22.615, MDH Theory of Fusion Systems Prof. Final Exam Solution. ψ ψ. 1a. Solution: 1+κ κ

Transcription

1 .65, MDH Theor of Fusion Sstems Prof. Finl Em Solution. ψ ψ + +μ R J ψ ( S S: + Solution: ψ A + A + ψ A + R J μ A μ R J + μr J ψ , MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge of

2 ψ ψ + ( + μ R J ψ, μ J ψ + R ψ ψ b. Bp ψ ez e R R R e B ψ ψ R R B ψ ψ R R. B ξ B B ξξ B B ξξ e B ξ p B p B B B e e J ξ + μ z μ J B δ F dr J B p ( p ξ + γ ξ + ξ ξ μ B δ F dr J B ξ μ B B B ξ 4ψξ μr μ μ B B ξ J B ξe ( Jzez ξ e ξ e ξ B Jz ξj B + ξ 4 + μ μ ψ ψ ψξ 4 4 R R R (.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge of

3 4ψξ ψξ F R μr μr δ dr π π 4πψξ F μr δ 4π ψξ μr 3. B A e B z B A e e A A e e A + A e A 3b. δ B μ dr z z B A e A e A z z z z z A A δ ( A dr + dr μ μ 3c. (See ttched sheet for summr of coordinte trnsformtion A n B n A ez n ez A n ez A t A ρθ A n B S b θ ρ b A b, θ Boundr Conditions n B n A e Sp z B n B n ξ ξ C B coshu cos B + sinhu sin C ψ ψ ξ coshu cos + sinhu sin R R.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 3 of

4 C ψ R ξ ( coshu cos ψξ cos R C ψξ n B R C cos n A e n e A n e A t A ( C A z z z A ψξ cos R u A u, ψ ξ sin R Vcuum Vector Potentil A A + ψξ A u, A( u, sin R ( u ( ψξ sinh u A K sinh ( u u sin sin R sinh u u A ψξ cosh ( u u u R sinh( u u sin Vcuum Energ δ B dr ( A dr ( A A A A dr μ μ μ μ n AAdS p δ An A μ ds p.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 4 of

5 A An AdSp A πr C d C Then A R π A d π A δ πr A μ u d u Complete δ πr 4 μ π ψξ cosh u sin R sinh u 4πψξ cosh u μr δ + πψξ R ( u ( u u 4 δ + coth μ Mrginl Stbilit ( u coth u ( u u sin d coshu u coshu coshu sinhu sinhu sinhu u sinhu coshu coshu sinhu coth u ( u tnhu tnhu tnhu tnhu Therefore +.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 5 of

6 + Equte Epression for ( ( + ( ( + 4 ( , MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 6 of

7 Elliptic Coordintor Bsic Prmeters,, b Epress C,u,u,b, in terms of these prmeters C nd u Csinhucos C coshu sin Csinhu Ccoshu tnhu u,b, C.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 7 of

8 Csinhu Ccoshu b C b b ( ( b ( ( + b + ( + b + tnhu Derition + Csinhucos Ccoshusin d C coshu cos du c sinhu sin d coshu cos sinhu sin d C sinhu sin du + c cosh cos d sinhu sin + coshu cos C cosh u cos + sinh u sin du coshu cos d + sinhu sin d.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 8 of

9 C cosh u cos + sinh u sin d sinhu sin d + coshu cos d cos h u cos + sinh u sin cos h u cos + cos h u sin sin du d coshu cos d + sinhu sin d ( sin u C cosh u sinhu sin d + coshu cos d ( sin C cosh u cos h u sin u coshu cos sinhu sin u C cosh u sin C cosh u sin ( ( sinhu sin coshu cos C cosh u sin C cosh u sin ( ( u + u + coshu cos sinhu sin C cosh u ( sin sinhu sin + coshu cos C cosh u ( sin C ( cosh u sin + u u u + + u coshucos u sinhusin + coshucos sinhusin u + u + + u sinhusin + u coshucos.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge 9 of

10 + sinhusin + coshu cos Use u u u u + u + u u coshucos + u sinhusin u { ( + u coshucos u sinhusin + ( u sinhusin u coshu cos + u coshu cos + u sinh usin u cosh u cos + sinh u sin + + C cosh usin n, t n u u e + u e coshu cos sinhu sin e + e C n coshucose sinhusine C n coshucos sinhusin + C coshucos coshucos sinhusin + sinhu sin sinhu sin + coshu cos.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge of

11 n ( C C t ez n coshucose sinhusine C t coshu cos sinhu sin C coshu cos sinhu sin coshu cos + sinhu sin coshu cos sinhu sin t ( C Plsm Are u u du d d d ( u u d d cosh u cos sinh u sin d d + d d C d d C du d Surfce Are ds π R dl π R + d π R C sinh u sin + C cos h u cos d p ds πr C d.65, MDH Theor of Fusion Sstems Finl Em Solution Prof. Pge of