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 0 (x,y,z) Z Induced Surface Charge q d d - 个 Image Charge Φp= q 1 4πe0 ye-y ax 9 z(xI.tyi,e-0T.)Zxiyi(+ 40 z[2+y+-7z[x+y++形 q (-d) =eg (perpendicular to equipotential ground plane) 6.641,Electromagnetic Fields,Forces,and Motion Lecture 5 Prof.Markus Zahn Page 1 of 116.641, Electromagnetic Fields, Forces, and Motion Lecture 5 Prof. Markus Zahn Page 1 of 11 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 ( ) ( ) p 22 22 2 2 0 q1 1 4 x y zd x y zd ⎡ ⎤ ⎢ ⎥ Φ= − π + +− + ++ ⎣ ⎦ ε ___ ppp E iii p xyz p xyz ⎡ ⎤ ∂Φ ∂Φ ∂Φ = −∇Φ = − + + ⎢ ⎥ ∂∂∂ ⎣ ⎦ 0 2 q 4 = πε ( ) __ _ xy z xi yi z d i 2 ⎛ ⎞ ⎜ ⎟ + +− ⎝ ⎠ ( ) 3 2 2 2 2 2 x y zd − ⎡ ⎤ + +− ⎢ ⎥ ⎣ ⎦ ( ) __ _ xy z xi yi z d i 2 ⎛ ⎞ ⎜ ⎟ + ++ ⎝ ⎠ ( ) 3 2 2 2 2 x y zd ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎡ ⎤ + ++ ⎥ ⎢ ⎢ ⎥ ⎣ ⎦ ⎥ ⎣ ⎦ ( ) (− ) = = π ⎡ ⎤ + + ⎣ ⎦ ε _ p z 3 0 2 2 22 q d Ez0 i 2 xyd (perpendicular to equipotential ground plane)