University Electromagnetism: Integral types: linear, vectorial, scalar, 2D and 3D

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

2. History Electricity Magnetism EM-fields EM-waves (Technical) Applications Franklin Coulomb Galvani Volta Ampère Oersted Gauss Faraday Helmholtz Maxwell Lorentz Hertz Millikan Marconi 1700 1800 1900 2000

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

5. Line-integral (scalar) (2) Problem : which integration path ? In general: Result of integration depends on choice of integration path. f x i x j P 1 P 2 f f = f (….., x i , x j ,….) Multi-dimensional

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

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

8. Surface-integral (scalar) T x y P 1 P 2 T Temperature field: T = f (x,y) = c(2x+y) Problem: determine < T > over S; Find formula and calculate: S

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

10. Surface Integral (vectorial): Example 1 Contribution from  -component ( // e z ) only ! Calculate surface integral over x = 1..2 and y = 1..3 A x y A S A e n  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) = x e x + 2y e y + 3 e z

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  R.sin  u v Spherical surface element Calculate surface integral of B over A at radius r over octant (  = 0 .. ½  ;  = 0.. ½  )

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

13. Volume integral: Example Suppose: charge density  =3ar. sin  .cos  [C/m 3 ] Calculate charge Q in region: ( 2<r<3 ; 0<  < ½  ; - ½  <  < ½  ) the end r  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