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An electron is moving round the nucleus of a hydrogen atom in a circular orbit of radius r. The Coulomb force F⃗ between the two is (where k=1/(4πε₀))

Asked in CBSE AIPMT 2003 · Coulomb's law and superposition

Answer: (2) -k(e²)/(r³)r⃗

Step-by-step solution

The electron carries -e and the hydrogen nucleus +e, so

F⃗=k(q₁q₂)/(r²)r̂=k((-e)(+e))/(r²)r̂=-k(e²)/(r²)r̂

The minus sign says the force is directed inward, toward the nucleus, which is what keeps the electron in orbit.

To write it with r⃗ instead of r̂, use r̂=(r⃗)/r:

F⃗=-k(e²)/(r²)·(r⃗)/r=-k(e²)/(r³)r⃗

The extra power of r in the denominator is the price of using the un-normalised vector; the magnitude is still an inverse square.

Why the other options are wrong

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