Problems in curvilinear coordinates

Transcript

1 Poblems in cuvilinea coodinates Lectue Notes by D K M Udayanandan Cylindical coodinates. Show that ˆ φ ˆφ, ˆφ φ ˆ and that all othe fist deivatives of the cicula cylindical unit vectos with espect to the cicula cylindical coodinates vanish. Answe We ve ˆ ˆx cos φ + ŷ sin φ ˆφ ˆx sin φ + ŷ cos φ ˆ 0, ˆ φ ẑ ẑ ˆx sin φ + ŷ cos φ ˆφ, ˆ z 0 ˆφ φ ˆφ 0, ˆx cos φ ŷ sin φ ˆx cos φ + ŷ sin φ ˆ

2 ˆφ z 0 ẑ 0, ẑ φ 0, ẑ z 0 2. Show that ˆ ˆ + zẑ. Woking entiely in cicula cylindical coodinates, show that. 3 and 0. Answe Fom figue OP So we ve ˆ + zẑ. V. ˆ + zẑ V V φ 0, V z z V + φ V φ + z V z. + 2 φ 0 + z z 2 + z z

3 2 + 3 ˆ ˆφ ẑ φ 0 z z In ight cicula cylindical coodinates a paticula vecto function is given by V, φ ˆV, φ + ˆφV φ, φ. Show that V has only a z component. Answe ˆ ˆφ ẑ V φ z V, φ V φ, φ 0 ˆ0 + ˆφ0 + ẑ ẑ Thus V has only z component. V φ, φ V, φ φ, V, φ V, φ φ 4. A igid body is otating about a fixed axis with a constant angula velocity ω. Take ω to lie along the z axis. Expess in cicula cylindical coodinates and using cicula cylindical co-odinates. a Calculate V ω b Calculate V Answe We ve ω ωẑ a and ˆ + zẑ ω ˆ ˆφ ẑ 0 0 ω 0 z ω ˆφ 3

4 b V ˆ ˆφ ẑ φ z 0 ω 0 ˆ0 + ˆφ0 + ẑzω zω ẑ zωẑ 5. A paticle is moving though space. Find the cicula cylindical components of its velocity and acceleation. We ve V a φ 2 V φ φ a φ φ + z φ V z ż a z z t ˆtt + ẑzt ˆx cos φt + ŷ sin φt t + żzt d dt ˆx cos φt +ŷ sin φt +t ˆx sin φt φ+tŷ cos φt φ+ẑż ˆx cos φ + ŷ sin φ + t φ ˆx sin φ + ŷ cos φ + żẑ d 2 dt d cos φˆx + sin φŷ + 2 dt ˆ + φ ˆφ + żẑ V, V φ φ, V z ż φ sin φˆx + φ cos φŷ + żẑ cos φˆx+ sin φŷ+ sin φˆx φ+ cos φŷ φ+ φ sin φˆx+ φ cos φŷ + φ sin φˆx + φ cos φŷ + φ 2 cos φˆx φ 2 sin φŷ + zẑ cos φˆx + sin φŷ + φ sin φˆx + cos φŷ + φ sin φˆx + cos φŷ + φ sin φˆx + cos φŷ + φ 2 cos φˆx + sin φŷ + zẑ 4

5 φ 2 cos φˆx + sin φŷ + 2 φ + φ sin φˆx + cos φŷ + zẑ φ 2 ˆ + φ + 2 φ ˆφ + zẑ a φ 2, a φ φ + 2 φ, a z z 6. Solve Laplace s equation 2 ψ 0 in cylindical coodinates fo ψ ψ. Answe We ve 2 ψ We ve Laplace equation 2 ψ 0 ψ + 2 ψ 2 φ + 2 ψ 2 z 2 ie, ψ ψ ψ 0 ψ 0 ψ constant k ψ k ψ k k log + log c ifψ 0, log c log 0 ψ klog log 0 ψ k log 0 7. A conducting wie along the z axis caies a cuent I. The esulting magnetic vecto potential is given by A ẑ µi 2π ln 5

