Electricity and Magnetism Motion of Charges in Magnetic Fields

Transcription

1 Electricity and Magnetism Motion of Charges in Magnetic Fields Lana heridan De Anza College Feb 21, 2018

2 Last time introduced magnetism magnetic field Earth s magnetic field force on a moing charge

3 Oeriew charged particles motion in magnetic fields

4 Force on a Moing Charge The force on a moing electric charge in a magnetic field: F = q where is the magnetic field, is the elocity of the charge, and q is the electric charge. The magnitude of the force is gien by: F = q sin θ if θ is the angle between the and ectors.

5 Magnetic field direction reminder Magnetic field lines coming out of the paper are indicated by dots, representing the tips of points arrows out coming of theoutward. page. Magnetic field lines going into the paper are indicated by crosses, representing the feathers of arrows going points inward. into the page. out in a b is directed perpendicularly into the page, we use green crosse

6 . sented a magnetic field directed into that plane.) If the charge of the particle were negatie, the magnetic deflecting force would be directed in the opposite direction that is, horizontally from east to west. This is predicted automatically For example: by Eq. here28-2 the dots if we indicate substitute the field a negatie is directed alue upward for q. out of the slide. Force on a Moing Charge N Path of proton W + F E Fig An oerhead iew of a proton moing from south to north The force with on elocity the particle : in a is chamber. to its A elocity magnetic and field the field. is directed ertically upward in the chamber, as represented by the array of dots (which resemble the tips of arrows). The proton is deflected toward 1 Figure the from east. Halliday, Resnick, Walker, 9th ed.

7 Circular Motion of a Charge If a charge enters a magnetic field with a elocity at right angles to the field, it will feel a force perpendicular to its elocity. This will change its trajectory, but not its speed. Uniform Circular Motion! The radius of the circle will depend on 4 things: mass of the particle charge of the particle initial elocity strength of the field

8 the charge is always directed toward the center of the circle. Circular Motion of a Charge in q F q r F F q Figure 29.8 a charged part to a uniform m particle moes a plane perpen

9 Circular Motion of a Charge Electrons in a uniform magnetic field: 28-6 A CIRCULATING CHAR F Fig Electrons circulating in a chamber containing gas at low pressure (their path is the glowing circle). A uniform magnetic field :, pointing directly out of the plane of the page, fills the chamber. Note the radially directed magnetic force F : ;for circular motion to occur, F : must point toward the center of the circle.use the right-hand rule for cross products to : : : 1 Photo from Halliday, Resnick, Walker 9th ed, John Le P. Webb, ussex Uniersity.

10 Circular Motion of a Charge To find the radius: F net = F c = F ince and are perpendicular F = q: m 2 r = q r = m q The sign of q will determine whether the charge circulates clockwise or counter-clockwise.

11 Circular Motion of a Charge To find the radius: F net = F c = F ince and are perpendicular F = q: m 2 r = q r = m q The sign of q will determine whether the charge circulates clockwise or counter-clockwise. m 2 = q r

12 the left and right sides indicate that Question e field at an end is strong enough, the le reflects from both ends, it is said to The figure here shows the circular paths of two particles that trael at the same speed in a uniform magnetic field, which is directed into the page. One particle is a proton; the other is an electron (which is less massie). Which particle follows the smaller circle? particles that trael :,which is directed ther is an electron s the smaller circle, nterclockwise? (A) proton () electron 1 Halliday, Resnick, Walker, 9th ed., page 746

13 the left and right sides indicate that Question e field at an end is strong enough, the le reflects from both ends, it is said to The figure here shows the circular paths of two particles that trael at the same speed in a uniform magnetic field, which is directed into the page. One particle is a proton; the other is an electron (which is less massie). Which particle follows the smaller circle? particles that trael :,which is directed ther is an electron s the smaller circle, nterclockwise? (A) proton () electron 1 Halliday, Resnick, Walker, 9th ed., page 746

14 the left and right sides indicate that Question e field at an end is strong enough, the le reflects from both ends, it is said to The figure here shows the circular paths of two particles that trael at the same speed in a uniform magnetic field, which is directed into the page. One particle is a proton; the other is an electron (which is less massie). Which particle traels clockwise (as iewed in the diagram)? particles that trael :,which is directed ther is an electron s the smaller circle, nterclockwise? (A) proton () electron 1 Halliday, Resnick, Walker, 9th ed., page 746

15 the left and right sides indicate that Question e field at an end is strong enough, the le reflects from both ends, it is said to The figure here shows the circular paths of two particles that trael at the same speed in a uniform magnetic field, which is directed into the page. One particle is a proton; the other is an electron (which is less massie). Which particle traels clockwise (as iewed in the diagram)? particles that trael :,which is directed ther is an electron s the smaller circle, nterclockwise? (A) proton () electron 1 Halliday, Resnick, Walker, 9th ed., page 746

