Magnetic Fields - Engineering Physics - Lecture Slides, Slides of Engineering Physics

Download Magnetic Fields - Engineering Physics - Lecture Slides and more Slides Engineering Physics in PDF only on Docsity! Today’s agenda: Magnetic Fields. You must understand the similarities and differences between electric fields and field lines, and magnetic fields and field lines. Magnetic Force on Moving Charged Particles. You must be able to calculate the magnetic force on moving charged particles. Motion of a Charged Particle in a Uniform Magnetic Field. You must be able to calculate the trajectory and energy of a charged particle moving in a uniform magnetic field. Magnetic forces on currents and current-carrying wires. You must be able to calculate the magnetic force on currents. docsity.com Magnetism Recall how there are two kinds of electrical charge (+ and -), and likes repel, opposites attract. Similarly, there are two kinds of magnetic poles (north and south), and like poles repel, opposites attract. S N S N S N S N Repel Attract S N S N S N Repel Attract S N docsity.com Magnetic field lines point in the same direction that the north pole of a compass would point. Magnetic field lines are tangent to the magnetic field. The more magnetic field lines in a region in space, the stronger the magnetic field. Outside the magnet, magnetic field lines point away from N poles (*why?). *The N pole of a compass would ―want to get to‖ the S pole of the magnet. Huh? Later I’ll give a better definition for magnetic field direction. docsity.com For those of you who aren’t going to pay attention until you have been told the secret behind the naming of the earth’s magnetic poles… The north pole of a compass needle is defined as the end that points towards the ―Santa Claus‖ north pole, which experts in the field of geomagnetism call ―the geomagnetic north‖ or ―the Earth’s North Magnetic Pole.‖ If you think about it, the people to whom compasses meant the most— sailors—defined magnetic north as the direction the north poles of their compass needles pointed. Their lives depended on knowing where they were, so I guess it is appropriate that we acknowledge their precedence. Thus, by convention, the thing we call Earth’s North Magnetic Pole is actually the south magnetic pole of Earth’s magnetic field. For those of you who are normal humans, just ignore the above. docsity.com Here’s a ―picture‖ of the magnetic field of a bar magnet, using iron filings to map out the field. The magnetic field ought to ―remind‖ you of the earth’s field. docsity.com Today’s agenda: Magnetic Fields. You must understand the similarities and differences between electric fields and field lines, and magnetic fields and field lines. Magnetic Force on Moving Charged Particles. You must be able to calculate the magnetic force on moving charged particles. Motion of a Charged Particle in a Uniform Magnetic Field. You must be able to calculate the trajectory and energy of a charged particle moving in a uniform magnetic field. Magnetic forces on currents and current-carrying wires. You must be able to calculate the magnetic force on currents. docsity.com A charged particle moving in a magnetic field experiences a force. Magnetic Fields and Moving Charges The magnetic force equation predicts the effect of a magnetic field on a moving charged particle. F=qv B force on particle magnetic field vector velocity of charged particle What is the magnetic force if the charged particle is at rest? Oh nooo! The little voices are back. What is the magnetic force if v is (anti-)parallel to B? docsity.com  Vector notation conventions:  is a vector pointing out of the paper/board/screen (looks like an arrow coming straight for your eye).  is a vector pointing into the paper/board/screen (looks like the feathers of an arrow going away from eye).  docsity.com ―Foolproof‖ technique for calculating both magnitude and direction of magnetic force. F=qv B ˆˆ ˆ                x y z x y z i j k F = q det v v v B B B All of the right-hand rules are just techniques for determining the direction of vectors in the cross product without having to do any actual math. http://www.youtube.com/watch?v=21LWuY8i6Hw docsity.com Example: a proton is moving with a velocity v = v0j in a region of uniform magnetic field. The resulting force is F = F0i. What is the magnetic field (magnitude and direction)? ^ ^ To be worked at the blackboard. Example: in the above example, what is the minimum magnetic field that can be present (magnitude and direction)? Blue is +, red is -. Image from http://www.mathsisfun.com/algebra/matrix-determinant.htm docsity.com A charged particle moving along or opposite to the direction of a magnetic field will experience no magnetic force. Homework Hint θF= q vB sin Conversely, the fact that there is no magnetic force along some direction does not mean there is no magnetic field along or opposite to that direction. It’s OK to use if you know that v and B are perpendicular, or you are calculating a minimum field to produce a given force (understand why). F= q vB docsity.com Example: an electron travels at 2x107 m/s in a plane perpendicular to a 0.01 T magnetic field. Describe its path. Motion of a charged particle in a uniform magnetic field docsity.com Example: an electron travels at 2x107 m/s in a plane perpendicular to a 0.01 T magnetic field. Describe its path. The above paragraph is a description of uniform circular motion. The electron will move in a circular path with a constant speed and acceleration = v2/r, where r is the radius of the circle. The force on the electron (remember, its charge is -) is always perpendicular to the velocity. If v and B are constant, then F remains constant (in magnitude). -                                                                 