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