- 33 Extra Credit 5 Pts Fill In The Steps Leading From Equation 7 To Equation 11 1 Compare Electric Dipole And P 1 (15.67 KiB) Viewed 30 times
- 33 Extra Credit 5 Pts Fill In The Steps Leading From Equation 7 To Equation 11 1 Compare Electric Dipole And P 2 (24.47 KiB) Viewed 30 times
- 33 Extra Credit 5 Pts Fill In The Steps Leading From Equation 7 To Equation 11 1 Compare Electric Dipole And P 3 (17.75 KiB) Viewed 30 times
Using these in (2) gives the unsightly expression, Ep=kq{(r2−rLcosθ+L2/4)3/2(r−2Lcosθ)r^+2Lsinθθ˙−(r2+rLcosθ+L2/4)3/2(r+2Lcosθ)r^−2Lsinθθ^} 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−rLcosθ+4r2L2)−3/2≈r31(1+23rLcosθ+Order(L2/r2)) Neglecting terms of order L2/r2 gives, (r2−rLcosθ+L2/4)3/21≈r31(1+23Lcosθ)
MoDEL A CHARCE DISTRIHUTION AS A MONOROLR OR A DIPOLE Similarly, (r2+rLcosθ+L2/4)3/21≈r31(1−23rLcosθ) Using (9) and (10) in (7) and neglecting additional terms of order L2/r2 gives, EP≈2r3kqLcosθr^+r3kqLsinθθEp≈r32kpcosθr^+r3kpsinθθ 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.