Magnetism Worksheet

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 A certain wire carries a current I such that at point P a distance r from the wire, the magnetic field is measured to be 1.0 T. A second wire is now added such that it is parallel to the first wire, and is placed a distance r from point P and 2r from the original wire. What will be the magnetic field at point P if the new wire carries a current I, but in the opposite direction as the first wire? 

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Lecture Set 07—Magnetism G.P.II L07 Overview: Magnetism Magnetism is closely related to electricity: ● Moving charges produce magnetic fields ● Magnetic fields can affect moving charges ● Magnetic fields can create electric fields Magnetic fields have as their ultimate source charges in motion (electric current) G.P.II Fig. 7.1.1: A variety of magnets. L07OV Part 1: Particle in a Magnetic Field (Openstax 22.1-22.5) Magnetic fields (B) are associated with moving charges, but there are permanent magnets possible Permanent magnets are associated with poles—they have a north pole and a south pole ● The poles are the magnetic equivalent of charges—same poles repel, opposite poles attract ● There are theoretically no magnetic monopoles ● Fig. 7.1.2: A rail gun uses magnetic fields to launch a projectile at very high speed G.P.II L07P1 Hard Magnetic Materials Hard magnetic materials are used in permanent magnets ● Hard to magnetize, but then keep their magnetization ● These are “permanent” magnets ● Provide permanent magnetic field without electricity ● Typically made from alloys of cobalt-iron-nickel or aluminumnickel-cobalt (“alnico”), some rare earth elements ● Used in loudspeakers, magnetic motors, computer hard drives G.P.II Fig. 7.1.3: Field lines from a permanent magnet L07P1 Soft Magnetic Materials Soft magnetic materials can be temporarily magnetized when brought into contact with a magnet ● They are easily magnetized but lose their magnetization easily ● Examples: iron, nickel, nickel-iron alloys, ferrites (ferric oxide + magnesium, nickel, etc) ● Commonly used as magnetic shields to prevent signals from escaping from or being added to a cable (e.g. a coaxial cable) G.P.II Fig. 7.1.4: a) A coaxial cable and b) a sketch of the (external) magnetic field lines approaching the shield L07P1 Earth's Magnetic Field The earth has a magnetic field which can be treated as similar to the field generated by a bar magnet: Magnetic south corresponds to geographic north, and magnetic north corresponds to geographic south ● Result: a regular bar magnet's north pole will point toward the north and a regular bar magnet's south pole will point towards the south ● Magnetic field of the earth is likely generated by the movement of charges in the earth's liquid core ● Direction reverses over time (~millions of years) ● G.P.II Fig. 7.1.5: The earth's magnetic field (same as the field produced by a bar magnet) along with geographic poles (axis of rotation) L07P1 Magnetic Fields A charged particle interacts with magnetic fields if it is moving: ● A stationary charged particle is not affected by magnetic fields ● It does not generate magnetic fields either Magnetic field can be defined in an analogous way to electric field: ● Electric field defined via E = F /q. E ● Magnetic field defined via magnetic force and electric charge, but also via particle velocity (magnitude and direction): FB B≡ q v sin θ G.P.II L07P1 Units of Magnetic Field Magnitude The SI unit for magnetic field strength (magnitude) in mks units is the Tesla (T) ● ● ● This is a very large field, as shown in Table 7.1.1. Alternative unit is the cgs unit Gauss (G) Equivalent to 1 T = 1 N/(C•m/s) and 1 T = 104 G. Table 7.1.1: Some approximate magnetic field strengths; field strengths at atomic nucleus are ~100 T, surface of a neutron are ~108 T. G.P.II L07P1 Magnetic Force The magnetic force exerted on a moving charge is thus: ⃗ B =q ⃗v × B ⃗ F ⇒ F B =qvB sin θ Magnetic force is in a direction which is perpendicular to both the magnetic filed direction and the particle's velocity ● Force magnitude is maximized if the velocity is perpendicular to the magnetic field, and minimized (=0) if the two are parallel ● Opposite charges experience opposite forces ● G.P.II Fig. 7.1.6: The force must be perpendicular to the magnetic field and to the particle's velocity. Direction can be found from the right-hand rule. L07P1 Right Hand Rule The direction of the magnetic force acting on a moving positivelycharged particle is found using the right-hand rule ● If the charge is negative, reverse the direction of the force Fig. 7.1.7: Two versions of the right-hand rule for determining direction of the magnetic force (FB) given the velocity (v) of a positive charge and magnetic field (B). G.P.II L07P1 Longer Example—Proton in Earth's Magnetic Field LE7.1.1. A proton is traveling perpendicularly to the earth's magnetic field near the earth's surface (strength: ~0.5 G). The field points due north and the proton is moving straight down. a) What is the magnitude of the magnetic force if the proton's speed is 1.00×104 m/s? b) What is the magnitude of the acceleration of the proton? c) In which direction will the magnetic force act? d) What changes in your answers if the particle is an electron instead? N.B. A