6 Show that the magnetic induction B is given by B ˆφ µi 2π Answe B A ˆ ˆφ ẑ φ z µi 0 0 ln 2π ˆ0 + ˆφ 0 µi 2π ln + ẑ0 µi 2π ˆφ µi 2π ˆφ 8. A foce is descibed by y F ˆx x 2 + y + ŷ x 2 x 2 + y 2 a Expess F in cicula cylindical coodinates opeating entiely in cicula cylindical coodinates fo b. b Calculate the cul of F Answe We ve ˆx ˆ cos φ ˆφ sin φ Then ŷ ˆ sin φ + ˆφ cos φ ẑ ẑ y F ˆx x 2 + y + ŷ x 2 x 2 + y 2 ˆ cos φ + ˆφ sin φ sin φ + ˆ sin φ + ˆφ cos φ cos φ 2 ˆ sin φ cos φ + ˆφ sin 2 φ + ˆ sin φ cos φ + ˆφ cos 2 φ 2 6

7 b F ˆφ ˆφ F ˆ ˆφ ẑ fac φ z 0 0 ˆ0 + ˆφ0 + ẑ A calculation of the magneto-hydonamic pinch effect involves the evaluation of B. B. If the magnetic induction B is taken to be B ˆφB φ Show that Answe B. B ˆ B2 φ ˆ + ˆφ φ + ẑ z B ˆφB φ B. B φ φ B. B Bφ ˆφBφ φ B2 φ B2 φ B2 φ φ ˆφ sin φˆx + cos φŷ φ cos φˆx + sin φŷ B2 φ ˆ ˆ B2 φ 7

8 Spheical pola coodinates. A igid body is otating about a fixed axis with a constant velocity ω. Take ω to be along the z axis. Using spheical pola coodinates a Calculate V ω b Calculate We ve sin θ cos φˆx + sin θ sin φŷ + cos θẑ Then b V ω ω ωẑ ˆx ŷ ẑ sin θ cos φ sin θ sin φ cos θ ˆx ω sin θ sin φ + ŷω sin θ cos φ ω sin θ sin φˆx + cos φŷ ω sin θ ˆφ V ˆ ˆθ sin θ ˆφ 2 sin θ θ φ 0 0 sin θ ω sin θ ˆ2 2 ω sin θ cos θ + ˆθ 2ω sin 2 θ + sin θ 2 sin θ ˆφ0 2 sin θ2ω cos θˆ 2ω sin θˆθ 2 sin θ 2. With A,any vecto 2ωcos θˆ sin θˆθ 2ωẑ A. A aveify this esult in Catesian coodinates. bveify this esult using spheical pola coodinates. Answe a A A x î + A y ĵ + A zˆk A. A x x + A y y + A z z 8

9 b A. A x x + A y y + A z xî + yĵ + zˆk z A x î + A y ĵ + A zˆk A A ˆA + ˆθA θ + ˆφA φ ˆ + ˆθ θ + ˆφ sin θ φ A. A + A θ θ + A φ sin θ φ A A. + A θ θ + A φ sin θ A + A θ θ + A φ sin θ φ φ. ˆ sin θ cos φˆx + sin θ sin φŷ + cos θẑ A sin θ cos φˆx+sin θ sin φŷ+cos θẑ+a θ cos θ cos φˆx+cos θ sin φŷ sin θẑ+ A φ sin θ sin φˆx + sin θ cos φŷ sin θ A ˆ + A θ ˆθ + Aφ ˆφ A 3. Fom the above poblem show that i x y y x i φ This is the quantum mechanical opeato coesponding to the z component of obital angula momentum. Answe We ve then x sin θ cos φ + cos θ cos φ y sin θ sin φ + cos θ sin φ z cos θ sin θ θ i x y y x θ sin φ sin θ θ + cos θ sin θ φ φ 9

