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1 MIT OpenCouseWae Electomagnetic Fields, Foces, and Motion, Sping 5 Please use the following citation fomat: Makus Zahn, Electomagnetic Fields, Foces, and Motion, Sping 5. (Massachusetts Institute of Technology: MIT OpenCouseWae). (accessed MM DD, YYYY). License: Ceative Commons Attibution-Noncommecial-Shae Alike. Note: Please use the actual date you accessed this mateial in you citation. Fo moe infomation about citing these mateials o ou Tems of Use, visit:

2 6.641, Electomagnetic Fields, Foces, and Motion Pof. Makus Zahn Lectue 5: Method of Images I. Point Chage Above Gound Plane 1. Potential and Electic Field Φ p q 1 1 4πε x + y + (z d) x + y + (z + d) Φ Φ Φ E Φ p i + p i + p i p p x x y z y z xi x + yi y + (z d) i z xi x + yi y + (z + d) i z q 4πε 3 3 x + y + (z d) x + y + (z + d) q ( d ) E ( z ) i p πε x + y + d 3 z (pependicula to equipotential gound plane) Pof. Makus Zahn Page 1 of 11

3 . Gauss s Law Bounday Condition ε S Ei da ρdv V ε E i da ( ε E i n + ε ) σ 1 1 S σ ε n i E E s 1 E i n ds s ds (total chage inside pillbox) 3. Back to Point Chage Above Gound Plane At z: σ ε n i E E ε qd qd s i i E ε E 1 z 1 z 3 3 π x + y + d π + d x + y T ( ) + + π qd q z σ dxdy σ dd φ ( s s π π ) d 3 y x φ + d Pof. Makus Zahn Page of 11

4 u + d du d 3 d + d du u 3 1 u 1 + d T ( ) q z + qd + d q f q πε q ( ) 4 d i z q 16 d πε II. Point Chage and Sphee 1. Gounded Sphee Coutesy of Kiege Publishing. Used with pemission. Pof. Makus Zahn Page 3 of 11

5 1 q q' Φ + 4πε s s ' 1 1 b + θ s + D Dcos θ, s' bcos q q' Φ ( R) q q' s s ' s s ' q s' q' s q' R + D RDcos θ q b + R Rbcos θ q' (R + D ) q (b + R ) +q' RDcos θ +q Rbcos θ q' b q D b (R + D ) b + R b R b D D + D + R ( b D ) b R D R b D b R q' q q D q' qr D D foce on sphee qq' f x 4πε (D b) q R D q RD 4πε D R 4πε (D R ) D. Isolated Sphee [Put additional Image Chage +q' +qr D at cente] (zeo chage) q' Φ ( R) 4π ε R q 4π ε D Pof. Makus Zahn Page 4 of 11

6 foce on sphee f qq' D (D b) q R bd b q q' q' 4πε D 3 D R D 4πε (D b) D 4πε D (D b) x 3 q RD R R f x D 4πε D 3 (D R D ) D qr D R 4πε D 3 (D R ) Pof. Makus Zahn Page 5 of 11

7 III. Demonstation Chage Induced in Gound Plane by Ovehead Conducto Coutesy of Hemann A. Haus and James R. Melche. Used with pemission. 1 λ (a x) + y λ Φ ln ln (a x) + y πε (a x) + y 1 4πε (a+ x) + y + C' Φ (x l λ R,y ) λ λ a l+ R π, Φ (x l R,y ) U ln l R + l πε a + l R ln R ε Pof. Makus Zahn Page 6 of 11

8 σ ε E ( x ) ε Φ s x x x + ε λ d ln (a x) + y ln (a + x) + y 4πε dx ( ) a + x) λ a x ( 4π (a x) + y (a x) + + y x λa π (a + y ) Total Chage pe unit length on gound plane is: λ T (x ) y σ s dy λa dy λ a 1 1 y tan π (a + y ) π a a π λ i s dq dσs aa dλ aac ' du A dt dt π (a + y ) dt π (a + y ) dt take U U cos ωt v i R s s C'Aa U ω sin ωt π (a + y ) Pof. Makus Zahn Page 7 of 11

9 IV. Point Electic Dipole 1. Potential q 1 1 Φ 4πε + + x + y + z d d x + y + z + Note: Φ (z ) Pof. Makus Zahn Page 8 of 11

10 . Point Electic Dipole (>>d) Coutesy of Kiege Publishing. Used with pemission. + d d cos θ 1 cos θ + d d cos θ 1 + cos θ Φ q 4 πε 1 1 d cos θ d cos θ q πε d cos θ 1 d cos θ qdcos θ πε 4 lim p qd (dipole moment) Φ d q pcos θ 4 πε E Φ Φ i + 1 Φ i θ θ + 1 Φ sin θ φ i φ p πε 3 4 cos θ i + sin θ i θ Pof. Makus Zahn Page 9 of 11

11 d E cos θ 3. Field Lines: cot θ dθ E θ sin θ d cot θd θ ( ) ln ln sin θ + C sin θ (θ π ) q 4 d 1 πε Coutesy of Hemann A. Haus and James R. Melche. Used with pemission. Pof. Makus Zahn Page 1 of 11

12 V. Line Cuent Above a Pefect Conducto Fom Electomagnetic Field Theoy: A Poblem Solving Appoach, by Makus Zahn, Used with pemission. f I I (µ H) Newton/mete [foce pe unit length] I µ I Ii µ i i z x y 4π d 4π d Pof. Makus Zahn Page 11 of 11