proton has mass 1.67×10-27 kg and charge +1.60×10-19 C, whereas an electron has mass 9.11×10-31 kg and charge -1.60×10-19 C. G.P.II L07P1 Motion of a Charged Particle in a Uniform Magnetic Field The force acting on a moving charged particle is perpendicular to the instantaneous velocity This causes the particle to “turn,” so the force is a type of centripital force ● In a constant, uniform field this force results in the particle's moving in a helix ● “End-on” view is a circle, so can consider in terms of circular motion ● If v ┴ B, then motion will be circular only (not helical) ● G.P.II Fig. 7.1.8: The motion of a charged particle in a uniform and constant magnetic field is a helix L07P1 Motion of a Charged Particle in a Uniform Magnetic Field: Circular Component Let us consider the circular component of the particle's motion: Magnetic force supplies the centripital force, FC = FB. ● Recall that centripital force is given by FC = mac = m v2/r. ● Magnetic force is given by F =qvB. B ● Thus, the radius of the circular path is ● r=mv /qB ● Period is given by: T =2 π /ω=2 π m/qB G.P.II Fig. 7.1.9: End-on view: particle moves in a circle L07P1 Nonuniform Magnetic Field: Magnetic Mirrors The motion of a particle in a nonuniform magnetic field is more complex: Magnetic mirror: particle reflects from a region of high magnetic field strength to a region of low magnetic field strength ● It's typically a combination of the radial component of the field and the azimuthal (circular) component of the particle's velocity ● Perpendicular force will be in the opposite direction as the direction along which the particle is translating z ● G.P.II Fig. 7.1.10: A pair of magnetic mirrors make a magnetic bottle. Note that vz anf Fz are in opposite directions when moving from low field density to high field density. L07P1 Magnetic Bottles and the Van Allen Belt Two opposing magnetic mirrors can a) be combined to create a magnetic bottle: ● Particle become trapped between the two mirrors ● b) As each is mirror is approached, the particle's translational (z-) velocity is reversed ● The earth forms a magnetic bottle ● Charged particles ejected from the sun (solar wind, Fig. 7.1.11a) get trapped by the magnetic fields running between the poles of the earth (Fig. 7.1.11b) ● Fig. 7.1.11: Van Allen belts These are the Van Allen belts G.P.II L07P1 Aurora Borealis (1) The Aurora Borealis is is caused when charged ions from the solar wind approach the earth near the poles: ● Velocity component parallel to field lines carries the particles towards poles (until reflected) ● Perpendicular component confines the particles Fig. 7.1.12: So much space! In this photo of the night sky over Ifjord, Finland, we see stars, the Milky Way, the Aurora Borealis, and (look closely) some shooting stars. G.P.II L07P1 Aurora Borealis (2) Concentration of ions is greatest near poles: ● Because the field lines are most dense near the poles, this means that there will also be a greater concentration of particles here ● Hence, more collisions between charged particles and air molecules Fig. 7.1.12: So much space! In this photo of the night sky over Ifjord, Finland, we see stars, the Milky Way, the Aurora Borealis, and (look closely) some shooting stars. G.P.II L07P1 Aurora Borealis (3) High particle concentration means more collisions: ● Collisions between these charged particles and the air molecules cause ionization (electrons ripped free) ● Electrons re-combine with atoms in the air, releasing photons (light)—hence the northern lights! Fig. 7.1.12: So much space! In this photo of the night sky over Ifjord, Finland, we see stars, the Milky Way, the Aurora Borealis, and (look closely) some shooting stars. I.P.C. II L07P1 Lorentz Force The combined force from an electric field and a magnetic field is called the Lorentz force (FL): ⃗ L=q [ ⃗ ⃗ )] F E +(⃗v × B Motion of a particle in such a combination of fields is a corkscrew motion. ● However, the contribution to the motion from the E-field causes the particle to speed up (or slow down) ● If particle speeds up, gyration radius must increase FL = FE + FB FB ● G.P.II +q E F E B θ v Fig. 7.1.13: Lorentz force acting on a charged particle L07P1 Application: Cyclotron The cyclotron is an early type of particle accelerator invented by E.O. Lawrence and M.S. Livingston: Accelerates charged particles to very high speed ● Used both E-field and B-Field ● Charged particle moves in a spiral through two semi-circular, DShaped containers (called “dees”). ● E-Field accelerates the particle as it moves in the gap between the two dees ● B-Field confines the particle ● G.P.II Fig. 7.1.14: Cyclotron design L07P1 Application: Cyclotron (2) There are two parameters of interest for cyclotrons: ● Cyclotron frequency, which is also the frequency at which the electric field should alternate ωC =q B /m ● The exiting energy for an ion: K =mv 2 / 2=( q 2 B2 R 2 ) /(2 m) R is the radius if the dees Cyclotrons are often used in hospitals to produce radioactive substances for diagnosis and for treatment (e.g. radiation therapy) ● G.P.II Fig. 7.1.14: Cyclotron design L07P1 Application: Velocity Selector A velocity selector can be build by making the E-Field and B-Field perpendicular to each other: ● Both are also perpendicular to the initial particle velocity ● Particles pass