Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

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1 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27

2 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field exets a foce on any othe chage q that is pesent in the field. What is coming up fo magnetic fields 1. A MOVING chage (o chages) poduce a magnetic field in the space aound it. 2. The magnetic field exets a foce on any othe MOVING chage o cuent that is pesent in the field.

3 Some Simple Phenomenology

4 The wold is a big magnet

5 Magnetic and Electic Foces Just Replace + & - with N & S???? This is an extemely good idea that is, unfotunately WRONG The basic poblem is that thee ae no Fee N & S poles.

6 Basic elationship between electic fields and magnetic cuents Demonstated in Oested s Expeiment Place a compass nea a wie N Compass deflects when an electic cuent flows in the wie

7 Motion of a chaged paticle in a magnetic field 1. A moing chage o a electic cuent poduces a magnetic field in the suounding space (It also poduces an electic field) 2. The magnetic field exets a foce on any othe moing chage o cuent that is in the field. Stategy: We will begin with a discussion of the foce on a moing chage (pat 2.) Then we will discuss how a moing chage makes the field (pat 1.)

8 Some examples of the foce on a moing chage in a magnetic field obseation #1- The magnetic foce is always pependicula to the magnetic field obseation #2- The magnetic foce is always pependicula to the paticle elocity

9 Magnetic Foces Fou obseations about a chage q moing in a magnetic field B The foce is: popotional to the chage q popotional to the elocity pependicula to both and B popotional to sinφ, whee φ is the angle between and B This can be summaized as: F = q " B The symbol epesents a coss poduct (not a multiplication) of the elocity ecto of the chaged paticle and the magnetic field ecto.

10 Moe on magnetic foces The magnetic foce is zeo if the elocity is eithe paallel o anti-paallel to the magnetic field. sin(0) = sin(180) = 0 The foce has its maximum alue when the elocity and magnetic field ae pependicula sin(90) = 1 The foce on a negatie chage is in the opposite diection

11 The ecto o coss poduct If C = A B then the magnitude of C = A B sin, whee θ is the angle between A and B. The diection of C is gien by the Right Hand Rule : Adice on using the Right Hand Rule: 1) Fist detemine the plane that contains A and B. The coss poduct will point pependicula to that plane. Thee ae only two choices. 2) Use the Right Hand Rule to pick which choice is coect. 3) If you ae using F = q B, Remembe that a negatie chage will eese the diection of the coss poduct

12 F Magnetic Foce and Magnetic Field = q " B F = q B Units of Magnetic Field: 1 Tesla = 1 T = 1 Newton/(Ampee mete) 10-4 T =1 Gauss ~ Magnetic field of the eath A steady 50 T field is ey lage (about the lagest possible today in a lab). The suface of a neuton sta is belieed to be ~10 8 T

13 Example A unifom magnetic field points into the sceen. The diection is indicated by the cosses, dots would be coming out of the sceen (imagine aows, you see the tail feathes, not the points). A positie chage moes fom point A to point C, the diection of the magnetic foce is: a) up and ight, b) up and left, c) down and ight, d) down and left

14 A unifom magnetic field points into the sceen. Example A positie chage moes fom point A to point C, the diection of the magnetic foce is: a) up and ight, b) up and left, c) down and ight, d) down and left The magnetic foce is gien by F = q B The coss poduce of the elocity and the magnetic field ecto is up and to the left. As the chage is positie, the foce is in the same diection

15 Motion of chaged paticles in EM fields F = F = ma q( E + B) + d = m dt 2 2 EM foce Loentz Foce Newton s 2 nd Law d q[ E( ) + B( )] = dt m d dt 2 2 Diffeential equation (t) Solution = Equation of Motion Fo constant foce thee ae two impotant simple cases: F is paallel to F is pependicula to Unifom linea acceleation + at Unifom cicula motion = 0

16 Unifom cicula motion (see Y&F chapte 3) Simila tiangles (i.e. same angle) gies t s R t s R R s = = = Aeage acceleation R a t s R t a t t lim lim = = = " " Unifom Cicula Motion implies a acceleation that is always diected towads the cente of the cicle. Amplitude of acceleation is constant: Diection of acceleation changes with time R m m a F 2 = =

17 Angula Velocity (see Y& F chapte 9) Unifom Cicula Motion is descibed by an Angula Velocity Since = s = 2 a = = F = m 2 Fequency Peiod and 2 2 = m f T " = 2 1 = f = ds dt " Angula Velocity (adians/s) cycles/s=hetz seconds " = d dt

18 Angula elocity as a ecto

19 Motion due to a Magnetic Foce What is the motion like if the elocity is not pependicula to B? Beak up the elocity into components along the magnetic field and pependicula to it The component pependicula will still poduce cicula motion The component paallel will poduce no foce, and this motion will be unaffected The combination of these two types of motion esult in a helical motion

20 Velocity Selecto A neat deice fo selecting ions by thei elocities (actually used in eseach) has cossed electic and magnetic fields. A chaged paticle (ion) expeiences both the E and the B field. The foces acting on the ion ae: F E = qe F B = q " B Fo a positie chage, foce due the electic field is to the left and foce due to the magnetic field is to the ight If the elocity of the ion is pecisely ight then the foces cancel out F E = F B = E B

21 Thomson s e/m Expeiment (1897) J.J. Thomson used the idea of a elocity selecto to measue the atio of chage to mass fo the electon. The hot cathode eleases electons which ae acceleated towads the two anodes. 1 2 m2 = ev = E B e m = E 2 2VB 2 Measue E,V and B, find e/m = x C/kg egadless of mateial on cathode. Discoey of the electon

22 Magnetic foce on a cuent-caying conducto We e seen that thee is a foce on a chage moing in a magnetic field Now we e going to conside multiple chages moing togethe, such as a cuent in a conducto We stat with a wie of length l and coss section aea A in a magnetic field of stength B with the chages haing a dift elocity of d. The total numbe of chages in this section is then nal whee n is the chage density. The foce on a single chage is gien by F=q d B. So, the total foce on this segment is: F = nq d AlB

23 Magnetic foce on a cuent-caying conducto We e found: F = nq d AlB; howee we aleady know J=nq d. So F=JAIB = IlB The foce is popotional to the cuent though the wie, the length of the wie in the field and the magnetic field stength

24 Magnetic foce on a cuent-caying conducto But what if the magnetic field and the wie ae not pependicula? Only the component of B ( B " = Bsin# ) pependicula to the wie exets a foce. F = IlB " F = IlBsin# F = Il $ B

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