10 i sin θ cos φ sin θ sin φ + cos θ sin φ i sin 2 θ sin φ cos φ + sin θ cos θ sin φ cos φ i φ θ + cos θ sin θ sin θ sin φ sin θ cos φ φ θ + cos2 φ φ sin2 θ sin φ cos φ 4. Show that the following thee foms spheical co-odinates of 2 ψ ae equivalent. a d 2 dψ 2 d d b d 2 d 2 ψ c d2 ψ d 2 Answe + 2 dψ d a d 2 dψ 2 dψ 2 d d 2 d + d2 ψ 2 d 2 d2 ψ d dψ d b d 2 d d ψ ψ d 2 d d d dψ d d + ψ d2 ψ d + dψ 2 d + dψ d d2 ψ d + 2dψ 2 d d2 ψ d + 2 dψ 2 d 5. Find the spheical coodinate components of the velocity and accele- 0

11 ation of a moving paticle. Answe We ve xt ˆt t ˆx sin θt cos φt + ŷ sin θt sin φt + ẑ cos θt t d dt ṙ sin θ cos φˆx + cos θ cos φ θŷ + sin θ cos φ φŷ + ṙ cos θẑ sin θ θẑ ṙsin θ cos φˆx + sin θ sin φhaty + cos θż + θcos θ cos φˆx + cos θ sin φŷ sin θẑ + φ sin θ sin φˆx + cos φŷ ṙˆ + θˆθ + sin θ φ ˆφ V ṙ, V θ θ, V φ sin θ φ d 2 dt dˆ ˆ+ṙ 2 dt dˆθ +ṙ θˆθ+ θˆθ+ θ dt +ṙ sin θ φ ˆφ+ cos θ θ φ ˆφ+ sin θ φ ˆφ+ d ˆφ sin θ φ dt ˆ + ṙ d dt sin θ cos φˆx + sin θ sin φŷ + cos θẑ + ṙ θˆθ + θˆθ+ θ d dt cos θ cos φˆx + cos θ sin φŷ sin θẑ + ṙ sin θ φ ˆφ + cos θ θ φ ˆφ+ sin θ φ ˆφ + sin θ d sinφˆx + cos φŷ dt ˆ+ṙ cos θ cos φˆx θ sin θ sin φˆx φ + cos θ sin φŷ θ + sin θ cos φ φŷ sin θẑ θ + ṙ θˆθ+ θˆθ+ θ sin θ cos φˆx θ cos θ sin φ φˆx sin θ sin φ θŷ + cos θ cos φ φŷ cos θ θẑ + ṙ sin θ φ ˆφ + cos θ θ φ ˆφ + sin θ φ ˆφ + sin θ φ cos φ φˆx sin φ φŷ ˆ + ṙ θˆθ + ṙ φ sin θ sin φˆx + sin θ cos φŷ + ṙ θˆθ + θˆθ θ 2ˆ+ θ φ cos θ sin φˆx + cos θ cos φŷ + ṙ sin θ φ ˆφ + cos θ φ ˆφ+ sin θ φ ˆφ sin θ cos φ φ 2ˆx sin θ sin φ φ 2 ŷ ˆ θ 2ˆ + 2ṙ θˆθ + θˆθ + θ φ sin θ ˆφ + θ φ cos θ ˆφ + ṙ φ sin θ ˆφ+ θ φ cos θ ˆφ + sin θ φ ˆφ

