Chapter. 4 Solution of Electrostatic Problems ChinWook Chung Dept. of Electrical Engineering Hanyang University Plasma Electronics Laboratory Hanyang University, Seoul, Korea Summary • Electrostatic cases (, V, E) Charge density, V 1 4 0 V R dv E 2V 0 1 4 0 V Rˆ R2 dV E / 0, E 0 E V Electric Field, E Potential, V V E dl Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Poisson's & Laplace’s eqn. 2V 2V 2V V 2 2 2 x y z In generalized coordinate, 2 2V V 1 h1h2 h3 (Laplacian) h2 h3 V h1h3 V h1h2 V u h u u h u u h u 1 2 2 2 3 3 3 1 1 E V , D= Ε D E V 2V (Poisson Eqn.) 0 2V 0 Laplace's equation Plasma Electronics Laboratory Hanyang University, Seoul, Korea The solutions of Laplace's equation • One dimensional case – V(x) is a function of only one of the three coordinates. 2V 2V 2V 2 2 0 2 x y z 1 V 1 2V 2V r 2 2 2 r r r r z 1 2 V 1 V 1 2V R 2 sin 2 2 2 R R R R sin R sin 2 2 1 d dV 3. 1 d V d 2V 1. 0 2. 2 2 4. r 2 r d r dr dr dx 1 d 2 dV 1 d dV R 5. sin R 2 dR dR R 2 sin d d Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Example 4.1 a) The potential at any point between the plates b) the surface charge densities on the plates d 2V y dy 2 0 dV C1 dy y 0 V y C1 y C2 E1n at y 0, V 0 at y d , V V0 Surface charge density a n a y , sl 0 E V V 0 y d E y a y y 0 a n a y , su 0 E V V a y 0 y d s 0 yd 0V0 d 0V0 d sl su in this case Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Example 4.2 – By solving Poisson’s and Laplace’s equations for V, – Determine the E field both inside and outside a spherical cloud of electrons with a uniform volume charge density, = -0 (where =0 is positive) for 0<R<b and =0 for R>b – Cf. example 3-23 1 d 2 dV R R 2 dR dR 0 E V aR dV dR Plasma Electronics Laboratory Hanyang University, Seoul, Korea Uniqueness Theorem • A solution of Poisson’s equation that satisfies the given boundary conditions is a unique solution. V1 , V2 : two solutions 2 V1 , V2 2 Assume that both V1 and V2 satisfy the same boundary condition on S1 , S 2 , S3 ,...S n Vd V1 V2 2Vd 0 Vd 2Vd 0 (Vd Vd ) Vd 2Vd Vd 2 fA f A A f Plasma Electronics Laboratory Hanyang University, Seoul, Korea Uniqueness Theorem V V a ds S d d n Vd d 2 1 1 S Vd Vd an ds 0 Vd R , Vd R 2 Vd 0 Vd const or 0 Vd 0 on the surfaces Vd 0 everywhere • A unique solution The solution obtained even by intelligent guessing !! Plasma Electronics Laboratory Hanyang University, Seoul, Korea Method of image • Point charge and ground plate Plasma Electronics Laboratory Hanyang University, Seoul, Korea Method of image V Q 1 Q 1 4 0 R 4 0 R R x 2 y d z 2 , R x 2 y d z 2 2 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Method of image V Ez y 0 y S 0 Ez y 0 Qd 2 x 2 z 2 d 2 Qtotal S 2 rdr Qd 0 0 rdr 2 r d 2 2 2 , x z r 3/ 2 2 3/ 2 Q Plasma Electronics Laboratory Hanyang University, Seoul, Korea Method of image Negatively charged conduction plane s Q Q Q d 2 r 2 d 2 3/ 2 Q • What if the conduction plane is not grounded ? Plasma Electronics Laboratory Hanyang University, Seoul, Korea Method of image • Example 4-3 F1 a y F2 a x F3 Q2 4 0 2d2 2 Q2 4 0 2d1 2 Q2 2 3/ 2 4 0 2d1 2d2 2 a x 2d1 a y 2d2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Line charge & line images • Conducting cylinder and line charges E • l 1 2 0 r Potential from a line charge (r0 = reference point) l r0 V Er dr ln r 2 0 r r r l V? 0 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Line charge & line images Two line charges V l r ln 0 2 0 r Finding equipotential surfaces VM l r r r ln 0 l ln 0 l ln i 20 r 20 ri 20 r Equipotential surface ri /r =const. PM OPi OM i PM OM OP ri di a a2 const. di r a d d Plasma Electronics Laboratory Hanyang University, Seoul, Korea Two wire transmission line • Example 4-4 – Capacitance ? C = Q / V V= 220 Volts V=0 D Plasma Electronics Laboratory Hanyang University, Seoul, Korea Two wire transmission line V2 C l a a ln , V1 l ln 2 0 d 2 0 d l 0 V1 V2 ln d / a a2 1 d D di D d D D 2 4a 2 d 2 C 0 ln D / 2a D / 2a 0 D 1, C 2a ln D / a 2 1 0 ln x x 2 1 cosh 1 x cosh D / 2a 1 F/m Plasma Electronics Laboratory Hanyang University, Seoul, Korea Point charge & conducting sphere • Example 4-5 P – OMQ, OMP : similar triangles a di d a 1 Q Qi 0 4 0 r ri r Q a i i Qi Q r Q d VM Plasma Electronics Laboratory Hanyang University, Seoul, Korea Point charge & conducting sphere • Example 4-5 P VM r a, , 1 Q Qi 4 0 r ri , r a 2 d 2 2ad cos , ri a 2 d i2 2ad i cos For grounded sphere Qi 1 Q V a, , 4 0 a 2 d 2 2ad cos a 2 d i2 2ad i cos V a, 0 0, V a, 0 0 a2 a di , Qi Q d d Plasma Electronics Laboratory Hanyang University, Seoul, Korea Point charge & conducting sphere • Example 4-5 Q 1 a 1 V R, , 4 0 RQ d RQi RQ R 2 d 2 2 Rd cos , 2 a2 a2 2 RQi R 2 R cos d d Electric field V R Charge density ER R, s 0 ER a, Q d 2 a2 4 a a 2 d 2 2ad cos 3/ 2 Total induced charge on the sphere Qind s ds 2 0 S 0 a d s a 2 sin d d Q Qi Plasma Electronics Laboratory Hanyang University, Seoul, Korea Point charge & conducting sphere When V a, , V0 , additional image charge is required Qi V0 Qi 4 0 aV0 4 0 a Qi V=V0 additional image charge Electric field profile Plasma Electronics Laboratory Hanyang University, Seoul, Korea Charged Sphere and conducting plane • 4 - 4.4 2 Q Q0 Q1 Q2 ... Q0 1 .... 2 1 a 2c V0 Q0 4 0 a , C Q 2 4 0 a 1 ... 2 V0 1 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Laplace’s equation in cylindrical coordinates 1 V 1 2V 2V r 2 2 2 0 r r r r z • Cylindrical symmetry and the lengthwise dimension is very large 2V 2V 0, 2 0 2 z 1 dV r r dr dr • 0 V r C1 ln r C2 If the problem is such that electric potential changes only in the circumferential direction and not in r- and z-directions, 1 r2 d 2V 2 d 0 V K1 K 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Example (Problem 4-23) Determine the potential distribution fo the regions : a) 0 b) 2 d 2V For 0 , 0 2 d V 0 0 V V0 V V A B V0 , 0 For 2 , V V0 K1 +K 2 V 2 0 2 K1 +K 2 V0 2 V0 , K2 2 2 Finally, V0 V 2 , 2 2 K1 E? E 1 dV 1V a 0 a r d r Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Example – spherical capacitor – Determine the potential distribution Spherical symetry, V is independent of and C1 d 2 dV dV C1 2 V C2 R 0 dR dR dR R R C1 at R Ri , V Ri V1 C2 Ri C1 at R Ro , V Ro V2 C2 Ro Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems C1 R0 Ri V1 V2 R0 Ri V R R0V2 RiV1 , C2 R0 Ri 1 Ri R0 V V R V R V 1 2 0 2 i 1 , Ri R R0 R0 Ri R V is independent of the dielectric constant of the insulating material C ? 