1 / 122

Physics 441

Physics 441. Electro-Magneto-Statics M. Berrondo. 1. Introduction. Electricity and Magnetism as a single field (even in static case, where they decouple) Maxwell: * vector fields * sources (and sinks) Linear coupled PDE’s * first order (grad, div, curl)

ganesa
Download Presentation

Physics 441

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 441 Electro-Magneto-Statics M. Berrondo Physics BYU

  2. 1. Introduction • Electricity and Magnetism as a single field (even in static case, where they decouple) Maxwell: * vector fields * sources (and sinks) • Linear coupled PDE’s * first order (grad, div, curl) * inhomogeneous (charge & current distrib.) Physics BYU

  3. Tools Math Physics trajectories: r(t) FIELDS: * scalar, vector * static, t-dependent SOURCES: charge, current superposition of sources => superposition of fields unit point sources Maxwell equation field lines potentials charge conservation • trigonometry • vectors (linear combination) dot, cross, Clifford • vector derivative operators • Dirac delta function • DISCRETE TO CONTINUUM • INTEGRAL THEOREMS: * Gauss, Stokes, FundThCalc • cylindrical, spherical coord. • LINEARITY interpretation of equations and their solutions Physics BYU

  4. 2. Math. Review • sum of vectors: A , B => A + B • dilation: c , A => c A • linear combinations: c1A +c2B • scalar (dot) product: A , B => A  B = A B cos ( ), a scalar whereA2=A  A (magnitude square) • cross product: A , B => A  B = n A B |sin( ) |, a new vector, with n A and B, and nn = n2 = 1 • orthonormal basis: {e1 , e2, e3} = {i , j , k} Physics BYU

  5. Geometric Interpretation:Dot product Physics BYU

  6. Cross Product and triple dot product Physics BYU

  7. Rotation of a vector (plane) • Assume s in the x-yplane • Vector s‘= k x s • Operation k x rotates s by 90 degrees • k x followed by k x again equivalent to multiplying by -1 in this case!! y s’ s x Physics BYU

  8. Rotation of a vector (3-d) n unit vector: n2 = 1 defines rotation axis  = rotation angle vector r  r’ where Physics BYU

  9. Triple dot product is a scalar and corresponds to the (oriented) volume of the parallelepiped {A, B, C} Physics BYU

  10. Triple cross product • The cross product is not associative! • Jacobi identity: • BAC-CAB rule: is a vector linear combination of B and C Physics BYU

  11. Inverse of a Vector 0 1 | | n-1=n as a unit vector along any direction in R2or R3 Physics BYU

  12. Clifford Algebra Cl3(product) • starting from R3basis {e1 , e2 , e3 }, generate all possible l.i. products  • 8 = 23 basis elements of the algebra Cl3 • Define the product as: • with Physics BYU

  13. R3 R i R i R3 Physics BYU

  14. Clifford Algebra Cl3 • Non-commutative product w/ A A = A2 • Associative: (A B) C = A (B C) • Distributive w.r. to sum of vectors • *symmetric part  dot product *antisym. part  proportional cross product • Closure: extend the vector space until every product is a linear combination of elements of the algebra Physics BYU

  15. Subalgebras • R Real numbers • C = R + iR Complex numbers • Q = R + iR3 Quaternions Product of two vectors is a quaternion: represents the oriented surface (plane) orthogonal to A x B. Physics BYU

  16. Bivector: oriented surface b b a a sweep sweep • antisymmetric, associative • absolute value  area Physics BYU

  17. Differential Calculus • Chain rule: In 2-d: • Is an “exact differential”? • given Physics BYU

  18. In 3-d: • Geometric interpretation: Physics BYU

  19. del operator Examples: Physics BYU

  20. Gradient of r • Contour surfaces: spheres •  gradient is radial • Algebraically: and, in general, Physics BYU

  21. Divergence of a Vector Field • E (r)  scalar field (w/ dot product) • It is a measure of how much the filed lines diverge (or converge) from a point (a line, a plane,…) Physics BYU

