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# Question and Answer

Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges/masses.  Compare the strength of these forces by determining the ratio of their magnitudes for an electron and a proton and for two protons. $$(m_p=1.67\times 10^{-27}\;\mathit{kg},m_e=9.11\times 10^{-31}\;\mathit{kg})$$
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## QuestionPhysicsClass 12

Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges/masses.  Compare the strength of these forces by determining the ratio of their magnitudes for an electron and a proton and for two protons. $$(m_p=1.67\times 10^{-27}\;\mathit{kg},m_e=9.11\times 10^{-31}\;\mathit{kg})$$

## Answer

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## Solution

We know that the force between an electron and a proton separated at a distance  $$r$$  can be given as,
$$F_e=\frac{-1}{4\pi \varepsilon _0}\frac{e^2}{r^2}$$
where as here the negative sign indicates that the force is attractive. The corresponding gravitational force which is always attractive can be given by,
$$F_G=-G\frac{m_pm_e}{r^2}$$
where as  $$m_p$$  is the mass of a proton and  $$m_e$$  is the mass of an electron.
Known data:
$$e=1.6\times 10^{-19}\mathit{C.}$$
$$G=6.67408\times 10^{-11}m^3\mathit{kg}^{-1}s^{-2}.$$
mass of a proton,  $$m_p=1.67\times 10^{-27}\;\mathit{kg}$$
mass of an electron,  $$m_e=9.11\times 10^{-31}\;\mathit{kg}$$
$$\varepsilon _0$$- the permittivity of free space .
$$\varepsilon _0=8.854\times 10^{-12}\;C^2N^{-1}m^{-2}$$
Therefore, the ratio of the magnitudes of electric force  $$F_e$$ to the gravitational force  $$F_G$$ between a proton and an electron separated at a distance$$r$$is,
$$\frac{F_e}{F_G}=\frac{\frac{-1}{4\pi \varepsilon _0}\frac{e^2}{r^2}}{-G\frac{m_pm_e}{r^2}}=\frac{e^2}{4\pi \varepsilon _0Gm_pm_e}=\frac{9\times 10^9 \times (1.6\times 10^{-19})^2}{6.67\times 10^{-11}\times 1.67\times 10^{-27} \times 9.11\times 10^{-31}}=2.4\times 10^{39}$$
The ratio of the magnitudes of electric force  $$F_e$$ to the gravitational force  $$F_G$$ between two protons separated  at a distance  $$r$$  is,
Using all known data,
$$\frac{F_e}{F_G}=\frac{e^2}{4\pi \varepsilon _0Gm_pm_p}=\frac{9\times 10^9 \times (1.6\times 10^{-19})^2}{6.67\times 10^{-11}\times 1.67\times 10^{-27} \times 1.67\times 10^{-27}}=1.3\times 10^{36}$$
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