)\)around the proton nucleus of the hydrogen atom, with the gravitational force between the same electron and proton. What is the ratio of these two forces?

Short answer

The ratio of electric force between the electron and proton to the gravitational force between them is \(2.27 \times {10^{39}}\).

Step-by-step solution

Understanding the electric force and gravitational force

The electric force between two charged particles relies on the charges on both the particles, whereas the gravitational force relies on the mass of these particles.

 Step 2: Given Data

The separation distance is \(r = 0.53 \times {10^{ - 10}}\;{\rm{m}}\).

Determination of the ratio of electric force and gravitational force

The expression for the electric force is given as:

\(F = k\frac{{{Q_1}{Q_2}}}{{{r^2}}}\) … (i)

Here, k is the Coulomb’s constant,\({Q_1},\;{Q_2}\)are the charges and r is the separation between the charges.

The expression for gravitational force is given as:

\({F_{\rm{G}}} = G\frac{{{m_1}{m_2}}}{{{r^2}}}\) … (ii)

Here, G is the universal gravitational constant,\({m_1},\;{m_2}\)are the masses and r is the separation distance.

Divide equation (i) by (ii).

\(\begin{aligned}{c}\frac{F}{{{F_{\rm{G}}}}} = \left( {\frac{{\frac{{k{q^2}}}{{{r^2}}}}}{{\frac{{Gmm'}}{{{r^2}}}}}} \right)\\\frac{F}{{{F_{\rm{G}}}}} = \left( {\frac{{k{q^2}}}{{Gmm'}}} \right)\end{aligned}\)

Substitute the values in the above expression.

\(\begin{aligned}{l}\frac{F}{{{F_{\rm{G}}}}} = \left( {\frac{{\left( {9.0 \times {{10}^9}\;{\rm{N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right){{\left( {1.6 \times {{10}^{ - 19}}\;{\rm{C}}} \right)}^2}}}{{\left( {6.67 \times {{10}^{ - 11}}\;{\rm{N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/k}}{{\rm{g}}^{\rm{2}}}} \right)\left( {9.1 \times {{10}^{ - 31}}\;{\rm{kg}}} \right)\left( {1.67 \times {{10}^{ - 27}}\;{\rm{kg}}} \right)}}} \right)\\\frac{F}{{{F_{\rm{G}}}}} = 2.27 \times {10^{39}}\end{aligned}\)

Thus, the ratio of electric force between the electron and proton to the gravitational force between them is\(2.27 \times {10^{39}}\).