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Magnetism PHY232 Remco Zegers [email protected] Room W109 – cyclotron building http://www.nscl.msu.edu/~zegers/phy232.html quiz (extra credit) which of the following cannot be correct? a) V-I1R1-I3R3-I6R6-I5R5=0 b) -I3R3-I2R2-I5R5-I4R4=0 c) I4+I5+I6=0 d) I1R1+I3R3+I4R4=0 e) I3+I2+I6=0 V PHY232 - Remco Zegers - magnetism I1,R1 I3 R3 I4 R4 I2 R2 I6 R6 I5 R5 2 magnetism magnetic fields are produced by moving electrical charges (current) macroscopic level (e.g. currents in a wire) microscopic (electrons in atomic orbit and rotating around their own axis) PHY232 - Remco Zegers - magnetism 3 magnets the magnetic field produced by electrons tend to cancel each other, so most materials are not magnetic in certain ‘ferromagnetic’ materials (iron) neighboring electrons can couple and form domains (< 1mm) that are magnetic. Since there are many domains that have different orientation, the material is overall not magnetized when an external magnetic field is applied the fields in the different domains align and the whole object becomes magnetic after the external field is removed, a material like iron becomes unmagnetized quickly, but some remain magnetized and can be used as ‘permanent’ magnets. PHY232 - Remco Zegers - magnetism 4 para and ferro magnets strawberry in a B-field do not retain any magnetism in absence of external field retains domains in which magnetic field remain in the absence of external fields PHY232 - Remco Zegers - magnetism 5 magnetic poles and fields magnets have ‘north’ and ‘south’ poles and field lines run from north to south unlike the case of electrical fields, where positive charges can exists separate from negative charges, north and south poles always come together. There are no monopoles discovered so far. demo: magnetic field lines (ohp) broken magnet PHY232 - Remco Zegers - magnetism 6 One big magnet! demo: compass needles compass Why is it higher here? Note that the geographical North pole is in fact the magnetic south pole B=0.3-0.6 Gauss PHY232 - Remco Zegers - magnetism 7 question If you are standing exactly at the (magnetic) south Pole (I.e. near the geographical north pole), and are holding a compass parallel with the earth’s surface, in which direction would the needle point? a) It would point roughly to the geographical south b) It could point anywhere c) It would rotate with constant angular speed The compass needle in fact wants to point into the earth (along the direction of the field line). But if hold parallel to earth, it can’t do that and will point wherever. There is no reason for it to rotate though. PHY232 - Remco Zegers - magnetism 8 charged particles moving in a magnetic field A charged particle q that is moving with a velocity v in a magnetic field B will feel a force where q: charge of particle v: velocity of paticle B: magnetic field : angle between velocity vector and field direction PHY232 - Remco Zegers - magnetism unit: N/(Cm/s)=T 1 T = 104 Gauss 9 direction of force on charged paricles in B-field magnitude of the force demo: bending the beam I you can find the direction of the force using the right hand rule. It holds for positive charges. For negative charges switch the direction of the force the use of the fingers is different from the book (but more straightforward, I think) but the result the same PHY232 - Remco Zegers - magnetism 10 example: electron in magnetic field an electron with v=1x106 m/s is entering a area with B=1 T. The field is directed into the screen. a) in which direction will the electron be bent, if at all? b) how large is the force? what is the acceleration? x x x x x x x x a) use right hand rule: x x x thumb is velocity (initially to the right) index finger is field (in the screen) middle finger is force perpendicular to both switch direction because negative charge b) F=|q|vBsin=1.6x10-19 x 1x106 x 1=1.6x10-13 N a=F/m=1.6x10-13 N/9.11x10-31 kg =1.76x1017 m/s2 PHY232 - Remco Zegers - magnetism x 11 question a Magnesium ion (Z=12) with all its electrons