Question
Compare the electrostatic force and gravitational force between a proton and an electron. Show that the electrostatic force is far stronger, and calculate the ratio Fe/Fg.
Given: mass of proton mp=1.67×10−27 kg, mass of electron me=9.11×10−31 kg, charge on each =1.6×10−19 C, k=9×109 N m² C⁻², G=6.67×10−11 N m² kg⁻².
Solution — Step by Step
Both forces follow the inverse square law, so the r2 terms will cancel when we take the ratio — which is why we don’t need to know the actual separation distance.
Fe=kr2e2,Fg=Gr2mpme
Divide Fe by Fg. The r2 cancels cleanly:
FgFe=Gmpmeke2
This ratio is a pure number — same regardless of where the proton and electron are sitting.
ke2=9×109×(1.6×10−19)2
=9×109×2.56×10−38
=23.04×10−29=2.304×10−28 N m2
Gmpme=6.67×10−11×1.67×10−27×9.11×10−31
Work step by step:
1.67×9.11≈15.21, so mpme≈15.21×10−58=1.521×10−57 kg²
Gmpme=6.67×10−11×1.521×10−57=10.15×10−68≈1.015×10−67 N m2
FgFe=1.015×10−672.304×10−28≈2.27×1039
The electrostatic force is approximately 2.27×1039 times stronger than the gravitational force between a proton and electron.
Why This Works
Both Coulomb’s law and Newton’s law of gravitation are inverse square laws — the force drops as 1/r2 in both cases. When you form the ratio, the distance r completely cancels. This is why the answer is universal: it doesn’t matter if the proton and electron are 1 nm apart or 1 km apart, the ratio stays the same.
The enormous factor of 1039 comes from the constants themselves. The Coulomb constant k≈9×109 is huge, while G≈6.67×10−11 is tiny. Electric charges (∼10−19 C) are small but their square is still far larger than the product of electron and proton masses (∼10−57 kg²) when scaled by their respective constants.
This result has deep physical meaning: gravity is completely negligible at the atomic and molecular scale. Atoms, chemical bonds, and everyday materials are held together by electromagnetic forces — gravity simply doesn’t participate at that level.
Alternative Method
Instead of computing the ratio symbolically, you can compute each force separately at a fixed separation (say r=1 Å =10−10 m) and then divide.
Fe=(10−10)29×109×(1.6×10−19)2=10−202.304×10−28=2.304×10−8 N
Fg=(10−10)26.67×10−11×1.67×10−27×9.11×10−31=10−201.015×10−67≈1.015×10−47 N
FgFe=1.015×10−472.304×10−8≈2.27×1039
Same answer. The symbolic cancellation method is cleaner in an exam, but this numerical route is a good sanity check.
In NCERT Class 12 Chapter 1 and JEE theory questions, they often ask you to “justify why gravity is neglected at atomic scale.” The answer is this ratio — ∼1039. Memorise the order of magnitude, not the exact digits.
Common Mistake
Students forget to square the charge when computing ke2. They write k×e instead of k×e2. Both forces have the form constant×r2property2, so the charge and mass each appear squared in their respective force formulas. Writing Fe=ke/r2 is dimensionally wrong and will cost you full marks in board exams.