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So, I assumed a pair of electron and proton to behave as a dipole

Classicaly, this is correct. It would behave as a dipole.

But in quantum mechanics we don't talk about a localised electron ,but about orbitals. This is because particles have an associated wavefunction $\psi(t)$, which tells you the probability of finding your particle at a particular place.

In the case of an electron and a proton (hydrogen atom), the orbital has spherical symmetry. This means that the will be no net charge.