removed is moving in a field of 0.1 T as shown. What direction will the force act? a) into the screen b) out of the screen c) parallel to the B field lines and the screen d) perpendicular to the B field lines and parallel to the screen e) in the direction of motion Mg 45o v thumb is velocity, index finger is field so force comes out of the screen (the ion is positive) PHY232 - Remco Zegers - magnetism 12 loncapa do problem 1 from set 5 PHY232 - Remco Zegers - magnetism 13 Charged particle in a magnetic field Let’s assume a charged particle is moving in a uniform magnetic field so that the velocity is perpendicular to the field. The particle will follow a curved path and is directed towards the center Use Newton’s second law and the equation for centripetal acceleration demo: bending the beam II PHY232 - Remco Zegers - magnetism 14 Magnetic spectrometers Beam from cyclotrons target chamber S800 spectrometer At the cyclotron Bending angle ~ 150o PHY232 - Remco Zegers - magnetism 15 question In a nuclear reaction two types of fully ionized particles are created. 120Sn with Z=50 and v=12.8814x107 m/s (Tin) 120Sb with Z=51 and v=13.099x107 m/s (Antimony) Both have a mass of 1.991x10-25 kg and pass through a 180o magnetic spectrometer with B=1T. If the detector used to locate the particles can separate events that are 2 mm away from each other, are 120Sn and 120Sb separated? r= mv/qB For 120Sn: M=1.991x10-25 kg v=12.8814x107 m/s B=1T q=50x1.6x10-19 C. RSn=3.2060 m For 120Snb: M=1.991x10-25 kg v=13.0990x107 m/s B=1T q=51x1.6x10-19 C. RSb=3.1961 m RSn-RSb=3.206-3.1961=9.9x10-3 m = 9.9 mm thus separated PHY232 - Remco Zegers - magnetism 16 loncapa do problem 8 from set 5 Note: a particle with charge q gains kinetic energy (0.5mv2) qV when accelerated by a potential of V volts. PHY232 - Remco Zegers - magnetism 17 What we did so far Moving charged particles make magnetic field Field lines go from North to South North and South poles cannot exist independently The magnitude of a force on a charge particle in a magnetic field: F=qvBsin where is the angle between v and B. The direction of the force is given by the (first) right-hand rule for + particles: use directly for – particles: after using the right hand-rule, reverse the direction of the force For a particle moving in a direction perpendicular to a magnetic field PHY232 - Remco Zegers - magnetism 18 quiz (extra credit) a proton is moving from left to right into a field of which the field lines point into the screen. As a result, the proton will a) continue along its original trajectory b) bend upwards x x x x c) bend downwards d) bend into the screen x x x x e) bend out of the screen proton x x x x PHY232 - Remco Zegers - magnetism 19 MOVIE magnetic force on a conducting wire consider positive charges moving through a wire. Each particle feels a force, hence there is a net force on the wire N: total number of charges n: charges per unit volume Use: see earlier To get I More general: where : angle between I and B vectors PHY232 - Remco Zegers - magnetism I 20 question: a floating wire a 1 m long copper wire of unknown mass is held horizontally with a current of 1 A going through it. It is placed in a horizontal magnetic field whose field lines are perpendicular to the wire. When the magnetic field is 1 T, one can let go of the wire without if falling down. top view What is its mass? I B electrons are moving left to right, so force due to B is up (out of the screen). When floating Fgravity=FB-field mg=Bil so m=Bil/g=1x1x1/9.81=0.102 kg PHY232 - Remco Zegers - magnetism 21 question x x a rectangular looped copper x x wire carrying a current is placed horizontally in a B-field x B x pointing down. Disregarding any other forces, it will move x x a) in direction of vector A b) in direction of vector B x x c) in direction of vector C d) in direction of vector D e) none of the above correct answer: e) A x x x x C x top view x D x I x x x It will not move at all. Forces on left and right sides will cancel and likewise for top and bottom sides