4 0 Ri Ro 4 0 C R0 Ri 1 1 Ri R0 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems • Example (problem 4-26) 1 d dV sin 0 2 R sin d d excluding R 0 & 0 or dV A d d V A B A ln tan B sin 2 sin ln tan V / 2 0 2 V V0 V / 2 V0 ln tan 2 E 1 dV a R d V0 a R sin ln tan 2 S V0 R sin ln tan 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Electrostatic Boundary Value Problems V0 2 2V0 R sin ddR Q dR R sin ln tan 0 0 ln tan 0 2 2 infinity ! Q Q 2V0 2R1 R1 C V0 ln tan ln tan 2 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cartesian coordinates • Problems governed by partial differential equations with prescribed boundary conditions are called boundary value problems (BVPs) • BVPs for potential can be classified into three types: – Dirichlet problems • The value of the potential is specified everywhere on the boundaries – Neumann problems • The normal derivative of the potential (electric field) is specified everywhere on the boundaries – Mixed boundary-value problems • The potential is specified over some boundaries and normal derivative of the potential is specified over the remaining ones. • The solutions of Laplace’s equation are often called harmonic functions. Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cartesian coordinates • Laplace’s equations – Separation of variable 2V 2V 2V 2 2 0 2 x y z V x , y, z X x Y y Z z d2 X d 2Y d 2Z 1 d 2 X 1 d 2Y 1 d 2Z YZ XZ XY 0 0 dx 2 dy 2 dz 2 X dx 2 Y dy 2 Z dz 2 – A function of only one coordinate variable, each of the three terms must be a constant 1 d2 X d2 X 2 2 k k X, x x 2 2 X dx dx d 2Y d 2Z 2 2 Similarly, k Y , k Z y z 2 2 dy dz Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cartesian coordinates • The condition for the separation constants kx2 ky2 kz2 0 • Possible solutions for X(x) 1 d2 X 2 k x X dx 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cartesian coordinates • Example 4-6 V 0, y V0 , V , y 0 V x , 0 V x ,b 0 kx2 ky2 0 ky2 kx2 k 2 1 d2 X 2 2 k k , x 2 X dx 1 d 2Y 2 2 k k y Y dy 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cartesian coordinates • Example 4-6 V 0, y V0 , V , y 0 V x , 0 V x ,b 0 ky2 kx2 k 2 X x D2e kx ,Y y A1 sin ky Vn x , y Cne kx sin ky V x , b 0 Vn x , b Cn e kx sin kb 0 k Vn x , y Cn e n x / b n n sin y V x , y Cn e n x / b sin y b b n 1 n V 0, y V0 Cn sin y 0 yb b n 1 4V0 1 n x / b n V x, y e sin y n odd n b n b 4V0 if n is odd Cn n 0 if n is even 1 sin A sin B cos A B cos A B 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • Laplace’s equation for V in cylindrical coordinate 1 V r r r r 2 2 1 V V 2 0 Bessel functions 2 2 r z • In such cases, 𝜕 2 𝑉/𝜕𝑧 2 = 0. After separation of variables V r , R r r d dR 1 d r d dR 1 d 2 2 r 0 r k , k R dr dr d 2 R dr dr d 2 k integer=n d2R dR r r n 2 R r 0 R r Ar r n Br r n 2 dr dr 2 • A sin n B cos n General solutions Vn r, r n An sin n Bn cos n r n A 'n sin n B 'n cos n Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • The simplest form when k=0 1 d 2 k A0 B0 B0 no circumference variation d 2 d dR r 0 R r C0 ln r D0 , for k 0 dr