THE MAGNETIC FIELD. This handout covers: The magnetic force between two moving charges. The magnetic field, B, and magnetic field lines

Size: px

Start display at page:

Download "THE MAGNETIC FIELD. This handout covers: The magnetic force between two moving charges. The magnetic field, B, and magnetic field lines"

Osborn Shaw
5 years ago
Views:

1 EM 005 Handout 7: The Magnetic ield 1 This handout coes: THE MAGNETIC IELD The magnetic foce between two moing chages The magnetic field,, and magnetic field lines Magnetic flux and Gauss s Law fo Motion of a chaged paticle in E and : the Loentz foce Impotant special cases: Note: - Motion pependicula to unifom - The elocity selecto - The Hall effect 1. You do not need to emembe the full ecto teatment of the magnetic foce between moing chages.. o exam puposes, the impotant elationship defining the magnetic field, is M Q( ) 3. Howee, it is impotant to undestand the ecto teatment so that you can gain a good conceptual gasp of the subject. 4. The key to undestanding the magnetic field is to be able to use the ecto coss poduct eise this if you ae not sue of it.

2 EM 005 Handout 7: The Magnetic ield The magnetic foce Up to now, we hae consideed the ELECTROSTATIC foce, due to chages at est. Coulombs Law: Q Q Q 4πε 1 o 1 Q 1 1 If the chages ae OTH moing, anothe foce exists between them: the MAGNETIC ORCE. Q µ o Q1Q M ( 1 1 ) 4π Q Simple case: 1 and paallel 1 Magnitude Diection out of page Q M ( 1 ) Magnitude 1 1 Diection Towads Q 1 Q µ o Q1Q M 1 1 4π

3 EM 005 Handout 7: The Magnetic ield 3 Note: 1. M Q 1 Q as fo the electostatic foce. M 1 no foce unless OTH chages ae moing 3. M exists in addition to the electic foce 4. The constant µ o is called the PERMEAILITY CONSTANT o the PERMEAILITY O REE SPACE SI System: µ o 4π x 10-7 N s C - Relatie magnitudes of the electic and magnetic foces M o µ π 4 Q1Q 1 E Q Q 4πε 1 o M E µ ε 1 o o 1 c whee c 1 µ oεo 1/ c SPEED O LIGHT [To be explained late] M E 1 c M << E unless speeds ae close to speed of light. UT: The magnetic foce can still dominate een at ey low speed because E tends to be cancelled out due to the oeall chage neutality of matte. Example: Two wies caying cuents E 0 (neithe wie is chaged) M 0 (and can be quite lage) M M I 1 I

4 EM 005 Handout 7: The Magnetic ield 4 The magnetic field DEINITION: The magnetic foce expeienced by a chage Q moing with elocity is mag Q( ) Q (inwads) This equation defines, the MAGNETIC IELD Special case: Q Recall: µ o Q1Q M ( 1 4π 1 ) Q 1 1 Q ut, by definition of, 1 Q( 1) whee 1 magnetic field at Q due to Q 1. So µ oq π Magnetic field of a moing point chage SI Units fo Magnetic ield: 1 T (Tesla) ield which exets a foce of 1 N on a 1-C chage moing with elocity 1 m s -1 pependicula to. 1 T 1 N C -1 m -1 s Q

5 EM 005 Handout 7: The Magnetic ield 5 Magnetic field lines Recall: Electic field lines point in the diection of E Similaly, magnetic field lines point along Recall: µ oq π Magnetic field at P due to Q 1 Diection is outwads by ight hand ule P 1 (outwads) 1 1 iew along hee Q 1 Imagine we iew ALONG THE DIRECTION O MOTION, so that Q 1 appeas to be coming staight at us: 1 is OUT O PAGE P If we conside points on a cicle (e.g., P - P 5 ) then the diection of the unit ecto 1 etc. depends on which point we conside. 1 is always out of the page. P Apply the ight hand ule to each point is always tangential Lines of fom CLOSED LOOPS Conention: As befoe, line spacing is used to indicate the magnitude of (closely spaced lines stong field). P 4 P 5 Q moing outwads

6 EM 005 Handout 7: The Magnetic ield 6 Magnetic flux, ψ This is defined in exactly the same way as electic flux. Conside a small flat aea da. Let be the magnetic field at its cente. Assume that da is so small that can be egaded as unifom oe the whole of da. θ da Nomal to ecto ds DEINITION: The MAGNETIC LUX, dψ though the aea da is the poduct of da and the nomal component of. Aea ds da o dψ (cosθ)ds so dψ da whee d A is the NORMAL VECTOR of the aea da: Magnitude of da is: da Diection of d A is: pependicula to da Note: 1. Magnetic flux is a SCALAR.. Young & eedman uses the symbol Φ fo magnetic flux 3. The SI unit of magnetic flux is the Webe (Wb): 1 Wb (1 Tesla)(1 m ) o 1 T 1 Wb m -.

