Answer Happy

33. Extra Credit (5 pts): Fill in the steps leading from Equation (7) to Equation (11). 1. Compare Electric Dipole and Point Charge Electric Fields
Using these in (2) gives the unsightly expression, Ep​=kq{(r2−rLcosθ+L2/4)3/2(r−2L​cosθ)r^+2L​sinθθ˙​−(r2+rLcosθ+L2/4)3/2(r+2L​cosθ)r^−2L​sinθθ^​} This expression is correct but hard to interpret. It turns out that for r≫L this expression simplifies to a tidy expression that is easier to interpret. I'll do this in the next section. 2.2 Dipole Electric Field in the "Far Field" (r≫L) Often we are interested in the electric field of a dipole at a position such that r≫L. This region is sometimes called the far field. Note that for a molecular dipole this restriction is essentially just that r>10−9 m. To proceed with this approximation note that, using the Trylor series expansion formula, allows (1+x)n≈1+nx+order(x2) (r2−rLcosθ+L2/4)3/21​=r31​(1−rL​cosθ+4r2L2​)−3/2≈r31​(1+23​rL​cosθ+Order(L2/r2)) Neglecting terms of order L2/r2 gives, (r2−rLcosθ+L2/4)3/21​≈r31​(1+23L​cosθ)
MoDEL A CHARCE DISTRIHUTION AS A MONOROLR OR A DIPOLE Similarly, (r2+rLcosθ+L2/4)3/21​≈r31​(1−23​rL​cosθ) Using (9) and (10) in (7) and neglecting additional terms of order L2/r2 gives, EP​≈2r3kqL​cosθr^+r3kqL​sinθθEp​≈r32kp​cosθr^+r3kp​sinθθ​ Here p=∣p​∣=qL is the magnitude of the dipole moment p​. The far-field electrie fleld is proportional to the inverse of the distance-cubed and so folls off foster with distance than doen the fleld due to a point charge which falls off as the distance-squared. Also, note that the far-field electric field in proportional to the magnitude of the electric dipole moment.