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It does, if the test charge is located anywhere but in the plane perpendicular to the axis joining the electron and proton (which I presume you are holding fixed to form an electric dipole). The electric field of an electric dipole has the same shape as a magnetic dipole’s magnetic field, which you’ve probably seen. (If not, put a piece of paper over a bar magnet and sprinkle iron filings on it.)
But the dipole field is much weaker (at a distance) than the monopole fields of either charge, because it falls off as the inverse cube of distance.
Source- Jess H. Brewer(Quora).
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