through a slit to set up beam orientation ● F = qv×B acts to the left in Fig. B 7.1.15. ● F = qE acts to the right. E ● If (and only if) F = F , there will be E B no deflection ● This happens when v = E/B G.P.II Fig. 7.1.15: v selector L07P1 Application: Mass Spectrometer The velocity selector can be used to make a mass spectrometer: ● Particles pass through a second pinhole near the end of the velocity selector ● The particle's initial velocity and position are thus known ● An additional B-field is set up after the velocity selector ● This deflects the particle with a centripital acceleration a = v2/rm. ● Ratio of mass to charge is thus: m r B0 r B0 B = = q v E G.P.II ⃗v B ⃗ Ev Fig. 7.1.16: Mass deflection L07P1 Earth’s Magnetic Field Strength Fig. 7.1.17: Magnetic field strength contour lines G.P.II L07P1 Earth’s Magnetic Field Strength The earth’s magnetic field strength is not uniform everywhere at the earth’s surface ● The contour map to the right has magnetic field strengths near the surface ● Units are in nT, so 50000 on this map is 50000 nT or 0.5 G. ● The maximum field strength is just greater than 0.65 G, and the minimum is a bit less than 0.25 G Fig. 7.1.17b: Closeup of the magnetic contours for North America G.P.II L07P1 Summary of New Terms and Equations (1) ● ● ● ● ● G.P.II Magnetic fields are caused by charged particles in motion; they act upon other charged particle sin motion. The magnetic field strength is defined in terms of the magnitude of the magnetic force (FB) per unit charge (q) per speed (v) The magnetic force (FB) is the electrical charge (q) times velocity (v) crossed with the magnetic field (B) The magnetic force must be perpendicular to both the field and the charge velocity Charged particles moving in a magnetic field will experience a magnetic force which causes them to take on a motion with a circular component; the radius (r) and period (T) of the circular motion can be determined from the field strength (B), the mass (m), the charge (q), and the speed (v) of the particle FB B= q v sin θ ⃗ B =q ⃗v × B ⃗ F mv r= qB 2πm T= qB L07P1 Summary of New Terms and Equations (2) ● ● ● ● ● G.P.II Lorentz force (FL) is the total force experienced by ⃗ L=q E ⃗ + ( ⃗v × B ⃗) a charged particle moving through an electric ( E) F and magnetic (B) field. Soft magnetic materials are easy to magnetize when placed in an external magnetic field; however, they lose this magnetization quickly once the external magnetic field is removed Hard magnetic materials are more difficult to magnetize when placed in an external magnetic field; however, they retain their magnetization once the field is removed. The earth's magnetic field is comparable to that of a bar magnet with the south pole of the magnet at the north pole of the earth Opposite magnetic poles attract, and like poles repel [ ] L07P1 Lecture Set 07 Part 1 References (1) Fig. 7.1.1: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.3 (pp. 780). Fig. 7.1.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.1 (pp. 660). Fig. 7.1.3: S. Hannahs, “Permanent Magnet,” National High Magnetic Field Laboratory, 23 September 2014, web. Accessed 28 December 2015. Fig. 7.1.4: a) “Coaxial Cable Cutaway,” Wikimedia Commons, 02 January 2012, web. Accessed 28 December 2015. b) “Magnetic Shielding,” Excel @ Physics, 2014, web. Accessed 28 December 2015. Fig. 7.1.5: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 27-5 (pp. 709). Table 7.1.1: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Table. 29.1 (pp. 873). Fig. 7.1.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.6 (pp. 665). Fig. 7.1.7: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.5 (pp. 872). G.P.II L07P1 Lecture Set 07 Part 1 References (2) Fig. 7.1.8: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.9 (pp. 876). Fig. 7.1.9: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 27-17 (pp. 715). Fig. 7.1.10: “The Magnetic Bottle: How does a magnetic mirror reflect charged particles?” The XMM Satellite Schoolpage, Astrophysics and Space Research Group, The University of Birmingham, web. Accessed 31 December 2015. Fig. 7.1.11: a) E. Zolfagharifard, “Earth is being protected by a 'Star Trek-style invisible shield,': Scientists probe mysterious barrier blocking 'killer electrons,” Daily Mail, 26 November 2014, web. Accessed 31 December 2015. Image copyright I. Cuming, Ikon Images, Corbis. b) R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.12 (pp. 879). Fig. 7.1.12: “Spectacular displays of the northern lights or aurora borealis in northern Norway,” Telegraph, web. Accessed 31 December 2015. Photo by T. Eliassen, Barcroft Media. Fig. 7.1.14: “Cyclotron,” Hyperphysics, Department of Physics and Astronomy, Georgia State University, web. Accessed 31 December 2015. Fig. 7.1.16:R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.13 (pp. 880). G.P.II L07P1 Lecture Set 07 Part 1 References (3) Fig. 7.1.16: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.14 (pp. 880). Fig. 7.1.17: NOAA/NGDC and CIRES, “US/UK World Magnetic Model—Epoch 2015.0 Main Field Total Intensity,” Wikipedia/Wikimedia Commons, 14 February 2015, web. Accessed 18 May 2018. G.P.II L07P1 Part 2: Current-Carrying Conductors (Openstax 22.7-22.8) Magnetic fields are generated by moving charges, but they also influence other moving charges: ● This