Spring 2015 Eugene V. Colla

Transcription

1 Spring 015 Eugene V. Colla

2 The main goals of the Lab: Study of the magnetic field distribution created by various systems using Hall probe and Gauss meter. Calculating for simple systems the magnetic field profile and comparing it with experimental data. Getting understanding of the application of the Hall effect to measurements of the magnetic fields. This is one week Lab Physics 401 Spring 013

3 Magnetic field due to current loops Helmholt coils Solenoid Halbach magnets Hall effect. Measuring of the magnetic field Physics 401 Spring 013 3

4 Jean-Baptiste Biot ( ) db Félix Savart ( ) db o Idl r 3 I 4 r N/A, dl permeability of the free space Physics 401 Spring 013 4

5 db q db (0,0, 1 ) db q db 1 dl a d a db db cosq db r db 0 4 ad r 3 dl df r a I x dl r q a mia 0 B = db = dj= 3 4p r I a I r 1 a a Physics 401 Spring 013 5

6 Hermann Ludwig Ferdinand von Helmholt ( ) Magnetic field vector in a plane bisecting the current loops. (courtesy Wikipedia) Physics 401 Spring 013 6

7 Physics 401 Spring I I N turns 0 a a I a B a For single loop: For Helmholt coils total current equals NI, a For right hand coil and a for left hand coil B

8 a I 0 I B Finally: μ NIa a a 0 B= a a + +a - +a a N turns or μ NI B = a a a Physics 401 Spring 013 8

9 1. a=r 0.04 R R B (a.u.) (cm) Physics 401 Spring 013 9

10 1. a=1.5r 0.04 R R B (a.u.) (cm) Physics 401 Spring

11 3. a=r 0.04 R R B (a.u.) (cm) Physics 401 Spring

12 4. a=0.5r 0.04 R 0.03 R/ B (a.u.) (cm) Physics 401 Spring 013 1

13 a/r=0.5 a/r=1 a/r=1.5 a/r= B/B(0) a=r B (a.u.) (cm) (cm) In the range a/4 a/4 the field uniformity is better than 0.5% Physics 401 Spring

14 The results of D field mapping can be presented in 3D plot Step#1. Plugin your data in the worksheet Physics 401 Spring

15 Step#. Convert data to matrix Physics 401 Spring

16 Pixel = 1 Step#3. Plot (here is the color map chosen but you have many options how to plot) Z Axis Title Pixel = X Axis Title Y Axis Title Color map surface Contour map Physics 401 Spring

17 Helmholt coils can produce the pretty uniform magnetic field in large volume free of material. Helmholt coils are not very suitable to generate high magnetic fields. Helmholt coils in Rb optical pumping experiment. UIUC Physics 403 Helmholt coil from Brookhaven National Laboratory Physics 401 Spring

18 Solenoids are another source of the uniform magnetic field. Solenoids could be used to produce very high magnetic field. Physics 401 Spring

19 Magnetic field generated by length d: 0 q 1 q 0nId B a B a a 3 1 Here n is number of turns per unit length and I solenoid current 1 d To calculate the magnetic field generated by the whole length of the solenoid we need to perform the integrating from 1 to Physics 401 Spring

20 Field from current loop 0 q 1 q B a B 0nId a 3 1 a 1 d n turns per unit length I solenoid current q 0nI 0 ni B sinqdq cos q1 cosq q 1 where 1 cos( q1) ; cos( q) a a 1 Physics 401 Spring 013 0

21 L R B B/B(0) L=R /R Physics 401 Spring 013 1

22 R B B/B(0) L 0. L=R /R Physics 401 Spring 013

23 B R B/B(0) L 0. L=4R /R Physics 401 Spring 013 3

24 B/B(0) L=R L=R L=4R L=10R /R To create the uniform field in solenoid you need you need to wind a long coil with L>>R Physics 401 Spring 013 4

25 B/B(0) /R L=R Physics 401 Spring 013 5

26 1 4 N-turns 3 L B B dl NI 0 y André-Marie Ampère ( ) B =B (inside), B =0 (outside), B y =0 0 LB 0 0 oni and oni B oni L where n=n/l Physics 401 Spring 013 6

27 Physics 401 Spring y B x B B B x y B x B y x y x B B x y x

28 17T solenoid (4.K, 105A) from T magnet from Units: 1T=10 4 G; typical fields reachable in your experiments <100G Physics 401 Spring 013 8

29 The late Klaus Halbach of Lawrence Berkeley National Laboratory discovered an interesting permanent magnet configuration that concentrates magnetic flux on one side of the array and cancels it on the other Klaus Halbach Courtesy MatchRockets.com Physics 401 Spring 013 9

30 Edwin Herbert Hall ( ) The current in x direction could be written as: I x NqAv x where N is the concentration of carriers, q is carrier charge and A is a cross-section area of the bar and v x drift velocity. x Physics 401 Spring

31 qe y IB x yˆ yˆ 0 NA After the field application in direction the carriers experience a force: I x IB x F qv B q x B ˆ ˆ y NqA NA This force will produce the deflection of the carriers resulting in extra charges on the surfaces normal to y axis. Extra charges will give a rise to an electric field E y. The electric field will exerts a force on carriers in the direction opposite the magnetic force. Carriers will flow in y direction until both forces balance: or E y IB x qna The equilibrium field could be determined by measuring the potential a/ difference across the sample. a width of the bar V E dy E a H y y a/ Physics 401 Spring

32 Finally V H IB x qnb a b Let define (b is a thickness of the bar, A=ab) R H 1 as a Hall coefficient Nq And expression for Hall voltage could be rewritten as V H R H IB x b Physics 401 Spring 013 3

33 X-Y scanning equipment Hall probe scanning the iron box magnet Physics 401 Spring

34 Be smart! Do not forget about the symmetry of the investigated magnetic system. B B(r,) B(r,)=f(r) f() I r Magnetic field created with the circular loop (solenoid, Helmholt coil) depends only on radius r but not on the angle Physics 401 Spring