FdM 1 Electromagnetism First-year course on Integral types © Frits F.M. de Mul

FdM 2History Electricity Magnetism EM-fields EM-waves (Technical) Applications Franklin Coulomb Galvani Volta Ampère Oersted Gauss Faraday Helmholtz Maxwell Lorentz Hertz Millikan Marconi

FdM 3 Multidimensional integrations Integration of charges and currents over lines, planes and volumes 1.Line integrals (scalar, vectorial) +example 2.Surface-integrals (scalar, vectorial) + 2 examples 3.Volume-integrals (scalar only) + example

FdM 4 Line-integral (scalar) (1) y xx1x1 x2x2 f (x) One-dimensional: F F = area under curve Example: Calculate average temperature T(x) between x 1 and x 2 Meaning of F:

FdM 5 Line-integral (scalar) (2) Problem: which integration path ? f xixi xjxj P1P1 P2P2 f f = f (….., x i, x j,….) Multi-dimensional In general: Result of integration depends on choice of integration path.

FdM 6 T x y P1P1 P2P2 T T (x,y) = c (2x+y) P 1 (1,1) P 2 (2,3) Example Line-integral (scalar) (3) (1) (2) (1)(2) Different paths : different results; Calculate line-integrals first and check below: Special case: Conservative field: result independent of path

FdM 7 Line-integral (vectorial) A x y P1P1 P2P2 A (1) (2) dl along integration path Example: Work done by force: Definition: Consequence: if F  dl : W = 0 : (example: centripetal force)

FdM 8 T x y P1P1 P2P2 T Temperature field: T = f (x,y) = c(2x+y) Surface-integral (scalar) Problem: determine over S; Find formula and calculate: S

FdM 9 Surface-integral (vectorial) Definition: A enen  dx dy dS e n = normal unit vector, // to dS dS = e n dS dS A x y A S

FdM 10 Surface Integral (vectorial): Example 1 A x y A S A enen  dx dy dS e n = normal unit vector, // to dS dS = e n dS dS Suppose: Area S in z=0 plane ; there A(x,y,0) = xe x + 2ye y + 3e z Contribution from  -component ( // e z ) only ! Calculate surface integral over x = 1..2 and y = 1..3

FdM 11 Surface Integral (vectorial): Example 2 dA = u.v = (R.sin .d  ).(Rd  ) B Suppose: B = r.sin  e r + cos  e  + tan  e  Normal vector e n = e r everywhere ! R  dd  dd R.sin  u v Spherical surface element Calculate surface integral of B over A at radius r over octant (  =0.. ½  ;  =0..½  )

FdM 12 x y z V Block V : limited by points (1,1,1) and (2,4,5) Volume-integral (scalar only) Example: Charge density:  = c(3x-2y+5z) [C/m 3 ] Problem: determine total charge Q in V dV=dxdydz Define charge element dQ in volume element dV : dQ = .dV

FdM 13 Volume integral: Example r  dd  dd r.sin  u v Spherical volume element w R dV = u.v.w = (r.sin .d  ).(rd  ).dr 0<  < 2  ; 0<  <  ; 0 < r < R Suppose: charge density  =3ar.sin .cos  [C/m 3 ] Calculate charge Q in region: (2<r<3 ; 0<  <½  ; -½  <  <½  ) the end