means that a current-carrying conductor (e.g. a wire) can produce a magnetic field ● First observed by Hans Christian Oersted ● A current-carrying wire which is in a magnetic field which it does not produce will experience a force from that field ● If the wire make a loop, then it may also experience a torque G.P.II Fig. 7.2.1: a) Hans Christian Oersted and b) a close model of his compass experiment L07P2 Magnetic Force on a Wire A current-carrying wire contains many moving charged particles: ● Magnetic field can exert a force on the charges in the wire provided that current is not parallel to the magnetic field ● Consequently the magnetic field exerts a force on the wire proportional to the magnitude of charge and the speed at which the charges are moving G.P.II Fig. 7.2.1: Magnetic field acts on a current-carrying wire and causes the wire to bend. The magnetic field is into the page, the current is 0 (a), up (b) or down (c). L07P2 Force on a Wire in an Uniform Magnetic Field The total magnetic force magnitude is therefore proportional to current: ● The force is actually given by F=qvdBnAcsLsinθ, where n is the density of charge carriers and Acs is the crosssection area of the wire (see Fig. 7.2.2) ● The current I is given by I=nqv A . d cs ● Therefore, the force acting on a wire of length L and current I in a magnetic field of strength B (Fig. 7.2.3) is ⃗ B =I ⃗L× B ⃗ F G.P.II Fig. 7.2.2 Fig. 7.2.3 F B =BIL sin θ L07P2 Short Examples SE7.2.1. A certain power line has a current of 20.0 A and a length of 1.0 km. The power line (and hence the current) is running from east-to-west. What is the magnitude and direction of the magnetic force acting on this line? SE7.7.2. If the linear density of the Fig. 7.2.4: Power lines above wires is 0.100 kg/m, how much which are running eastcurrent would the wires need for the west magnetic force to have an equal magnitude to gravity? G.P.II L07P2 Force on a Wire in a semi-Uniform Magnetic Field The previous expressions assume a uniform magnetic field ● Wire must also everywhere make the same angle with the field ● A more general case is given by ∑ F⃗ =∑ BILi sin(θi ) b⃗i This works if, for example, the wire makes a loop with discrete sides (as in Fig. 7.2.5). ● Force is applied to each side from the magnetic field—to get the total force, add up the forces for each side! ● G.P.II Fig. 7.2.5: Non-uniform B; really, non-uniform L (or I). L07P2 Application: Electromagnetic Pump for Artificial Heart or Kidneys One application of magnetic field force on a current-carrying wire is the electromagnetic pump (Fig. 7.2.6): Blood or other conducting fluids flows into the pump in one direction ● Current passes through the fluid perpendicularly to the direction in which it is to be pumped ● A constant magnetic field is applied in a third direction, perpendicular to both current and velocity ● Result: fluid is accelerated out of pump ● Does not risk damaging bloods cells as a mechanical pump would Fig. 7.2.6: Electromagnetic ● pump G.P.II L07P2 Longer Example—Electromagnetic Pump LE7.2.2. Blood has a density of 1060 kg/m3. Suppose you are designing an electromagnetic pump which has a cross-section area of 1.00 cm2 and a pumping length of 2.00 cm. Blood will slowly flow into the pump with vi=1.00 cm/s, but needs to leave with a speed of vf=2.00 cm/s. You are able to put a current of 1.00 mA through the blood without damaging it. Estimate the magnetic field strength will you need to use for this pump. G.P.II L07P2 Torque on a Current-Carrying Loop in a Uniform Magnetic Field When a current-carrying loop is placed in a magnetic field: Current direction is changing as it moves around the loop. ● Initially it may be in the +x direction, and after traversing 180o worth of loop it will be in the -x direction. ● Thus if the magnetic field is uniform, the force it exerts on the loop will not have a uniform direction ● Result: magnetic field exerts a net torque on the loop! ● G.P.II Fig. 7.2.8: A current-carrying loop may experience a net torque and thus begin to spin L07P2 Torque on a Current-Carrying Loop Consider the current loop in a uniform B-field in Fig. 7.2.9: ● Two sides are parallel to the B-field, so there is no force from the magnetic field on these sides ● Current in other two sides has a component which is perpendicular to the magnetic field: thus there is a nonzero force on each side of equal magnitudes Fig. 7.2.9 G.P.II L07P2 Torque on a Current-Carrying Loop (2) Since current is in opposite directions, the force points in opposite direction for these two sides, resulting in a torque: ● ⃗ ⇒ τ=r F sin θ Torque is given by ⃗τ =⃗r × F ● The forces are given by F =F =BIb (Fig. 7.2.9a) 1 2 ● If the wire rotates about point O (Fig. 7.2.9b), then the lengths will add to a. This is true for any axis of rotation! Fig. 7.2.8 G.P.II L07P2 Torque on a Current-Carrying Loop (3) The total torque is therefore given by: ● If the axis of rotation is some distance x from the end of the side of length a, then the torque is given by τ=τ1 + τ2=F 1 x+F 2 (a−x)=BIb x+BIb(a−x)=BIba ● The product of the lengths b and a is the area Aloop. Fig. 7.2.9 G.P.II L07P2 Torque on a Current-Carrying Loop (4) The magnetic field need not be perpendicular to one pair of sides—it may make an oblique angle as in Fig. 7.2.9c. ● τ=B I A loop sin θ The total