PHY232 - Remco Zegers - magnetism 22 lon-capa do problems 2,3,9 from set 5. PHY232 - Remco Zegers - magnetism 23 Torque on a current loop Top view Rotation axis Consider a current loop with dimension a x b in a B-field parallel b I to the loop. B The force F on the right side (length b): F=BIb (pointing into the screen in the top view or downward in the frontal view) a The force F on the left side (length F b): F=BIb (pointing out of the screen in the top view or upward in the x frontal view) force on up/down side (length a) is F frontal view zero With the given rotation axis: Torque: =Fd=(BIb x a/2) + (BIb x a/2) If there is a net torque, =BIba=BIA with A=axb: surface of the loop will rotate! loop. PHY232 - Remco Zegers - magnetism 24 Torque on a current loop Top view Now the loop makes an angle with the B-field as shown right To calculate the torque we only need the force perpendicular to the rotating loop: FL=Fsin =FLd=(BIb x a/2)sin + (BIb x a/2) sin =BIbasin=BIAsin If there would be N loops: =BIANsin F b PHY232 - Remco Zegers - magnetism I B a F frontal view frontal view FL sin=F/FL Rotation axis x F 25 So… The general equation for a torque on a loop of N windings of wire is: with B: magnetic field strength I: Current through the loop A: area of the loop (also holds for non-rectangular loops) N: number of windings : angle between B and line perpendicular to loop =IAN magnetic moment of the coil: it is a vector perpendicular to the coil. is also the angle between and B. Note that is independent of B and , so it describes the properties of the coil when placed in a field. Unit: Am2 B I PHY232 - Remco Zegers - magnetism A N 26 note for loncapa: area of an ellipse: ab with a, b radii in the two directions example A circular coil of 5 windings is placed in a B-field of 2 T that B o makes and angle =60 with A N I the line perpendicular to the coil. The radius of the coil is 3 cm, and the current through the coil is 0.5 A. What are: A=r2= (0.03)2=2.82x10-3 m2 a) the area of the coil? b) the magnetic moment of =IAN=0.5 x 2.82x10-3 x 5=7.1x10-3Am2 the coil? c)the torque one the coil? =Bsin= 7.1x10-3 x 2 T x 0.866=1.23x10-2 Nm PHY232 - Remco Zegers - magnetism 27 question a b c d four loops with the same magnetic moment but of different shape are place in the same B-field under the same angle. Which one will feel the largest torque? a) loop a b) loop b c) loop c same , , B: so same d) loop d e) all the same PHY232 - Remco Zegers - magnetism 28 electric motor By supplying electricity we can get some work done! PHY232 - Remco Zegers - magnetism 29 lon-capa do problems 4,6 from set 5. PHY232 - Remco Zegers - magnetism 30 creating magnetic field with current So far, we have seen that magnetic field can affect the motion of charged particles. However, the reverse is also true: moving charge can create magnetic fields. First seen by Hans Oersted who noted that a current through a wire creates a magnetic field. A second right-hand rule can be used to find the direction of the magnetic field demo: Oersted experiment field of a current PHY232 - Remco Zegersmagnetic - magnetism 31 How to quantify the field Ampere’s law states that for an arbitrary closed path around a current: B//: magnetic field parallel to l 0: permeability of free space 4 x 10-7 Tm/A If we make the arbitrary closed path circular then: I B// l B where r: radius of loop B//I=Bi=B since B is fixed at fixed r Rewrite to find the B-field at distance r: PHY232 - Remco Zegers - magnetism 32 an electron passing a wire a) b) c) d) an electron with v=1x106 m/s is moving parallel to a wire carrying a current I=1A at a distance of 2 cm, in the same direction as the current What is the direction of the magnetic field near the electron due to the wire? what is the magnitude of the magnetic field near the electron? what is the direction of the force on the electron? what is the magnitude of the force on the electron? I=1A q=-1.6x10-19C 2 cm a) use 2nd right hand rule B-field goes into the screen b) =4 x 10-7 x 1/(20.02)=1x10-5 T c) use 