Forces between electron and proton

The electrostatic (Coulomb) force is $F_e = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}$

The gravitational force is $F_g = G\frac{m_e m_p}{r^2}$

Take the ratio

$\frac{F_e}{F_g} = \frac{\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r^2}}{G\dfrac{m_e m_p}{r^2}}$

The $r^2$ cancels out: $\frac{F_e}{F_g} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{G m_e m_p}$

Substitute approximate values

$\frac{1}{4\pi\varepsilon_0} = 9\times 10^9\ \text{N m}^2/\text{C}^2$ $e = 1.6\times 10^{-19}\ \text{C}$ $G = 6.67\times 10^{-11}\ \text{N m}^2/\text{kg}^2$ $m_e = 9.1\times 10^{-31}\ \text{kg}$ $m_p = 1.67\times 10^{-27}\ \text{kg}$

Compute numerator

$9\times 10^9 \times (1.6\times 10^{-19})^2$ $= 9\times 10^9 \times 2.56\times 10^{-38}$ $= 23.04\times 10^{-29}$ $= 2.304\times 10^{-28}$

Compute denominator

$6.67\times 10^{-11} \times 9.1\times 10^{-31} \times 1.67\times 10^{-27}$

First, $9.1\times 1.67 = 15.197$

So, $6.67\times 15.197 \times 10^{-69}$ $\approx 101.36\times 10^{-69}$ $= 1.0136\times 10^{-67}$

Now take the ratio

$\frac{F_e}{F_g} = \frac{2.304\times 10^{-28}}{1.0136\times 10^{-67}}$ $\approx 2.27\times 10^{39}$

Order of magnitude

The ratio is of the order of $10^{39}$

Match with options

• A: $10^{19}$
• B: $10^{39}$
• C: $10^{29}$
• D: $10^{36}$

Hence, the correct option is B.