Edward C. Jordan Memorial Offering of the First Course under the Indo-US Inter-University Collaborative Initiative in Higher Education and Research: Electromagnetics for Electrical and Computer Engineering by Nannapaneni Narayana Rao Edward C. Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois, USA Amrita Viswa Vidya Peetham, Coimbatore July 10 – August 11, 2006

5.3 Poisson’s and Laplace’s Equations

5.3-2 Poisson’s Equation For static electric field, Then from If  is uniform, Poisson’s equation

5.3-3 If  is nonuniform, then using Thus Assuming uniform , we have For the one-dimensional case of V(x),

5.3-4 Anode, x = d V = V 0 Cathode, x = 0 V = 0 Vacuum Diode D5.7 (a)

5.3-5 (b)

5.3-6 (c)

5.3-7 Laplace’s Equation Let us consider uniform  first Parallel-plate capacitor If  Poisson’s equation becomes x = d, V= V 0 x = 0, V = 0

5.3-8 Neglecting fringing of field at edges, General solution

5.3-9 Boundary conditions Particular solution

5.3-10

area of plates For nonuniform  For

Example x = d, V = V 0 x = 0, V = 0

5.3-13

5.3-14