6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 5: Method of Images Φ Φ Φ. x y z

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1 6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 5: Method of Images I. Point Charge Above Ground Plane 1. Potential and Electric Field q 1 1 Φ p 4π x + y + z d x + y + z + d Φ Φ Φ E i i i x y z p p p Φ p p + + x y z xi + yi + ( z d) i q 4 π 3 x + y + ( z d) x y z xi + yi + ( z+ d) i x y z x + y + z + d 3 ( d) q E ( z ) i p π x y d (perpendicular to equipotential ground plane) z Prof. Markus Zahn Page 1 of 11

2 . Gauss s Law Boundary Condition S Ei da ρdv V ( 1 1 ) Eida E in + E in ds σsds (total charge inside pillbox) S σ ni E E s 1 3. Back to Point Charge Above Ground Plane At z: qd qd σ ni E E i ie E π x + y + d π r + d s 1 z 1 z 3 3 r x + y + + π ( π π rdr ) 3 qd qt ( z ) σ sdxdy σsrdrdφ y x r φ r r + d Prof. Markus Zahn Page of 11

3 u r + d du rdr r rdr du 1 1 u d u r + d + qd qt ( z ) q r + d f q q i 4π d q z 16π d II. Point Charge and Sphere 1. Grounded Sphere Courtesy of Krieger Publishing. Used with permission. Prof. Markus Zahn Page 3 of 11

4 1 q q' Φ + 4π s s' 1 1 s r + rcos θ, s' b + r rbcos θ q q' q q' Φ ( r R) s s' s s' q s' q' s q' R + Rcosθ q b + R Rbcosθ ( + ) ( + ) q' R q b R + q' Rcosθ + q Rbcosθ q' q b R R + b + R b b + + R b R b b b R b R q' q q q' qr force on sphere f qq' q R q R 4π b R 4π R 4π x. Isolated Sphere [Put additional Image Charge + q' + qr at center] (zero charge) q' q Φ ( r R) 4π R 4π Prof. Markus Zahn Page 4 of 11

5 force on sphere f ( ) qq' b q q' q' qr b b 4 x 4π ( b) 4 π b 3 R π 3 qr R R qr f R x 3 3 4π R 4π R Prof. Markus Zahn Page 5 of 11

6 III. emonstration Charge Induced in Ground Plane by Overhead Conductor Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. 1 a x + y λ λ ( a x) + y Φ ln ln π 1 4 ( a x) y π + + a x y + + λ λ π C', Φ ( x l R,y ) U Φ ( x l R,y ) λ a l+ R ln l R + l π + a l R ln R Prof. Markus Zahn Page 6 of 11

7 σ E x s x + Φ 4π 4 π x x λ d + ( + ) + ln a x y ln a x y dx λ a x a+ x a x + y a+ x + y x λa ( a y ) π + Total Charge per unit length on ground plane is: λa λ T ( x ) σ sdy dy π + y ( a y ) λ a π 1 a 1 y tan a π λ i s dq dσs aa dλ aac' du A dt dt π + dt π + dt ( a y ) ( a y ) take U U cosω t C'Aa v isrs Uωsinωt π + ( a y ) Prof. Markus Zahn Page 7 of 11

8 IV. Point Electric ipole 1. Potential q 1 1 Φ 4 r r π + r+ x + y + z d r x + y + z + d Note: Φ ( z ) Prof. Markus Zahn Page 8 of 11

9 . Point Electric ipole (r>>d) Courtesy of Krieger Publishing. Used with permission. d d r+ r cosθ r 1 cos θ r d d r r + cosθ r 1 cos + θ r q 1 1 q d d Φ 1 cos 1 cos 4 d d + θ θ πr 4π r r r 1 cos 1 cos θ + θ r r qdcos θ π 4 r pcosθ lim p qd (dipole moment) Φ 4 π r d q Φ 1 Φ 1 Φ E Φ i + i + i r θ r r θ rsinθ φ φ p cosθ i + sinθ i 3 r θ 4π r Prof. Markus Zahn Page 9 of 11

10 dr Er cos θ 3. Field Lines: cotθ rdθ E sin θ dr cot d lnr ln( sin ) C r θ θ θ + θ r r sin θ r π r θ q 4 d 1 π Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. Prof. Markus Zahn Page 1 of 11

11 V. Line Current Above a Perfect Conductor I ( ) f I µ H Newton/meter [force per unit length] I µ I Ii µ i i 4πd 4πd z x y Prof. Markus Zahn Page 11 of 11