12 θ 2 sin 2 θ φ 2 ˆ + θ + 2ṙ θ sin θ cos θ φ 2 ˆθ+ sin θ φ + 2ṙ sin θ φ + 2 cos θ θ φ ˆφ 6. A magnetic vecto potential is given by A µ 0 m 4π 3 Show that this leads to the magnetic induction B of a point magnetic dipole with dipole moment m. Answe sin θ cos φˆx + sin θ sin φŷ + cos θẑ m ˆx ŷ ẑ 0 0 m sin θ cos φ sin θ sin φ cos θ ˆx m sin θ sin φ + ŷm sin θ cos φ m sin θ sin φˆx + m sin θ cos φŷ A µ 0 m 4π µ 0 m sin θ sin φˆx + m sin θ cos φŷ 4π3 µ 0 m sin θ ˆφ 4π 2 B A ˆ ˆθ sin θ ˆφ 2 sin θ θ φ 0 0 sin θ µ 0 m sin θ 4π 2 µ0 m ˆ 2 sin θ 4π 2 sin θ cos θ µ0 m sin 2 θ + ˆθ 4π 2 µ0 m 2 sin θ 2π sin θ cos θˆ + µ 0m 4π sin2 θˆθ 2

13 µ 0m 2π cos θˆ + µ 0m sin θˆθ 3 4π3 µ 0 4π 2m cos θˆ + µ 0 m sin θˆθ 3 4π3 7. The magnetic vecto potential fo a unifomly chaged otating spheical shell is A ˆφ µ 0a 4 σω sin θ 3, > a 2 a adius of spheical shell, σ suface chage density and ω angula velocity Find the magnetic induction B A Answe µ0 a 4 σω ˆ 2 sin θ A ˆ ˆθ sin θ ˆφ 2 θ φ sin θ 0 0 sin θ µ 0a 4 σω sin θ 3 2 µ0 a 4 σω 2 sin θ cos θ + ˆθ sin 2 θ µ0 a 4 σω sin θ cos θˆ + µ 0a 4 σω sin 2 θˆθ sin θ 2µ 0a 4 σω 3 3 cos θˆ + µ 0a 4 σω 3 3 sin θˆθ 8. At lage distances fom its souce, electic dipole adiation has fields, sin θe E ik ωt a E ˆθ, Show that Maxwell s equations, B sin θe ik ωt ab ˆφ ae satisfied if we take E B t and B E ɛ 0 µ 0 t a E a B ω k c ɛ 0, µ 0 /2 3

14 Answe E 2 sin θ ˆ ˆθ sin θ ˆφ θ φ 0 a E sin θe ik ωt 0 ˆ0 + ˆθ0 + sin θ 2 sin θ ˆφa E sin θe ik ωt ik 2 sin θ a E sin 2 θe ik ωt ik ˆφ sin θe ik ωt a E ik ˆφ sin θeik ωt ia E k ˆφ sin θeik ωt ia B ω ˆφ Thus, B t ia sin θeik ωt Bω ˆφ E B t 2 sin θ B 2 sin θ ˆ ˆθ sin θ ˆφ θ sin θe 0 0 sin θa ik ωt B ˆ a B e ik ωt 2 sin θ cos θ + ˆθ a B sin 2 θe ik ωt ik 2 sin θ φ a B e ik ωt 2 sin θ cos θˆ a B sin 2 θe ik ωt ikˆθ a Be ik ωt 2 cos θ ˆ a B sin θe ik ωt ik ˆθ 2 Since is lage, highe powe can be neglected. B a B sin θe ik ωt ik ˆθ 4

15 But Replacing k by ωµ 0 ɛ 0 /2 B a Ek ω a B a Ek ω sin θe ik ωt ik ˆθ k ω µ 0ɛ 0 /2 B /2 sin θeik ωt ia E kµ 0 ɛ 0 ˆθ B sin θeik ωt ia E ωµ 0 ɛ 0 ˆθ Then E t ia sin θeik ωt Eω ˆθ E µ 0 ɛ 0 t ia sin θeik ωt Eωµ 0 ɛ 0 ˆθ B µ 0 ɛ 0 E t 9. ashow that A cot θ ˆφ is a solution of A ˆ 2 b Show that this spheical pola co-odinate solution agee with the solution given below. yz A ˆx x 2 + y 2 ŷ xz x 2 + y 2 cfinally show that A ˆθφ sin θ Answe is a solution. a A 2 sin θ ˆ ˆθ sin θ ˆφ fac θ φ 0 0 sin θ cot θ 5

