50. An electric diple consists of point charges ±q a distance 2a apart. The potential at an arbitrary point P can be expressed as V(P) = = , where k is the Coulomb constant, r+ is the distance to the point charge +, r− is the distance to the point charge −q. If r is the distance to the dipole center, for r ≫ a, the quantities r+, r−, and r are nearly the same as r² ≈ r+r−. The difference between the distances from the two charges to P, that is, r₋ − r₊, is approximately equal to 2a cos θ, where θ is the angle as shown in the figure. Eventually, the dipole potential for r ≫ a becomes V(r, θ) = . What is the electric field at an arbitrary position P for r ≫ a? Note that and are unit vectors.

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