16 Circular Motion of a Charge Can you find the period T for the orbit? (Time for the particle to make a full circle?)

17 Circular Motion of a Charge Can you find the period T for the orbit? (Time for the particle to make a full circle?) T = 2πr

18 Circular Motion of a Charge Can you find the period T for the orbit? (Time for the particle to make a full circle?) T = 2πr T = 2πm q

19 Circular Motion of a Charge Can you find the period T for the orbit? (Time for the particle to make a full circle?) T = 2πr Also, angular frequency T = 2πm q ω = q m

20 More general case What if the elocity ector of a charge particle is not perpendicular to the magnetic field?

21 More general case What if the elocitythe ector elocity of acomponent charge particle is not perpendicular to the magnetic field? perpendicular to the field causes circling, which is There will be somestretched component upward of theby elocity the in the direction of p the magnetic field. parallel component. φ For the cross product: (a) φ q + + Fig (a) A charged particle moes in a uniform magnetic field, the particle s elocity making an angle f with the field direction. (b) The particle follows a helical path of radius r and pitch p.(c) A charged particle spiraling in a nonuniform magnetic : ector. = sin : φ = ( sin φ) = soled q φ

22 More general case What if the elocitythe ector elocity of acomponent charge particle is not perpendicular to the magnetic field? perpendicular to the field causes circling, which is There will be somestretched component upward of theby elocity the in the direction of p the magnetic field. parallel component. For the cross product: φ φ (a) q φ + + Fig (a) A charged particle moes in a uniform magnetic field, the particle s elocity making an angle f with the field direction. (b) The particle follows a helical path of radius r and pitch p.(c) A charged particle spiraling in a nonuniform magnetic : ector. = sin : φ = ( sin φ) = soled The force will not depend on the -component and the -component of elocity will not be changed. q

23 Helical Trajectories APTER 28 MAGNETIC FIELD onent the field hich is by the nt. p Partic φ q + + a) A charged particle moes : q φ r (b) F

24 Helical Trajectories MAGNETIC FIELD The pitch, p, of the helix is p = T = 2π m q where T is the time period. d particle moes :, the particle s f with the field ollows a helical p φ q + r (b) F Particle The radius is F r = m q (c) piral path using our equation from earlier. ector. Figure 28-11a,for example,shows the elocity ector : of such a particle resoled into two components, one parallel to : and one perpendicular to it: F

25 Non-Uniform Fields: Magnetic ottle F Particle F piral path F trong trong Weaker field field field 11a,for example,shows the elocity ector : of such a particle remponents, one parallel to : and one perpendicular to it:, cos and sin. (28-20) (c)

26 Applications Inoling Charged Particles Moing in a Non-Uniform Fields: Van Allen elts Earth s magnetic field acts as a magnetic bottle for cosmic rays.

27 Non-Uniform Fields: Van Allen elts When these charges particles in the belts are disturbed by the solar wind they can drop down into the atmosphere. 1 Figure by NAA.

28 Non-Uniform Fields: Van Allen elts When these charges particles in the belts are disturbed by the solar wind they can drop down into the atmosphere. The resulting glow is the aurora borealis. 1 Photo by Donald R. Pettit, Expedition ix NAA I science officer, 2013.

29 The Lorentz Force A charged particle can be affected by both electric and magnetic fields. This means that the total force on a charge is the sum of the electric and magnetic forces: F = q E + q This total force is called the Lorentz force. This can always be used to deduce the electromagnetic force on a charged particle in E- or -fields.

30 Figure A elocity selector. Velocity elector: Using both electric and magnetic fields Charges are accelerated with and electric field then trael down a apter 29 Magnetic Fields channel with uniform electric and magnetic fields. in F E F e lit P r Detector array in ource Velocity selector

31 Velocity elector: Using both electric and magnetic fields The particles only reach the end of the channel if F = 0. F = q E + q so that means qe = q supposing and are perpendicular: = E

32 Mass pectrometer After selecting particles to hae elocity = E/ along the channel, they are fed into a magnetic field. 0, in F e lit locity selector. arged particle P Velocity selector r Detector array in E q Figure A mass spectrome-

33 Mass pectrometer F ource E F e lit A elocity selector. positiely charged particle g with elocity the path, in the r. presa magnetic field directed page and an electric field to the right, it experiences ric force q E to the right and tic force q 3 to the left. P Velocity selector r Detector array in 0, in E q Where they collide with the detector allows us to find the radius of Mass-to-charge ratio: Figure A mass spectrometer. Positiely charged particles are sent first through a elocity selector and then into a region where the magnetic field 0 causes the particles to moe min a semicircular path and strike a detector array at P. q = r 0

34 ummary magnetic field lines charged particles in crossed-fields properties of the electron Homework erway & Jewett: PREVIOU: Ch 29, Obj Qs: 1, 3, 5; Conc. Qs: 1, 7; Problems: 1, 8, 9 Ch 29, Obj Qs: 7; Problems: 13, 15, 23, 73, 80