PHY481: Electromagnetism

Transcription

1 PHY481: Electromagnetism Vector tools Sorry, no office hours today I ve got to catch a plane for a meeting in Italy Lecture 3 Carl Bromberg - Prof. of Physics

2 Cartesian coordinates Definitions Vector x is defined relative to the origin of primary coordinate system (x,y,z) In Cartesian coordinates x = x î + y ĵ + z ˆk Arbitrary rotation of the coordinate system changes vector direction but not the magnitude x = x ˆ i + y ˆ j + z k ˆ x = x = x 2 + y 2 + z 2 Example: rotation about the z-axis by angle φ ˆ i = îcosφ + ĵsinφ ˆ j = îsinφ + ĵcosφ k ˆ = ˆk x = x ˆ i = xcosφ + ysinφ y = x ˆ j = xsinφ + ycosφ z = z Lecture 3 Carl Bromberg - Prof. of Physics 1

3 Rotation matrix A rotation can be expressed as a 3x3 unitary matrix R Rotation of coordinates x x y = R y z z Rotation of vector components E x Ey E z = R E x E y E z Rotation Matrix is Unitary Maintains vector magnitude Transpose = Inverse R ij T = R ji = R ij 1 Rotation matrix properties R = BCD (3 rotations) Rotations do not commute B = C = D = Euler angle rotations cosφ sinφ 0 sinφ cosφ cosθ sinθ 0 sinθ cosθ cosψ sinψ 0 sinψ cosψ rotation about z rotation about x rotation about z Lecture 3 Carl Bromberg - Prof. of Physics 2

4 Vector multiplication Vector addition and multiplication by a constant is OK Scalar (or dot) product A B = A x B x + A y B y + A z B z = 3 A i B i i=1 (1,2,3) = (x,y,z) Einstein summation convention (repeated indices) A B = A i B i = A 1 B 1 + A 2 B 2 + A 3 B 3 Σ is superfluous Cross product (new techniques) C = A B C i = B k A k B j i,j,k are cyclic permutations of 1,2,3 C x = A y B z A z B y C y = A z B x A x B z C z = A x B y A y B x C 1 = A 2 B 3 A 3 B 2 C 2 = A 3 B 1 A 1 B 3 C 3 = A 1 B 2 A 2 B 1 Lecture 3 Carl Bromberg - Prof. of Physics 3

5 Economy of notation we will NEED Levi-Chivita anti-symmetric 3x3x3 tensor ε ijk Learn to love it! ( A B) i = ε ijk B k i, j, k = (1, 2 or 3) If any 2 indices are the same Permutations of 123 Cyclic = even # of pair-wise Non-cyclic = odd # of pair-wise ε ijk = 0 21 of 27 elements are 0! 6 non-zero elements ε 123 = ε 312 = ε 231 = +1 ε 213 = ε 132 = ε 321 = 1 (A B) 1 = ε 123 A 2 B 3 + ε 132 A 3 B 2 = A 2 B 3 A 3 B 2 Verify the remainder for yourself Levi-Chivita tensor product ε ijk ε klm = δ il δ jm δ im δ jl sum over k, k = 1, 2, 3 Kronecker delta { δ ij = 1, i = j 0, i j Lecture 3 Carl Bromberg - Prof. of Physics 4

6 Vector identities (Levi-Chivita tensor) Prove A x (B x C) = B(A C) C(A B) Cross product using Levi-Chivita tensor: ( A B) i = ε ijk B k Levi-Chivita tensor product: ε ijk ε klm = δ il δ jm δ im δ jl Start proof here: i th component B C = D A ( B C) ( ) i = ε ijk D k D k = (B C) k = ε klm B l C m [ ] i = A D [ A ( B C) ] i = ε ijk ε klm B l C m = ( δ il δ jm δ im δ ) jl B l C m = B i C j B j C i A ( B C) = B ( A C ) C ( A B ) need indices l and m B j = A B Lecture 3 Carl Bromberg - Prof. of Physics 5