magnitude of torque is therefore ● Finally, the loop may consist of N coils, in which case the torque becomes τ=B I A loop N sin θ Fig. 7.2.9 G.P.II L07P2 Magnetic Moment The quantity NIA is the magnitude of the loop's magnetic moment. The magnetic moment μB is a vector ● It is always perpendicular to the direction of the plane of the loop or loops ● Direction defined by right hand rule: curl fingers in the direction of the current in the loop, and the thumb of the right hand points in the direction of the magnetic moment ● The angle θ is therefore the angle between the magnetic field and the magnetic moment ● ⃗ ⇒ τ=μ B sin θ=BIAN sin θ The torque is therefore ⃗τ =μ ⃗×B Although this discussion has been limited to a rectangular loop, the loop's geometry is not actually important ● ⃗ U B =−μ ⃗⋅B Magnetic potential energy is given by ● G.P.II L07P2 Rotation in the Magnetic Field A net torque causes an angular acceleration, so the loops will rotate in the B-field: ● The torque will as a consequence change magnitude (Fig. 7.2.10a-b) ● There will also be a pair of equilibrium orientations (Fig. 7.2.10c) at which the torque is 0. ● Finally, the torque will change directions as the loop passes through the equilibrium positions (Fig. 7.2.10d) G.P.II L07P2 Short Examples SE7.2.3. Calculate the initial magnetic moment of a circular loop (single coil) of radius 1.00 m carrying a current of 10.0 mA in a magnetic field of strength 1.00 mT. Assume that the field is pointing to the right and that the loop is lying flat on the page with a clockwise current. SE7.2.4. Calculate the initial torque for the above loop. SE7.2.5. How do your answers change if the magnetic field is point up out of the page instead? G.P.II L07P2 Application: DC Motor If the current and magnetic field are both constant, then the loop will behave like an oscillator: ● For a small maximum angle θ , the loop will undergo simple max harmonic oscillations about it equilibrium ● A motor requires that the loop continues to rotate in the same direction ● This is accomplished by changing the current direction every halfcycle ● Accomplished using Fig. 7.2.11: Brushes with split-ring brushes (Fig. 7.2.11) commutator reverse current every 180o. G.P.II L07P2 Application: Galvanometer In principle, we could also use an ammeter to measure the induced current. What's the difference? ● Galvanometer is used to measure relatively small currents; ammeter is needed for larger currents (~1 mA or more) ● Galvanometer can detect direction of current ● Based on torque applied to a wire carrying current in a Bfield ● Can convert a Galvanometer into an ammeter by connecting a small resistor (“shunt resistance”) in parallel with it. ● Can also convert to a voltmeter by connecting a relatively large resistor in series with it. G.P.II L07P2 Application: Galvanometer In principle, we could also use an ammeter to measure the induced current. What's the difference? ● Galvanometer is used to measure relatively small currents; ammeter is needed for larger currents (~1 mA or more) Galvanometer can detect direction of current ● Based on torque applied to a wire carrying current in a B-field (Fig. 7.2.12) ● Fig. 7.2.12: Basic schematic of a Galvanometer G.P.II L07P2 Application: Z Machine for Fusion Research and X-Ray Generation (1) One interesting application of the Lorentz force is found in Sandia National Laboratory's ZMachine (Fig. 7.2.13a): ● A series of wires surrounds a DT fuel pellet ● A large current (Fig. 7.1.13 bd, orange lines) is driven through the wires, which vaporizes and then ionizes them to make a plasma ● The current continues through the plasma (an excellent conductor) G.P.II a) b) c) d) Fig. 7.1.13: a) The Z Machine, and b)-d) the physics of z-pinch L07P2 Application: Z Machine for Fusion Research and X-Ray Generation (2) The current runs in the Z- a) direction: ● Current generates a B-field (Figs. 7.1.13a-c, blue) ● Lorentz force compresses the wires or plasma (Fig. 7.1.13c) by applying a net “inward” force on it. ● You can check this by taking b) c) d) the cross product: z×θ = -r (cylindrical) or z×y=-x (Cartesian) using RHR ● This heats the plasma and generates x-rays (Fig. 8.1.13d) Fig. 7.1.13: a) The Z Machine, and b)-d) the physics of z-pinch G.P.II L07P2 Summary of New Terms and Equations ● ● ● ● ● G.P.II The force (FB) which a magnetic field (B) exerts on a current-carrying wire is determined by the current (I) in and length (L) of the wire The magnetic torque on a current-carrying loop of wire is given by the magnetic moment (μ) crossed with the magnetic field (B) The potential energy (UB) of a current-carrying loop of wire is given by the magnetic moment ( μ) dotted with the magnetic field (B) The magnetic moment (μ) of a current-carrying loop is determined by the current (I) in the loop, the number of turns (N), and the enclosed area (A) of each complete turn Thus, a current-carrying loop of N turns and enclosed area A per turn placed in a magnetic field will experience a torque (τ) ⃗ B =I ⃗L× B ⃗ F ⃗ ⃗τ =μ ⃗×B ⃗ U B =−μ ⃗⋅B μ A ⃗ =N I ⃗ τ=BIAN sin θ L07P2 Lecture Set 07 Part 2 References (1) Fig. 7.2.1: a) “Hans-Christian Oersted,” Indian Centre for Ocean Information Services (ESSO), web. Accessed 29 December 2015. b) A. Privat-Deschanel, “Elementary Treatise on Natural Philosophy, Part 3: Electricity and Magnetism” (1876), D. Appleton and Co.,New York, Tr. J.D. Everett, Fig. 456 (pp.656). Fig. 7.2.