16 b ˆ sin θ + ˆθ0 + sin θ 2 sin θ ˆφ0 2 sin θ sin θˆ ˆ 2 yz A ˆx x 2 + y 2 ŷ xz x 2 + y 2 ˆ sin θ cos φ+ˆθ cos θ cos φ ˆφ sin θ sin φ cos θ sin φ 2 sin 2 θ cos 2 φ + sin 2 θ sin 2 φ ˆ sin θ sin φ + ˆθ cos θ sin φ + ˆφ sin θ cos φ cos θ cos φ 2 sin 2 θ cos 2 φ + sin 2 θ sin 2 φ c ˆsin θ cos θ sin φ cos φ sin θ cos φ sin φ cos φ + ˆθcos 2 θ cos φ sin φ cos 2 θ sin φ cos φ sin θ sin θ ˆφsin 2 φ cos θ + cos 2 φ cos θ sin θ ˆφ ˆφ cos θ sin θ cot θ A ˆ ˆθ sin θ ˆφ 2 sin θ θ φ φ sin θ 0 0 ˆ sin θ ˆθ0 + sin θ 2 sin θ ˆφ0 ˆ 2 0. Show that L i i ˆθ sin θ φ ˆφ θ b Resolving ˆθ and ˆφ into Catesian components, detemine L x,l y,l z. Answe L p p i 6

17 L i Let i L i ˆ + ˆθ θ + ˆφ sin θ φ ˆ ˆθ sin θ ˆφ 2 sin θ 0 0 sin θ θ sin θ φ ˆθ + sin θ 2 sin θ φ ˆφ θ 2 2 sin θ φ ˆθ + 2 sin θ θ ˆφ sin θ L i i ˆθ + θ ˆφ sin θ φ ˆθ θ ˆφ b i L i sin θ φ ˆθ θ ˆφ ˆx cos θ cos φ + ŷ cos θ sin φ ẑ sin θ sin θ cos θ cos φ i sin θ φ + sin φ cos θ sin φ ˆx+i θ sin θ cos θ cos φ i sin θ φ + sin φ cos θ sin φ ˆx+i θ sin θ cos θ cos φ L x i sin θ φ + sin φ θ cos θ sin φ L y i sin θ φ + cos φ θ L z i φ 7 φ ˆx sin φ + ŷ cos φ θ φ cos φ ŷ i sin θ φ sin θ φ cos φ θ ŷ i φẑ φẑ

18 . With ê a unit vecto in the diection of inceasing q. Show that a.ê h 2 h 3 h h 2 h 3 q b Answe We ve. V h h 2 h 3 ê h h ê 2 ê 3 h h 3 q 3 h 2 q 2 V h 2 h 3 + V 2 h h 3 + V 3 h h 2 q q 2 q 3 Hee V, V 2 0, V 3 0 then V V ê + V 2 ê 2 + V 3 ê 3 V ê V.ê h h 2 h 3 q h 2 h 3 ê h ê 2 h 2 ê 3 h 3 ê h h 2 h 3 q q 2 q 3 h 0 0 h h h 2 ê 2 h 3 ê 3 h h 2 h 3 q 3 q 2 ê h h ê 2 ê 3 h h 3 q 3 h 2 q 2 2. With the quantum mechanical obital angula momentum opeato defined as L i.show that a L x + il y e iφ θ + i cot θ φ 8

