50. An electric diple consists of point charges ±? a distance 2a apart. The potential at an arbitrary point P can be expressed as V(P) = k? r+ + k(−?) r− = k?(r−−r+) r+r− , where k is the Coulomb constant, r+ is the distance to the point charge +?, r− is the distance to the point charge −?. 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 ?, where ? is the angle as shown in the figure. Eventually, the dipole potential for r ≫ a becomes V(r, ?) = k(2a?) cos ? r2 . What is the electric field ?⃗⃗ at an arbitrary position P for r ≫ a? Note that îand ?̂are unit vectors.
(A) ?⃗⃗ = k(2a?) r3 [(3 cos2 ? − 1)î+ sin ? cos ? ?̂]
(B) ?⃗⃗ = k(2a?) r3 [(3 sin2 ? − 1)î+ 3 sin ? cos ? ?̂]
(C) ?⃗⃗ = k(2a?) r [(3 cos2 ? − 1)î+ 3 sin ? cos ? ?̂]
(D) ?⃗⃗ = k(2a?) r2 [(3 cos2 ? − 1)î+ 3 sin ? cos ? ?̂]
(E) ?⃗⃗ = k(2a?) r3 [(3 cos2 ? − 1)î+ 3 sin ? cos ? ?̂]