2.Point Electric Dipole (r>>d) d cos *r+号co50 -9 p.gd Courtesy of Krieger Publishing.Used with permission. gr-9cos0rl1-号cos9 Lr+cos0r1+是cos0] 9 1 91+cos0- d cose 1- 1+ d 2 c0S0 4πor'2r ≈qdcos6 4πeor2 !imp=qd(dipole moment)-Φ≈pcos0 d→0 q→0 4πeor2 E=-VΦ=- ar 4πEr 2cos+sinei 6.641,Electromagnetic Fields,Forces,and Motion Lecture 5 Prof.Markus Zahn Page 9 of 116.641, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 9 of 11 2. Point Electric Dipole (r>>d) Courtesy of Krieger Publishing. Used with permission. d d r r cos r 1 cos 2 2r + ⎡ ⎤ ≈ − θ≈ − θ ⎢ ⎥ ⎣ ⎦ d d r r cos r 1 cos 2 2r − ⎡ ⎤ ≈ + θ≈ + θ ⎢ ⎥ ⎣ ⎦ 0 0 q 1 1 qd d 1 cos 1 cos 4 r 4 r 2r 2r d d 1 cos 1 cos 2r 2r ⎡ ⎤ ⎢ ⎥ ⎡ ⎛ ⎞⎤ Φ ≈ − ≈ + θ− − θ ⎢ ⎜ ⎟⎥ π π ⎢ ⎥ − θ+ θ ⎣ ⎝ ⎠⎦ ⎣ ⎦ ε ε 2 0 qd cos 4 r θ ≈ πε 2 d 0 q 0 p cos lim p qd (dipole moment) → 4 r →∞ θ = ⇒Φ≈ πε __ _ r 1 1 E ii i r r r sin θ φ ⎡ ⎤ ∂Φ ∂Φ ∂Φ = −∇Φ = − + + ⎢ ⎥ ∂ ∂θ θ ∂φ ⎣ ⎦ θ ⎡ ⎤ = θ ⎢ + θ ⎥ πε ⎣ ⎦ _ _ 3 r 0 p 2 cos i sin i 4 r