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.11 (pp. 668). Fig. 7.2.3: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.12 (pp. 668). Fig. 7.2.4: S. Salzberg, “Do High Voltage Power Lines Cause Cancer?”, Fighting Pseudoscience, Forbes, 01 September 2014, web. Accessed 28 December 2015. Fig. 7.2.5: My own drawing. Copyright J. Sanders, 2016. Fig. 7.2.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.14 (pp. 669). Fig. 7.2.7: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 27-52 (pp. 731). Fig. 7.2.8: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.15 (pp. 670). G.P.II L07P2 Lecture Set 07 Part 2 References (1) Fig. 7.2.9: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.35 (pp. 797). Fig. 7.2.10: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.36 (pp. 798). Fig. 7.2.11: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.37 (pp. 798). Fig. 7.2.12: R. Hamberly, “Laser Fusion: Lighting the Earth with Exploding Stars,” Stanford University, web. Accessed 04 January 2016. Fig a) from Fig 12, Figs. b)-d) from Fig. 10, of this source. Fig. 7.2.13: C. Freudenrich, “How Fusion Reactors Work—Fusion Reactors: Magnetic Confinement,” HowStuffWorks.com, 11 August 2005, web. Accessed 04 January 2016. G.P.II L07P2 Part 3: Sources of Magnetic Fields (Openstax 22.8-22.11, 22.6) We have so far considered the effects of magnetic fields on moving charges ● Now we turn to the sources of magnetic fields, which are other moving charges ● Can be produced by a single moving charge (does not effect this charge) ● Can be produced by a current, e.g. in a wire ● Can be produced a magnetized material G.P.II Yo dawg! Fig. 7.3.1: I heard you like magnets, so I put an electromagnet inside a horseshoe magnet…. L07P3 Magnetism Produced by Current A current carrying wire produces a magnetic field: ● First discovered by Hans Christian Oersted. Can use a compass to map magnetic field (Fig. 7.3.2). ● Direction can be found from the right-hand rule: point thumb in direction of current, and grab wire; fingers point in direction of B-field Fig. 7.3.2: Using a compass to map the magnetic field. When there is no current (a), the compasses all point northsouth, but when a current is applied (b) the compasses point in a circle around the wire. G.P.II L07P3 Magnetism Produced by Current (2) The strength (magnitude) of the magnetic field can be measured: ● A loop placed partially into the B-field can be used to measure field strength (Fig. 7.3.3). ● Alternatively, iron filings can be placed into the field to get relative field strength ● The magnetic field strength for a long straight wire is thus μ0 I B= 2πr Proportionality constant μ0=4π×10-7 T•m/A is permeability of free space G.P.II Fig. 7.3.3: A simple device for measuring magnetic field strength L07P3 Magnetic Force Between Two Long Parallel Wires Consider two long, straight wires carrying currents I1 and I2 as in Fig. 7.3.4: ● Each current produces a B-Field (the field for wire 1 wire shown, but not for wire 2) ● Magnitude of field produced by wire 1 at the location of wire 2 is B1 = μ0I1/(2πr) . ● This B-Field applies a force to the current (moving charges) in the other wire ● Can thus find force per unit length, F/L between the two wires Fig. 7.3.4a Fig. 7.3.4b μ0 I 1 I 2 F =I 2 B 1= L 2π r G.P.II L07P3 Short Examples SE7.3.1: A certain long wire carries a current of 1.00 A. What is the magnitude of the magnetic field 2.00 cm away from this wire? SE7.3.2: Two parallel wires are placed a distance of 2.00 cm apart. Each carries a current of 1.0 A, and both are 1.0 m long. What is the magnitude of the force between them? G.P.II Fig. 7.3.4a Fig. 7.3.4b L07P3 Operational Definitions for Units: Ampere and Coulomb The force between two parallel wires can be used to conveniently define units of current and charge: ● Current: one Ampere is the current needed to cause a force of 2×10-7 N/m to exist between two long parallel wires carrying equal current and separated by 1 meter ● Charge: one Coulomb is the charge passing through a cross section per second of a conduct carrying 1 A of steady current G.P.II Fig. 7.3.5: a) Currents in same direction pull wires together. b) Currents in opposite directions push wires apart L07P3 Ampere's Law The wire does not need to be straight to determine the magnetic field which it produces: ● Consider an arbitrary closed loop around a current (Fig. 7.3.6) ● The path consists of many small segments of length Δl ● The magnetic field has some component Bǁ which is parallel to the small segment of the loop ● The product of these can be added for all segments Fig. 7.3.6: An Amperian loop. G.P.II L07P3 Ampere's Law (2) Ampere's Law—discovered by Andre-Marie Ampere: ● States that the sum of these products ΔlBǁ is equal to the enclosed current Ienc times the permeability of free space: ∑ B ∥ Δ L=μ0 I enc ⃗ Note that Δ L B ∥ =Δ ⃗l⋅B ● Ampere's Law is only valid in this form if the currents and field are constant (not changing in Fig. 7.3.6: An Amperian loop. time) ● G.P.II L07P3 Right Hand Rules The right-hand rule can be used to determine the direction of the field: ● With a straight wire ● With an Amperian loop Fig. 7.3.7: The right-hand rule applied to two geometries