dr Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • Example 4-8 – coaxial cable V b 0, V a V0 – No z dependence and by symmetry, no dependence (k=0) C1 ln b C2 0 C1 ln a C2 V0 C1 V r V0 V ln b , C2 0 ln b / a ln b / a V0 b ln ln b / a r Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • Example 4-9 – infinite thin tube V0 for 0< < V b, V0 for < <2 – General solution Vn r, r n An sin n Bn cos n r n A 'n sin n B 'n cos n Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • Example 4-9 – infinite thin tube V for 0< < V b, 0 V0 for < <2 – Inside the tube ( r < b) V r, r n An sin n V finite @ r 0 & odd function of n 1 4V0 if n is odd V b, V0 for 0< < An n bn 0 if n is even – Outside the tube (r > b) V r, r n Bn sin n V finite @ r & odd function of n 1 V for 0< < At r b, V b, b n Bn sin n 0 n 1 V0 for < <2 4V0bn if n is odd Bn n 0 if n is even Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in Cylindrical coordinates • Example 4-9 – Inside the tube 4V0 n 1r V r, sin n , r b n odd n b – Outside the tube V r, 4V0 n 1 b sin n , r b n odd n r Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in spherical coordinates • Laplace’s equation for V in spherical coordinate with azimuthal symmetry 1 2 V 1 V R sin R 2 R R R 2 sin • 0 After separation of variables, V R, R 1 d 2 d 1 d d R sin 0 dR dR sin d d k2 k 2 d 2 d n 1 2 n 2 R 2 R k 0 R A R B R , n n 1 k n n n dR 2 dR 2 d d d sin n n 1 sin 0 Pn cos , Legendre polynomials d Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in spherical coordinates • General solution with no azimuthal variation V R, An R n Bn R n 0 n 1 P cos n Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in spherical coordinates • Example 4-10 – Uncharged conducting sphere of radius b is placed in an initially uniform electric field E0=z E0. Determine V(R,) and E(R,) Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in spherical coordinates • Example 4-10 – V(R,) for R>b V R, An R n Bn R n 0 V b, 0 n 1 P cos n V R, E0 z E0 R cos for R b V finite @ R & V E0 R cos V R, E0 RP1 cos Bn R n 1 n 0 Pn cos V R, B0 R B1R E0 R cos Bn R 1 2 n2 0 no charging – B1 E0b3 n 1 Pn cos 0 V b, 0 b 3 V R, E0 1 R cos R Plasma Electronics Laboratory Hanyang University, Seoul, Korea BVP in spherical coordinates • Example 4-10 – Electric field intensity E a R ER a E 3 V b ER E0 1 2 cos R R b 3 1 V E E0 1 sin R R ps P0 cos R b V R ? z – Surface charge density s 0 ER b, 3 0 E0 cos s cos p qd – Dipole p a zmoment 4 0b3 E0 V E0 R cos E0b cos R2 3 V qd cos 4 0 R 2 dipole potential term Plasma Electronics Laboratory Hanyang University, Seoul, Korea Numerical solution of Laplace’s equation • Finite difference method V V1 V2 x d 2V V2 2V1 V3 2V V4 2V1 V5 2 , Analogously, x d2 y 2 d2 2V 2V V2 V3 V4 V5 4V1 V 2 2 0 2 1 x y d 1 V1 V2 V3 V4 V5 4 2 Plasma Electronics Laboratory Hanyang University, Seoul, Korea Numerical solution of Laplace’s equation • Finite difference method – Potential profile ( +100V to 0V ) in a parallel plate Plasma Electronics Laboratory Hanyang University, Seoul, Korea H.W. • 4-1,2,3,4,7,5,9,15,17,22,23,24,27,29 Plasma Electronics Laboratory Hanyang University, Seoul, Korea
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