  22. Divergence as FLUX: ( ) dx Examples: Physics BYU

  23. Curl of a Vector Field The curl measures circulation about an axis. Examples: Physics BYU

  24. Clifford product del w/ a cliffor • For a scalar field T = T( r ), • For a vector field E = E( r ), • For a bivector field iB =i B( r ), Physics BYU

  25. Second order derivatives • For a scalar: • For a vector: Physics BYU

  26. What do we mean by “integration”? cdq c dr dw = c(cdq)/2dg = (2pr) dr | |dl rdq dr da= (rdq)dr Physics BYU

  27. Cliffor differentials • dkais a cliffor representing the “volume” element in k dimensions • k = 1  dl is a vector  e1 dx (path integral) • k = 2  inda bivector  e1 dx e2 dy (surface integral) • k = 3  i dtps-scalar  e1 dx e2 dy e3 dz (volume integral) Physics BYU

  28. Fundamental Theorem of Calculus Particular cases: Gauss’s theorem Stokes’ theorem Physics BYU

  29. Delta “function” (distribution) • 1-d: q step “function” 1 x Physics BYU

  30. Divergence theorem and unit point source apply to for a sphere of radius e  Physics BYU

  31. and Displacing the vector r by r‘: Physics BYU

  32. Inverse of Laplacian • To solve so • In short-hand notation: Physics BYU

  33. Orthogonal systems of coordinates • coordinates: (u1, u2, u3 ) • orthogonal basis: (e1, e2, e3 ) • scale factors: (h1, h2, h3 ) • volume: • area • displacement vector: Physics BYU

  34. Scale Factors • polar (s, f ): • cylindrical (s, f, z ): • spherical (r, q, f ): Physics BYU

  35. Grad: • Div: • Curl: • Laplacian: Physics BYU

  36. Maxwell’s Equations • Electro-statics: • Magneto-statics: • Maxwell: Physics BYU

  37. Formal solution separates into: and Physics BYU

  38. Electro-statics Convolution: where for point charge. Physics BYU

  39. Superposition of charges • For n charges {dq1, dq2, …, dqn } • continuum limit: • where Physics BYU

  40. linear uniform charge density l (x’) from –L to L • field point @ x = 0, z variable z q | | x -L L dq’ Physics BYU

  41. with so • Limits: Physics BYU

  42. Gauss’s law • Flux of E through a surface S: volume V enclosed by surface S. • Flux through the closed surface: • choose a “Gaussian surface” (symmetric case) Physics BYU

  43. Examples: • Charged sphere (uniform density) radius R Gaussian surface: a) r < R: b) r > R: as if all Q is concentrated @ origin Physics BYU

  44. Thin wire: linear uniform density (C/m) Gaussian surface: cylinder • Plane: surface uniform density (C/m2) Gaussian surface: “pill box” straddling plane CONSTANT, pointing AWAY from surface (both sides) Physics BYU

  45. Boundary conditions for E • Gaussian box w/ small area D A // surface w/ charge density s with n pointing away from 1 • Equivalently: • Component parallel to surface is continuous • Discontinuity for perp. component = s/e0 Physics BYU

  46. Electric Potential (V = J/C) • Voltage solves Poisson’s equation: • point charge Q at the origin Physics BYU

  47. Potential Difference (voltage) • in terms of E: and • Spherical symmetry: V = V (r) • Potential energy: U = q V (joules) • Equi-potential surfaces: perpendicular to field lines Physics BYU

  48. Example: spherical shell radius R, uniform surface charge density s Gauss’s law  E(r) = 0 inside (r < R) For r > R: and Physics BYU

  49. Example: infinite straight wire, uniform line charge density l Gauss’s law: and Physics BYU

  50. Electric-Magnetic materials • conductors • surface charge • boundary conditions • 2nd order PDE for V (Laplace) • dielectrics • auxiliary field D (electric displacement field) • non-linear electric media • magnets • auxiliary field H • ferromagnets • non-linear magnetic media Physics BYU

More Related