G.P.II L07P3 Longer Example—A Coaxial Cable LE7.3.1. Consider a coaxial cable: it consists of an insulated wire of radius r = 0.400 mm which carries a current of 4.00 A, surrounded by a cylindrical conductor of inner radius 2.40 mm and thickness 0.200 mm which carries a current of 4.00 A in the opposite direction. Determine the magnetic fields magnitudes (and directions) at the following locations: a) At r = 0.100 mm (inside the wire) b) At r = 0.500 mm (between the wire and the conductor) c) At r = 2.50 mm (inside the conductor) d) At r = 2.60 mm (immediately outside of the conductor) Assume that the wire and the conductor each have a uniform current density. G.P.II L07P3 Solenoid A solenoid consist of a long currentcarrying wire which is twisted in the shape of a helix: The magnetic field inside a solenoid is shown in Fig. 7.3.9. ● Inside the solenoid, the field lines are ~uniform straight lines running parallel to the solenoid's central axis ● Outside, the lines resemble though of a bar magnet ● Tighter coiling means more uniform (straight) interior fields ● Ideal solenoid: turns are infinitesimally close and solenoid's length >> radius of Fig. 7.3.9: Solenoid B-field solenoid's turns ● G.P.II L07P3 B-Fields in a Solenoid Use Ampere's Law to determine the value of the B-field inside the solenoid: ● Take the Amperian loop to be the dashed rectangle in Fig. 7.3.10. ● Dashed circle gives field outside the solenoid, which is very weak for an ideal solenoid! ● ⃗⋅Δ ⃗L =BL The closed loop gives ∑ B ● And the enclosed current is I = NI, enc where N is the number of loops enclosed ● Ampere's Law thus gives (for n = N/L) B L=μ 0 N I G.P.II ⇒ B=μ 0 n I Fig. 7.3.10 L07P3 Torroid A torroid is a solenoid which is wrapped in a circle as in Fig. 7.3.11a: ● We can use Ampere's law both inside and outside of the torroid ● Outside the torroid, we follow path 2 ● The total current enclosed is 0 ● Thus, the B-field outside the torroid is also 0! ● Inside the torroid, we follow path 1 ● Enclosed current is I =NI enc ● Path length is L = 2πr ● Total field inside the torroid is thus B=μ 0 N I /2 π r G.P.II Fig. 7.3.11 L07P3 Short Examples SE7.3.3: A certain solenoid has a total length of 1.00 m and 500 turns. The wire of the solenoid carries a current of 1.50 A. What is the magnitude of the magnetic field produced inside this solenoid? SE7.3.4: A certain torroid has a total diameter of 2.00 m and 1250 turns. The wire of the torroid carries a current of 0.50 A. What is the magnitude of the magnetic field produced inside of this torroid? Fig. 7.3.9: Solenoid B-field G.P.II L07P3 Longer Examples LE7.3.2: A certain solenoid has a total length of 1.00m, an inner radius of 2.50 cm, and an outer radius of 2.60 cm. The wire of the solenoid carries a current of 1.50 A, and it is wound such that the space between consecutive wires is equal to the wire's radiu. What is the magnitude of the magnetic field produced by this solenoid? Fig. 7.3.9: Solenoid B-field G.P.II L07P3 Gauss's Law in Magnetism Ampere's Law for B-Fields is the functional or operational equivalent to Gauss's law for EFields ● However, Gauss's Law can apply for any field, including B-Fields (Fig. 7.3.12a) ● Recall that the flux of a field is found using ( V is a vector field) ⃗ A ⃗ ΦV ≡V⋅ a) b) Unit of magnetic flux is T•m2 = Wb (weber) ● For magnetic field flux through a closed surface, Gauss's law gives Φ B =0 ● The implication is that there are no isolated magnetic monopoles in nature (Fig. 7.3.12b) ● Fig. 7.3.12: The flux G.P.II L07P3 Magnetic Materials The field B0 inside a a solenoid is given by B0 = μ0nI if the solenoid is filled with vacuum: ● Can fill the space of the solenoid with other material (like iron) ● The total magnetic field of the solenoid will be increased in this case. ● The total field can be written as: ⃗ =⃗ ⃗M B B0 + B ● G.P.II BM is the field due to the magnetic material Fig. 7.3.13: The total Bfield in an iron-core toroid as a function of Bo. L07P3 Magnetic Domains: Understanding the Enhancement of the B-Field (1) Certain materials like iron (or nickel, cobalt, and their alloy) can be made into strong magnets ● This enhances the externally produced magnetic field of the solenoid ● Such materials consist of small magnetic domains, each behaving like a tiny bar magnet with a north and south pole ● Bar magnet is a magnetic dipole, magnetic analogue to the electric dipole G.P.II Fig. 7.3.14: Magnetic domains in an unmagnetized and in a magnetized piece of material. Arrow heads point toward the north pole of the magnetic domain. L07P3 Magnetic Domains: Understanding the Enhancement of the B-Field (2) Domains' alignment determines whether the whole material will act as a magnet or not: ● If the domains are oriented randomly (Fig 7.3.14 left), the material will be unmagnetized ● Individual domains will cancel with each other ● If the individual domains are oriented in a common direction, the material will be magnetized ● Individual domains will largely add to each other to enhance the field strength. G.P.II Fig. 7.3.14: Magnetic domains in an unmagnetized and in a magnetized piece of material. Arrow heads point toward the north pole of the magnetic domain. L07P3 Magnetic Domains: Understanding the Enhancement of the B-Field (3) Unmagnetized pieces of ferromagnetic material can become magnetized