7 EM 005 Handout 7: The Magnetic ield 7 Magnetic flux fo the case of a non-unifom field passing though an abitay suface Poceeding exactly as we did fo the electic flux (see handout on electic flux and Gauss s Law), we can show that the total magnetic flux cossing a suface S is Ψ da What is Ψ fo a closed suface? S Recall: Gauss s Law fo E : Φ E da Q enclosed ε o Lines of E begin and end on electic chages. ut lines of fom closed loops (thee is no equialent of chage - i.e., no magnetic monopoles ). as many magnetic field lines will leae a gien olume as ente it (no enclosed magnetic chage ). THE TOTAL MAGNETIC LUX THROUGH A CLOSED SURACE IS ZERO o Ψ d A 0 THIS IS GAUSS S LAW OR THE MAGNETIC IELD MAXWELL S nd EQUATION This is sometimes witten as di 0 o 0.

8 EM 005 Handout 7: The Magnetic ield 8 oce on a chaged paticle moing in a unifom magnetic field Conside a unifom coming out of the page (usually epesented by dots): Let chage Q hae elocity pependicula to Q( ) is always pependicula to and no component of foce along the diection of THE SPEED NEVER CHANGES Q constant (out of page) Q Constant foce to elocity MOTION IN A CIRCLE oce equied fo cicula motion is m m mass adius Q m m Q Speed, Cicumfeence π equency is f Q π πm CYCLOTRON REQUENCY Note: 1. f is independent of. f depends only on and fundamental constants it can be used to find.

9 EM 005 Handout 7: The Magnetic ield 9 What if also has a component of motion along the diection of? Call this component paallel and paallel ae paallel, so paallel x 0 Q paallel no foce in this diection. motion along is not affected MOTION IN A SPIRAL Motion of a chaged paticle in combined electic and magnetic fields: the Loentz foce If a point chage Q moes with elocity in an electic field E and a magnetic field, the esultant foce on it is [ E + ( ) ] Q THE LORENTZ ORCE The elocity selecto Conside a chaged paticle moing with elocity pependicula to both an electic field, E, and a magnetic field,. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x E x x x x x x x x x x x x Q Sceen with hole (into the page) Q expeiences both electic and magnetic foces: Q( x ) Q QE Now so Q Q the two foces balance exactly if Q QE the paticle is not deflected if E/ only paticles fo which E/ pass though the hole the output beam is of unifom elocity (mono-enegetic).

10 EM 005 Handout 7: The Magnetic ield 10 The Hall effect Rectangula sample of conducto o semiconducto caying a cuent I Magnetic field applied pependicula to the diection of the cuent Cuent could be caied by eithe electons d d o holes Top ottom a -e +e b V 1 V Cuent I Voltmete between top and bottom sides measues V V 1 - V If I is caied by electons: If I is caied by holes: V 1 V 1 x V -e ~ (outwads) +e ~ (outwads) V -e( x ) e( x ) uild-up of negatie chage on the bottom side V V uild-up of positie chage on the bottom side V 1 > V V is positie V > V 1 V is negatie we can find out whethe the cuent is caied by electons (n-type semiconducto) o holes (p-type semiconducto). Chage build-up electic field, E, between top and bottom sides E opposes futhe chage build-up b E ee Equilibium established when ee e e ( V/b) e V b (b distance between top and bottom sides) e( x )

11 EM 005 Handout 7: The Magnetic ield 11 We can elate the speed,, to the cuent, I: length dt Let n no. of caies pe unit olume (each with chage e) Conside the amount of chage, dq, cossing aea ab in time dt: dq amount of chage in this olume a b So dq ne(ab)(dt) I dq/dt neab So I neab V I nea V is the HALL POTENTIAL y measuing the sign and magnitude of V we can : - ind n if is known - i.e., inestigate the sign and numbe density of the chage caies in a sample of mateial - O find if n is known - i.e., measue an unknown magnetic field with a HALL PROE

2/26/2014. Magnetism. Chapter 20 Topics. Magnets and Magnetic Fields. Magnets and Magnetic Fields. Magnets and Magnetic Fields

2/26/2014. Magnetism. Chapter 20 Topics. Magnets and Magnetic Fields. Magnets and Magnetic Fields. Magnets and Magnetic Fields Magnets and Magnetic ields Magnetism Howee, if you cut a magnet in half, you don t get a noth pole and a south pole you get two smalle magnets. ectue otes Chapte 20 Topics Magnets and Magnetic ields Magnets