1st right-hand rule and notice that the electron is negative. Force points to the right. d) F=qvBsin=1.6E-19x1E6x1E-5x1= = 1.6E-18 N (note sin(90)=1) PHY232 - Remco Zegers - magnetism 33 question a proton is passing by a wire carrying current and is moving perpendicular to the wire, into the screen 1) what is the direction of the B-field near the proton? into the screen out of the screen to the left to the right up 2) what is the direction of the force on the proton? to the left to the right up down no force at all I x proton moving into the screen 1) use 2nd right hand rule (same as example on previous slide) 2) use 1st right hand rule. velocity is into the screen, B-field is into the screen: no Force (sin(00)=0) PHY232 - Remco Zegers - magnetism 34 magnetic force between two parallel wires if we place two parallel wires next to each other, the current in wire 2 creates a field near wire 2, at distance d from wire 1: d The force on wire 1 due to wire 2 is then: Note so that the force per unit length is: PHY232 - Remco Zegers - magnetism attractive if same direction repulsive if opposite direction 35 question two wires are placed parallel, one carrying a current of 1A and the other of 2A, in the same direction. The distance between the two wires is 2 cm a) what is the magnitude of the B-field exactly in between the two wires? b) if a proton moves parallel to the two wires with v=1x105 m/s, exactly in between the two and in the same direction as the current, what is the magnitude of the force on the proton? c) what is the force per unit length between the two wires? a) B1=0I/(2r)=4x10-7x1/(20.01)=2x10-5 T 1A 2A -7 -5 B2= 4x10 x2/(20.01)=4x10 T B1: into the screen B2: out of the screen 2 Bnet=2x10-5 T out of the screen 1 b) F=qvBsin=1.6x10-19x105 x 2x10-5 x sin(90)=3.2x10-19 N (directed to the right, use 1st right-hand rule) c) F/l= 0I1I2/(2d)= 4x10-7x1x2/(20.02)=2x10-5 N 2cm PHY232 - Remco Zegers - magnetism 36 note the procedure of the previous slide can be used for any number of wires. In case of 4 wires (see lon-capa), one can calculate the force of one on the wires by adding the forces of each of the other three wires on that wire… I1 •B out of page due to I1 •B into page due to I2 •|Bnet|=|B1-B2| I2 •B into page due to I1 •B into page due to I2 •|Bnet|=|B1+B2 | •B into page due to I1 •B out of page due to I2 •|Bnet|=|B1-B2| • make little sketches on your equation sheets for the various cases PHY232 - Remco Zegers - magnetism 37 lon-capa do problems 5,7,10,12,13 from set 5 PHY232 - Remco Zegers - magnetism 38 other cases: the current loop magnetic field inside a current loop I I R X right-handed current through loop: B-field in the screen left-handed current through loop: B-field out of the screen PHY232 - Remco Zegers - magnetism example: A person wants to find the current in a superconducting coil with diameter of 2 cm. She measures the magnetic field at the center to be 1x10-5 T. What is the current? I=2RBcenter/0= 2x0.01x10-5/4x10-7= 0.16 A 39 other cases II: magnetic field of a solenoid a solenoid is a collection of coils stacked on top of each other Inside a perfect solenoid, the field lines are parallel and the field uniform outside the solenoid, the field pattern looks like that of a bar magnet. For the field inside of a solenoid: where I is the current and n is the number of turns (n) per unit length l of the solenoid note that the field at the center does not depend on the radius of the turns B-field of solenoid PHY232 - Remco Zegers - magnetism 40 example A perfect coil is 30 cm long and has 3000 windings. Its radius is 2cm. What is the field strength along the central line inside the coil if the current is 4 A? B=0nI=4x10-7 x 3000/0.3 x 4 = 1x10-3 T use n=N/L The field strength along a line parallel to the central line but 5mm away from the center is … along the central line? a) lower than b) the same as c) higher than inside the coil, the field is uniform PHY232 - Remco Zegers - magnetism 41 loncapa do problem 11 from set 5 PHY232 - Remco Zegers - magnetism 42