Question

Derive Variational functional of each partial differential equation the common PDEs of electromagnetics and their functionals at showing below: Variational Principle Associated with Common PDEs in EM Name of Equation PDE Variational Principle Inhomogeneous wave equation VD+k?D=8 op - k?o?+289)do Homogeneous wave equation v?0+k?o= 0 2(0)= ? Stva? – ko?jdv or Diffusion equation 030-40 =016=Suvep – +0}]dvdt 120-k0,=0 19=190p – koro, dvdt p?o=8 169= 151685 + 250]do 10) = Suvefjdv Poisson's equation Laplace's equation v =0 Note that|VOR = V0.50=0}+03+0.

Answer 1

Sdotio : a, z = stry) 010 (22) = m (A2) - lim z(x+Ax, y)-(2,4) olary Alo AX and Az = 2(2+0%, y+ Oy) - 26,9) Adding and substituting 7(x, y + Ay), we get → Az = 2(M+ An, y + Ay) – Z (A, Y+Ay ax + z(x,y+Ay)-z(2,4) → Taking Cimits x>0 and ayo dr = M dx+ wdy cchese M=(22)y and n= ( from the above Expression, we can conclude foom capation (1) dx = Cons)dy+ cesty dz –@

Eliminate de and combine 60 dz to and y can be constant (dy=0) the teoms in brackets must be equal to zero 6), (a), - → Recipoocity Relation → Sun CareC), -- > Cyclic Relation cyclic Relation PV = RT cu C), -. — @ P = Plu, T) - RT (21) = -PT V = V(0,7) - RI > GHDp - T = T(B,G) - PU (2) an 4 Becomes PV

Reciprocity Relation 음 음음 hence proved

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