7 Vector operators - Gradient (Cartesian) Gradient operates on a scalar function V (potential) Gives a vector in the direction of the maximum change in V V (x) = V ê i = V x î + V y Example: parallel plate capacitor ĵ + V z ˆk ê 1 = î = ê ê 2 = ĵ i ê 3 = ˆk V (x) = V (x) = Ex Maximum change in V is in the +x direction V (x) = V x î = E î Electric field E points in the direction opposite to maximum change in V, -x direction E(x) = V (x) = E î Lecture 3 Carl Bromberg - Prof. of Physics 6

8 Vector operators - Divergence (Cartesian) Divergence of a vector function is a scalar E(x) = E i = E x x + E y y + E z z Spreading of E at x. 0 if charge is at x Coordinate independent definition E(x) da = E i (x + ε 2 êi )ε 2 E i (x ε 2 S = = 3 i=1 êi )ε 2 3 E i (x + ε 2 êi ) E (x ε i 2 êi ) ε 3 ε i=1 3 i=1 E i ε 3 = E(x) = ê i tiny box centered on x da L ε 2 î [ ]ε 3 E(x) = lim V 0 1 V ε Del operator Divergence definition S ε ε da R ε 2 î E(x) da Lecture 3 Carl Bromberg - Prof. of Physics 7

9 Differential form of Gauss s Law Gauss s Law in terms of divergence of E Gauss s Law lim V 0 1 V 1 V Gauss s Law equivalent S S S E(x) da E(x) da E(x) da = q encl ε 0 = 1 V = lim V 0 q encl ε 0 E(x) = ρ(x) ε 0 q encl V 1 ε 0 a 2 î E(x) = lim V 0 Box centered on x a a a a 2 î ρ = q encl V ; V = a3 see previous slide 1 V S E(x) da Lecture 3 Carl Bromberg - Prof. of Physics 8

10 Vector operators - Curl (Cartesian) Curl of a vector function is another vector B [ B(x) ] i = ε k ijk x j Circulation of E around a loop ( B(x) ) 1 = B 3 B 2 x 2 x 3 Verify the remainder B(x) d = B i (x ε ê )ε B (x + ε ê )ε 2 j i 2 j P k Coordinate independent definition (1) (2) + B j (x + ε 2 êi )ε B (x ε j 2 = B j B i x j ε 2 = B(x) êi )ε [ ] k ε 2 Lecture 3 Carl Bromberg - Prof. of Physics 9 (3) (4) Curl definition ˆn [ B(x) ] = lim A 0 Path centered on x 1 A C B(x) d

11 Cartesian vector operators (Einstein notation) Summary of earlier slide 4 Del = ê i A B = A i B i ε ( A B) ijk i = ε ijk B k Permutations of 123 Cyclic = even # of pair-wise 6 non-zero elements ε 123 = ε 312 = ε 231 = +1 Non-cyclic = odd # of pair-wise ε 213 = ε 132 = ε 321 = 1 ê 1 = î, ê 2 = ĵ, ê 3 = ˆk Dot Product Tensor product ε ijk ε klm = δ il δ jm δ im δ jl Gradient V (x) = V ê i Levi-Chivita Tensor Divergence E(x) = E i V = 2 V Kronecker delta { δ ij = 1, i = j 0, i j Cross Product Vector differential operators in Einstein notation Curl B [ B(x) ] i = ε k ijk x j Laplacian of scalar function V Lecture 3 Carl Bromberg - Prof. of Physics 10

12 Physical interpretation of vector operators Characterize flow of field [area displayed (±3m, ±3m)] E(x) = ( x î + y ĵ ) V/m 2 B(x) = ( y î + x ĵ ) T/m 2 C 0 C 0 E = ρ(x) B = 0 B is solenoidal ε 0 = 2 V/m 2 B = µ 0 J(x) = 2 T/m 2 E = 0 E is irrotational (see Lecture 3 Carl Bromberg - Prof. of Physics 11