Lecture contents Magnetic field Ampere s law Lorentz force, cyclotron frequency, Hall effect Dipole moment, circulation electron, spin

Transcription

1 1 Lectue contents Magnetic field Ampee s law Loentz foce, cycloton fequency, Hall effect Dipole moment, ciculation electon, spin

2 Magnetostatics: Ampee s Law of Foce Ampee s law of foce is the law of action between cuent caying cicuits though magnetic field. Expeimental facts: The magnitude of the foce is invesely popotional to the distance squaed. The magnitude of the foce is popotional to the poduct of the cuents caied by the two wies. The foce acting on a cuent element I dl by a cuent element I 1 dl 1 is given by Expeimental facts: Two paallel wies caying cuent in the same diection attact. F 1 F 1 I 1 I Two paallel wies caying cuent in the opposite diections epel. F 1 F 1 F I l I1 l1 a Pemeability of fee space = {FW/m} ={N/A } 1 unit vecto in diection of I fom I 1 I 1 I A shot cuent oiented pependicula to anothe cuent expeiences no foce. I 1 F 1 = I

3 Magnetic field 3 The total foce acting on a cicuit having a cuent I by a cicuit 1 having cuent I 1 is given by Simila to oulomb s law, magnetic field can be intoduced F I l I1 l1 a F F 1 II dl dl a 1 1 I dl I dl 1 a 1 An infinitely small cuent element Idl immesed in a egion of magnetic flux density, expeiences a foce df 4 1 I dl a 1 T df Idl

4 The iot-savat Law 4 The contibution to the -field at a point P fom a diffeential cuent element Idl is given by d( ) 4 I dl 3 Idl ' P The total magnetic flux at the point P due to the entie cicuit is given by ( ) 4 I dl 3

5 Loentz Foce 5 df Idl A moving point chage placed in both electic and magnetic field expeiences a foce given by Idl Qv F F F q e v e m Q v The foce expeienced by the point chage is in the diection into the pape.

6 ycloton Fequency Electon ciculating in a unifom magnetic field Magnetic field causes a centipetal foce F q v qv mv Angula fequency does not depend on electon velocity ycloton esonance () used fo effective mass measuements, in paticle acceleatos (cyclotons, synchotons). c v q c m mv q q m Si ycloton fequency 6 Lage mean fee path of caies (long scatteing times) is needed fo measuements : c 1

7 Hall effect: caie chage and concentation detemination Loentz foce on a moving paticle: F e v 7 Steady state: balance of foces in y diection: eey evxz onside one type of caies (e.g. n >> p) uent density in x-diection: J x env x Hall coefficient (Hall and dift mobilities consideed equal): H E x y J z 1 ne 1 pe fo n type fo p type

8 Ampee s icuital Law in Integal Fom 8 Ampee s icuital Law in integal fom states that the ciculation of the magnetic field is popotional to the total cuent though the suface bounding the path ove which the ciculation is computed. dl I encl dl Just as Gauss s law follows fom oulomb s law, so Ampee s cicuital law follows fom Ampee s foce law. S ds Just as Gauss s law can be used to deive the electostatic field fom symmetic chage distibutions, so Ampee s law can be used to deive the magnetostatic field fom symmetic cuent distibutions. y convention, ds is taken to be in the diection defined by the ight-hand ule applied to dl. I encl S J ds Since volume cuent density is the most geneal, we can wite I encl in this way.

9 Applications of Ampee s Law known 9 Ampee s law in integal fom is an integal equation fo the unknown magnetic flux density esulting fom a given cuent distibution. dl unknown I encl Fo example, magnetic field aound a staight wie with a cuent I at a distance : I I Anothe example: field inside a solenoid is detemined by a numbe of tuns pe length n and cuent I (no coe): Field outside is small L If coe with elative pemittivity is inseted NI ni ni

10 Applications of Ampee s Law: oaxial cable 1 oaxial cable: thin intenal wie caies cuent I and coaxial metal cylinde caies I dl I encl Field is confined inside the oute cylinde

11 Magnetic dipole moment 11 A magnetic dipole compises a small cuent caying loop. The point chage (chage monopole) is the simplest souce of electostatic field. The magnetic dipole is the simplest souce of magnetostatic field. The magnetic dipole is analogous to the electic dipole. Toque on a cuent loop in a unifom filed: df Idl The magnetic dipole moment can be defined as cuent times the aea of the loop Simila to an electic dipole: U fom to IAn Am Diection of the dipole moment is detemined by the diection of cuent using the ight-hand ule.

12 iculating electon and spin Electon ciculating with unifom angula speed poduces cuent: I chage q peiod 1 Magnetic moment: IA 1 q an be expessed though angula momentum L m q L m Fo negative chage m and L ae in opposite diections A magnetic dipole pecesses about the diection of constant magnetic field dl dt d dt q m An electon also has an intinsic magnetic dipole moment associated with its spin S: with g s =.3 g s q m S Gyomagnetic atios

13 Test questions 13 onside Hall effect in two samples with diffeent caie concentation. Which sample will show highe magnitude of Hall voltage?