B v F v F - - docsity.com Motion of a proton in a uniform magnetic field +                                        Bout v FB + + v v FB FB r The force is always in the radial direction and has a magnitude qvB. For circular motion, a = v2/r so The rotational frequency f is called the cyclotron frequency The period T is 2mv F= q vB = r q rB mv v = r = m q B π π2 r 2 m T = = v q B π q B1 f = = T 2 m Thanks to Dr. Waddill for the use of the picture and following examples. Remember: you can do the directions “by hand” and calculate using magnitudes only. ocsity.com Apply B-field perpendicular to plane          docsity.com Lorentz Force Law If both electric and magnetic fields are present,  .F=q E+v B Applications See your textbook for numerical calculations related the next two slides. If I have time, I will show the mass spectrometer today. The energy calculation in the mass spectrometer example is often useful in homework. docsity.com Velocity Selector             E - - - - - - - - - - - - - - - -                                 v Bout + + q When the electric and magnetic forces balance then the charge will pass straight through. This occurs when FE = FB or Thanks to Dr. Waddill for the use of the picture. EF =qE BF =qv B E qE=qvB or v = B docsity.com Today’s agenda: Magnetic Fields. You must understand the similarities and differences between electric fields and field lines, and magnetic fields and field lines. Magnetic Force on Moving Charged Particles. You must be able to calculate the magnetic force on moving charged particles. Motion of a Charged Particle in a Uniform Magnetic Field. You must be able to calculate the trajectory and energy of a charged particle moving in a uniform magnetic field. Magnetic forces on currents and current-carrying wires. You must be able to calculate the magnetic force on currents. docsity.com So far, I’ve lectured about magnetic forces on moving charged particles. Magnetic Forces on Currents Actually, magnetic forces were observed on current-carrying wires long before we discovered what the fundamental charged particles are. F=qv B Experiment, followed by theoretical understanding, gives F=IL B. If you know about charged particles, you can derive this from the equation for the force on a moving charged particle. It is valid for a straight wire in a uniform magnetic field. For reading clarity, I’ll use L instead of the l your text uses. docsity.com Here is a picture to help you visualize. It came from http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/forwir2.html. Homework Hint  max F =ILB sin F =ILB For fixed L, B: this is the biggest force you can get from a given current I. For bigger F, you need bigger I. docsity.com Note: I generally use ds for an infinitesimal piece of wire, instead of dl. Font choice may make ―l‖ look like ―1.‖ It’s a pain to search the fonts for a script lowercase l: l. docsity.com Example: a wire carrying current I consists of a semicircle of radius R and two horizontal straight portions each of length L. It is in a region of constant magnetic field as shown. What is the net magnetic force on the wire?                                 B         L L R I x y There is no magnetic force on the portions of the wire outside the magnetic field region. docsity.com                                 B         L L R I x y First look at the two straight sections. F=IL B 1 2F =F =ILB L  B, so F1 F2 docsity.com                         B         L L R I x y F1 F2 dF  d ds Or, you can calculate the x component of F. Does symmetry give you Fx immediately?  xdF =-I R d B cos   x 0F =- I R d B cos   x 0F =-I R B cos d   0  xF =- I R B sin xF =0 dFx Sometimes-Useful Homework Hint Symmetry is your friend. docsity.com                                 B         L L R I x y Total force: F1 F2 dF  ds Fy 1 2 yF=F + F + F F=ILB + ILB + 2IRB  F=2IB L + R Possible homework hint: how would the result differ if the magnetic field were directed along the +x direction? If you have difficulty visualizing the direction of the force using the right hand rule, pick a ds along each different segment of the wire, express it in unit vector notation, and calculate the cross product. docsity.com Example: a semicircular closed loop of radius R carries current I. It is in a region of constant magnetic field as shown. What is the net magnetic force on the loop of wire?                                 B         R I x y We calculated the force on the semicircular part in the previous example (current is flowing in the same direction there as before). CF =2 I R B FC docsity.com                               B      S V +q x r Example: ions from source S enter a region of constant magnetic field B that is perpendicular to the ions path. The ions follow a semicircle and strike the detector plate at x = 1.7558 m from the point where they entered the field. If the ions have a charge of 1.6022 x 10-19 C, the magnetic field has a magnitude of B = 80.0 mT, and the accelerating potential is V = 1000.0 V, what is the mass of the ion? Radius of ion path: Unknowns are m and v. mv x =2r and r = qB Already derived today, so I can use them! docsity.com                               B      S V +q x r Conservation of energy gives speed of ion. The ions leave the source with approximately zero kinetic energy i i f fK +U =K +U  f i i fK =K + U -U     f f i f iK =- U -U =-q V - V =-q V 2 1 mv =-q V 2 -2q V v = m fK =-q V 0 A proton accelerates when it goes from a more positive V to a less positive V; i.e., when V is negative. That’s what this minus sign means. Caution! V is potential, v (lowercase) is speed. docsity.com                               B      S V +q x r -2q V v = m 2mv x =2r = qB 2m -2q V x = qB m 2 -2m V x = B q  2 2B x q m=- 8 V docsity.com