MIT OpenCourseWare http://ocw.mit.edu 6.013/ESD.013J Electromagnetics and Applications, Fall 2005 Please use the following citation format: Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J Electromagnetics and Applications, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms6.013, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn September 22, 2005 Lecture 5: The Electric Potential and the Method of Images I. Nonuniqueness of Voltage in an MQS System Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 1 of 12Φλ=Bda ∫i ScEds =v 1 + iR 1 = 0∫i C1 ∫i v2 2=Eds = −+ iR 0 C2 ∫Eds i = −dΦλ=(1 + R2 )iR dtCc 1dΦλi= −(R1 + R2 )dt +R1 dΦλv= −iR = 1 1 R1 +R2 dt −RdΦv=iR = 2 λ 2 2 R1 +R2 dt v1 = −R1 v2 R2 II. Point Charge Above Ground Plane 1. Potential Electric Field ⎡ ⎤ q ⎢ 1 1 ⎥Φ=p 4πε0 ⎢⎣⎢ x2 +y2 + − (z d)2 − x2 +y2 + + (z d)2 ⎥⎦⎥ 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 2 of 12_ _ _ _ _ _ ⎡∂Φ _ ∂Φ _ ∂Φ _ ⎤ E =−∇Φ =− ⎢ pi + pi + pi ⎥ pp ⎢∂xx ∂yy ∂zz ⎥ ⎣ ⎦ ⎢2 xi +yi + z −d i 2 xi +yi + z +d i q ⎢⎡ ⎝⎜⎛ x y ( ) z ⎠⎟⎞ ⎝⎜⎛ x y ( ) z ⎠⎟⎞ ⎥⎥⎤ ⎡ 2 2 2 ⎤ 2 = 4πε0 ⎢⎢ 2 ⎢⎣x + y +(z − d)⎥⎦ 3 − 2 ⎡⎢⎣x2 +y2 +(z +d)2 ⎤⎥⎦ 32 ⎥⎥⎥ ⎢⎣ ⎦ d_ Ez(= 0)= q (− ) i p2πε0 ⎡x2 +y2 +d2 ⎤ 32 z ⎣ ⎦ (perpendicular to equipotential ground plane) 2. Gauss’s Law Boundary Condition ∫ε0Eda =ρdV i ∫ S V ∫ i ( )=σ dS ε0Eda = ε0E1 i n1 + ε0E2 i n2dS s (total charge inside pillbox) S σ=ε n E i ⎡−E ⎤ s 0 ⎣ 1 2 ⎦ 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 3 of 12z=0: At z=0: σ=ε i ⎡ _ n E −E ⎤ =ε i i E =ε E =−qd =−qd s 0 ⎣ 1 2 ⎦ 0z 10 z 3 3 2xπ⎡ 2 + y2 + d2 ⎤ 22π⎡r2 + d2 ⎤ 2 ⎣ ⎦ ⎣ ⎦ r2 = x2 + y2 +∞ +∞ ∞ 2π ∞ 2 π)∫ rdr 3qz(= 0)= σ dxdy = σ rdrd φ= s r0 ⎡r2 + 2 2 T ∫∫ s ∫ ∫ 2 π−qd ( y=−∞ x=−∞ r=0 φ= 0 = d ⎤⎣ ⎦ ur2 + d2 ⇒ du = 2rdr = ∫ rdr 3 ∫ du 3 −12 1 = =− u =− ⎡r2 + d2 ⎤ 2 2u2 r2 + d2 ⎣ ⎦ ∞ T ( )+qdqz = 0 = =− q r2 + d2 0 _ −q2 _ −q2 f = i = q z4πε0 (2d )2 16πε0d2 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 4 of 12III. Point Charge and Sphere 1. Grounded Sphere Φ= 1 ⎛⎜q +q' ⎞⎟4πε0 ⎝s s' ⎠ 1 1 s =⎡⎣r2 + D2 − 2rDcos θ⎤⎦ 2 ⎣ 2 2 ⎦ 2, s' =⎡b +− ⎤r 2rbcos θ q q' ⎛⎞q2 q' 2− ⎛ ⎞ Φ(r = R)=⇒ = ⇒⎜⎟=⎜ ⎟ 0 s s' s ⎝ ⎠ '⎝⎠ s6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 5 of 12 From Electromagnetic Field Theory: A Problem Solving Approach,by Markus Zahn, 1987. Used with permission.2 2 2 2 2 2 2 2 2 2q s' =q' s ⇒q' ⎡⎣R +D −2RDcos θ⎤⎦ =q ⎡⎣b +R −2Rbcos θ⎤⎦ q'2 (R2 + D2 )= q2 (b2 + R2 ) 2+q'2 2 RDcos θ =+q2Rbcos θ⇒q'2 = b q2 D b (R2 +D2 )=b2 +R2 ⇒b2 −b ⎜⎛R2 +D⎟⎞ +R2 =0 D ⎜⎝D ⎟⎠ ( − )⎛⎜b −RD2 ⎟⎞⎟ =bD ⎜ 0 ⎝ ⎠ R2 b = D q'2 = q2 b = q2 R22 ⇒ q' =− qR D DD Force on sphere 2 2qq' −q R D =−q RD f = = x2 2 24πε0 (D − b) 4πε0 ⎛⎜⎜D −R2 ⎞⎟⎟ 4πε0 (D2 − R2 ) ⎝ D ⎠ 2. Isolated Sphere [Put additional Image Charge +q' =+qR D at center] (zero charge) Φ(r = R)= 4π q' ε0R = 4π q ε0D 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 6 of 12force on sphere q ⎢⎡ q' q' ⎥⎤ qq' ⎡⎢⎣D2 −(D − b)2 ⎤⎥⎦ 2 ⎣⎡ 2 ⎦⎤−qR 2bD − b fx = 4πε ⎢ 2 − D2 ⎥= 22 = 22 0 ⎣(Db)⎦ 4πε0D (D − b) 3R− ⎛ ⎞ 4πε0D ⎜⎜D − D ⎟⎟ ⎝ ⎠ 2 2 2 ⎡ 2 ⎤ 2 3 fx =−qRD 2R ⎢2D − R ⎥= −qR 2 ⎡2D 2 − R2 ⎤ 4πε0D3 (D2 − R2 ) D ⎣⎢ D ⎦⎥ 4πε0D3 (D2 − R2 )⎣ ⎦ IV. Demonstration 4.7.1 – Charge Induced in Ground Plane by Overhead Conductor Courtesy of Hermann A. Haus and James R. Melcher. Used with permission. 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 7 of 121 −λ ⎡⎢⎣(ax−)2 + y2 ⎤⎥⎦ 2 −λ ⎡(ax)2 + y2 ⎤ − Φ= ln = ln ⎢⎥ 2πε0 ⎡(ax)2 + y2 ⎤ 12 4πε0 ⎢(ax)2 + y2 ⎥⎣+ ⎦⎢⎣+ ⎥⎦ C' = Φ(xl= −λ R,y = 0)= −λ ln λ al− + R =⎡ l22π−ε R02 + l⎤ , Φ(x = l − R,y = 0)= U ⎥2πε0a + l − R ln ⎢⎢ R ⎥⎣ ⎦ 0 ( ) 0 ∂x σ=ε Ex = 0 = −ε ∂Φ s x x0 = + ε0 λ d ⎡ln ⎡(a − x)2 + y2 ⎤− ln ⎡(a + x)2 + y2 ⎤⎤= 4πε0 dx ⎣⎢ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥⎦⎥ ⎡− x (⎤λ−2a( ) 2a + x)=⎢ − ⎥ 4π ⎢⎣ ax2 + 2a 22 ⎥⎦(−) y (+ x)+ y x0= −λa = 2 2π(a + y ) Total Charge per unit length on ground plane is: ∞ ∞ ∞ −λa −λ a1 −1yλT (x = 0)=∫ σsdy = −∞ ∫ π(a2 + y2 ) dy =π a tan = −λ a y=−∞ π −∞ 6.013, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 8 of 12dq dσs −aA dλ −aAC' dU i = ≈ A = = s 2 2 2 2dt dt π(a + y )dt π(a + y )dt take UU0 cos ωt= C'Aa v =−iR =− U ωsin ωt0 s s 2 20π(a +y ) V. Point Electric Dipole 1. Potential q ⎡1 1 ⎤Φ= ⎢−⎥4πε0 ⎣r+ r−⎦ r+= x2 + y2 +⎛⎜z −d ⎞⎟ 2 ⎝ 2 ⎠ 2 2 ⎛ d ⎞2 r−= x +y +⎜z + ⎝ 2 ⎠⎟ …
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