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Asked in GUJCET 2024 · Dipole moment and dipole field
Idea: a point on the equatorial plane is equally far from +q and -q, so the two fields there have the same magnitude.
Resolve each along and across the dipole axis: the components across the axis are equal and opposite and cancel; the components along the axis add.
What is left points from the +q end towards the -q end, while P⃗ by definition points from -q to +q.
E⃗_eq=-(kP⃗)/((r²+a²)^3/2)
The minus sign says the two vectors are antiparallel, so the angle between E⃗ and P⃗ is 180°.
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