19 b Answe, We ve i L x il y e iφ θ i cot θ φ L i sin θ L i sin θ φ ˆθ θ ˆφ φ ˆθ θ ˆφ ˆx cos θ cos φ + ŷ cos θ sin φ ẑ sin θ sin θ cos θ cos φ i sin θ φ + sin φ cos θ sin φ ˆx+i θ sin θ cos θ cos φ i sin θ φ + sin φ cos θ sin φ ˆx+i θ sin θ φ ˆx sin φ + ŷ cos φ θ φ cos φ ŷ i sin θ φ sin θ φ cos φ θ Then cos θ cos φ L x +il y i sin θ φ + sin φ cos θ sin φ +i 2 θ sin θ cos θ cos φ i + i sin θ cos θ sin φ sin θ ŷ i φẑ φ cos φ θ φ + i sin φ + cos φ θ i cot θcos φ + sin φ φ + cos φ + i sin φ θ ie iφ cot θ φ + eiφ θ e iφ θ + i cot θ φ cos θ cos φ L x il y i sin θ φ + sin φ cos θ sin φ i 2 θ sin θ φ cos φ θ i cos θ cos φ cos θ sin φ i sin φ + cos φ φ sin θ sin θ θ i cot θ φ e iφ θ e iφ φẑ 9

20 e iφ θ cot θ φ 3. A cetain foce field is given by 2p cos θ F ˆ + ˆθ p 3 sin θ, p 3 2 Examine F to see if a potential exist. Answe F ˆ ˆθ sin θ ˆφ 2 sin θ θ φ 2p cos θ p sin θ sin θ 2 sin θ ˆφ 2 p sin θ + 2p 3 sin θ 3 2p sin θ 2p sin θ Fo the flow of an incompessible viscous fluid the Navie stokes equations lead to V V n 2 V Hee n is the viscosity and, the density of the fluid. Fo axial flow in a cylindical pipe we take the velocity V to be V ẑv Find the non linea tem V V Answe V V ẑ V ˆ ˆφ ẑ y z 0 0 V ˆφ 0 V V ˆφ 20

21 then V V V ˆ V ˆ ˆφ ẑ 0 0 V 0 V 0 V V ˆ V V 0 ˆ ˆφ ẑ φ z V V In Minkowiski space we define x x, x 2 y, x 3 z, and x 0 ct. This is done so that the space time inteval ds 2 dx 2 0 dx 2 dx 2 2 dx 2 3c velocity of light. Show that the metic in Minkowiski space is g ij Answe x x dx dx x 2 y dx 2 dy x 3 z dx 3 dz x 0 ct dx 0 c dt Space time inteval ds 2 ds 2 dx 2 0 dx 2 dx 2 2 dx 2 3 c 2 dt 2 dx 2 dy 2 dz 2 2

22 Then dx 2 0 dx 2 dx 2 2 dx 2 3 g ij c 2 dt 2 dx 2 dy 2 dz 2 6. Deive by diect application of equation Answe ψ ψ ψ ψ ê + ê 2 + ê 3 h dq h 2 dq 2 h 3 dq 3 ψ ψdσ lim dτ 0 dτ we ve ie, dσ ij ds i ds j h i h j dq i dq j dσ ê h 2 h 3 dq 2 dq 3 + ê 2 h h 3 dq dq 3 + ê 3 h h 2 dq dq 2 σ ê h 2 h 3 dq 2 dq 3 + q ê h 2 h 3 dq 2 dq 3 ê h 2 h 3 dq 2 dq 3 Fo simplifying all thee items dσ q ê h 2 h 3 dq 2 dq 3 + q 2 ê 2 h h 3 dq dq 3 + fo fist tem q 3 ê 3 h h 2 dq dq 2 ψ ψ ψ ψdσ ê h 2 h 3 dq dq 2 dq 3 +ê 2 h h 3 dq dq 2 dq 3 +ê 3 h h 2 dq dq 2 dq 3 2 q q 2 q 3 τ h h 2 h 3 dq dq 2 dq 3 3 substituting 2 and 3 in 22

23 ψ h ψ q ê + h 2 ψ q 2 ê 2 + h 3 ψ q 3 ê 3 23