PY3107 Experimental Physics II

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1 PY317 Eperimental Phsics II Magnetic fields and the Hall effect M.P. Vaughan and F. Peters

2 Related Eperiments Magnetic Lab Using a Hall-effect magnetic field sensor

3 Overview Ampere s Law The solenoid Lorentz Law Carrier scattering The Hall Effect

4 Ampère s Law

5 Ampère s Law Direction of field lines given b the right-hand-rule. A current I passing through a long, straight wire sets up a clindricall smmetric magnetic field B.

6 Ampère s Law In general, the line integral of the magnetic field B along a closed path C is given b Ampère s Law as C Bd l I, where is the permeabilit of free space.

7 Ampère s Law circular loop Using clindrical coordinates r d dl rd

8 Ampère s Law circular loop Since B and dl are parallel to each other, we have Bdl Bdl Brd. Hence C Bdl Brd 2 Br, so B I 2r.

9 The solenoid

10 Current carring coil A solenoid

11 Toroidal solenoid Consider a toroidal solenoid with N turns per unit length. We shall take the line integral around the circle C, centred on the ais of the toroid. C dl

12 Toroidal solenoid There is no net current flowing through the circle C, we have C Bdl, so the component of B parallel with dl, B, over this circle is zero. We obtain the same result for an circle centred on the toroidal ais. Since, b smmetr, B must be constant for an given circle, we conclude that everwhere outside the solenoid B. (Note that some tet books incorrectl use this reasoning to argue that B = everwhere outside the solenoid).

13 Inside a toroidal solenoid r dl C Inside the toroid, a current 2rNI passes through a circle of radius r centred on the ais.

14 Inside a toroidal solenoid Hence, we have C B d l 2 r NI. B smmetr, the magnitude of B is the same at all points on the curve. Thus, evaluating the integral, or 2rB 2r NI B NI.

15 Long, straight solenoid If the radius of the toroid is allowed to go to infinit, we obtain an infinitel long, straight solenoid. Assuming that the solenoid is long enough that we can neglect end effects, we ma use the previous result to assert that the magnetic field within the solenoid is constant and given b B NI.

16 Magnetic core Suppose the solenoid is filled with a rod of magnetic material. If the magnetic field in the absence of the rod is B, then the new magnetic field is B B M, where M is the magnetisation. Ampère s Law is given in terms of B, so C B B M dl. dl I C

17 Magnetic core For an isotropic material M B B, where is the magnetic susceptibilit. Thus C B Mdl 1 Bdl dl, B C C B where is the relative permeabilit.

18 Magnetic core We ma then write Ampère s Law C B d l I. For the long solenoid, this gives us B NI.

19 Lorentz Law

20 Lorentz Law Lorentz Law for the force on a charge q moving with a velocit v is given b F qe vb, where E is the electric field. This ields three coupled differential equations for the motion. Choosing coordinates such that B Be z,

21 Lorentz Law we have v v v z q E m q E m qez. m v v i B B,, Hence, the third equation for v z is uncoupled from the other two.

22 Lorentz Law Multipling the second equation b i, where and v iv i i is the cclotron frequenc. v, i v qe i m qb m,

23 Lorentz Law Defining v v iv and we have or v i, v iv i, v v iv.

24 Lorentz Law Solving via the integrating factor method, we obtain the general solution v i v i e i t, where v v.

25 Lorentz Law Setting the initial condition v, we obtain the particular solution v i it 1 e.

26 Lorentz Law Decomposing into real and imaginar parts, we find and v v sin t 1 cos t sin t 1 cos t. It can be shown that when =, this reduces to v t and v t.

27 Lorentz Law Integrating the velocit equations, we find and 2 sin 2 1 cos t t t cos t sin t t

28 Carrier scattering

29 Carrier scattering In a solid medium, we would have scattering of the charge carriers. We ma model this via a characteristic scattering time t, interpreted as the average time between scattering events. We then make the substitution and proceed as before. i i 1 t

30 Carrier scattering The particular solution, with v =, is found to be v t i t t 2 t i tt / 1 e. As t, we get v t i 2 1 t t 2 2.

31 Carrier scattering This gives us 2 2 t t t t v and v t 1 t 2 Note that we have used the angled brackets to indicate that this are averaged values. In particular 2 t is the average of the square of t, not the square of the average.

32 The Hall effect

33 The Hall Effect With an applied electric field in the direction, the charge deflected b the magnetic field builds up on one side generating a transverse electric field. E E z

34 energ Electrons and holes conduction band valence band

35 The Hall Effect The transverse electric field ma be found b setting v = (no further drift in the direction). This gives t t 2. In the absence of a magnetic field, the current densit in the (longitudinal) direction is given b j nq v nq t, where n is the densit of charge carriers.

36 The Hall Effect This ma also be given in the form j nq D E, where D is the drift mobilit. We ma now write the transverse field in terms of t t 2 j nq t t 2 2.

37 The Hall Effect scattering factor This gives the eplicit result for the transverse field E t t 2 2 jb nq r H jb nq. Here we have defined the Hall scattering factor r H t t

38 The Hall Effect Hall mobilit Substituting r j H D nq into the epression for the transverse field gives Note that all the terms on the right are directl measurable from eperiment. Thus, we measure the Hall mobilit H D E E E, B rh D..

39 The Hall Effect Hall constant We ma define the Hall constant R H via the general epression E T R H j B, where T and L stand for transverse and longitudinal respectivel. In the present case we have L R H j E B.

40 The Hall Effect Hall constant This ma be rewritten as R H rh nq, demonstrating that the sign of the Hall constant gives the sign of the charge carriers.

Physics 202, Lecture 13. Today s Topics. Magnetic Forces: Hall Effect (Ch. 27.8)

Physics 202, Lecture 13. Today s Topics. Magnetic Forces: Hall Effect (Ch. 27.8) Physics 202, Lecture 13 Today s Topics Magnetic Forces: Hall Effect (Ch. 27.8) Sources of the Magnetic Field (Ch. 28) B field of infinite wire Force between parallel wires Biot-Savart Law Examples: ring,