by placing them in a sufficiently strong B-field Domains align with the existing external field (Fig. 7.3.14 right) ● This also explains why a permanent magnet can pick up “unmagnetized” pieces of iron ● Domains in iron realign with external field ● Conversely, if a permanent magnet is jarred sufficiently, it demagnetizes ● G.P.II Fig. 7.3.14: Magnetic domains in an unmagnetized and in a magnetized piece of material. Arrow heads point toward the north pole of the magnetic domain. L07P3 The Hall Effect (1) Consider a conductor carrying a current in a B-field (Fig. 7.3.15): ● Magnetic field applies a force to the charge carriers in the current ● The charge-carriers will deflect as a result of this ● This ultimately means that net charges will begin to accumulate along edges of the conductor ● As net charges accumulate, a potential difference builds between the two opposite edges Fig. 7.3.15: Hall effect acting ● G.P.II This is the Hall voltage ΔVH on electrons in the conductor L07P3 The Hall Effect (2) The existence of a potential difference across a gap implies the existence of an electric field, ΔVH = Ehd: ● This E-field will apply a force on the charge carriers which acts in the opposite direction as the B-Field's force ⃗ B=− F ⃗ EH ⇒ q ⃗v d × B ⃗ =q E ⃗H F Where vd is the charge-carrier drift-velocity ● Therefore, the potential difference across the conductor (between points C and D in Fig. 7.3.15) is Δ V H =v d B d ● G.P.II Fig. 7.3.15: Hall effect acting on electrons in the conductor L07P3 Hall Probe We can use the hall effect to probe magnetic field strength: Need to measure the Hall voltage VH between points C and D (Fig.7.3.16) ● The drift speed can be expressed as (Lecture Set 05) ● I=n q v d A ⇒ v d =I /(nqA ) A is the cross-sectional area of the conductor, n is the charge carrier density ● A = wd in Fig. 7.3.17. ● Can measure the current I ● The Hall voltage is thus D C ● IB R H IB Δ V H= = nqw w G.P.II L d w Fig. 7.3.16: Hall effect acting on electrons in the conductor L07P3 Longer Example: Hall Probe LE7.3.3: A rectangular silver strip is 2.5 cm long and 0.095 cm thick. It carries a current of 4.0 A. Determine the following: D C a) Hall coefficient RH = 1/nq. b) Hall voltage when this strip is placed in a magnetic field of L magnitude 0.75 T such that the Hall voltage is developed between top and w bottom edges of the strip (e.g. d between the edges which are 2.5 cm Fig. 7.3.16: Hall effect acting apart). c) What magnetic field strength would on electrons in the conductor produce a Hall voltage of 100 nV? G.P.II L07P3 Summary of New Terms and Equations (1) ● ● ● ● ● G.P.II Magnetic fields are ultimately produced by moving charges (e.g. electrical current) The magnetic field strength (B) produced by a current-carrying wire is given by the current (I) at a distance r from the wire The constant μ0 is called the permeability of free space, and it is a fundamental constant of the universe, μ0=4π×10-7 T•m/A. The force per unit length of two current-carrying wires is dependent on their separation distance (r) and the currents in each wire ( I1 and I2 respectively) Ampere's Law states that the sum of the products of the magnetic field and the path length about any closed loop is equal to the total current (I) enclosed by the loop. ⃗ B =I ⃗L× B ⃗ F μ0 I B= 2πr F μ0 I 1 I 2 = L 2π r ∑ B⃗⋅Δ ⃗l =μ0 I enc L07P3 Summary of New Terms and Equations (2) ● ● G.P.II Magnetic field produced by a solenoid is dependent on the current (I) and the number of turns per unit length (n) for the solenoid Gauss's Law for magnetic field states the the flux of the magnetic field through a closed surface is zero, the implication of which is that there are no magnetic monopoles. B=μ 0 n I ⃗⋅⃗ ΦB≡ B A =0 L07P3 Lecture Set 07 Part 3 References (1) Fig. 7.3.1: M. Brain, “How Electric Motors Work: Electromagnets and Motors,” HowStuffWorks.com, 01 April 2000, web. Accessed 31 December 2015. Inset from Yo Dawg.com. Fig. 7.3.2: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.23 (pp. 676). Fig. 7.3.3: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 27-13 (pp. 713). Fig. 7.3.4: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.38 (pp. 799). and Fig. 22.42 (pp. 804) Fig. 7.3.5: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 28-6 (pp. 735). Fig. 7.3.6: R. Serway and C. Vuille, College Physics, 10th Ed. (2015), Cengage Learning (Brooks and Cole), Fig. 19.25 (pp. 678). Fig. 7.3.7: P. Urone, R. Hinrichs, K. Dirks, and M. Sharma, College Physics, OpenStax College, Fig. 22.38 (pp. 799). and Fig. 22.39 (pp. 800). Fig. 7.3.9: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 28-15 (pp. 741). G.P.II L07P3 Lecture Set 07 Part 3 References (2) Fig. 7.3.10: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 30.18 (pp. 916). Fig. 7.3.11: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 28-17 (pp. 742). Fig. 7.3.12: a) R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 30.19 (pp. 917). b) R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 30.22 (pp. 918). Fig. 7.3.13: D.C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 4th Ed. (2008), Prentice Hall (Pearson), Fig. 28-28 (pp. 748). Fig. 7.3.14: A.D. Elster, “Ferromagnetism: What is Ferromagnetism,” Questions and Answers in MRI, 2014, web. Accessed 05 January 2016. Fig. 7.3.15: R. Serway and J. Jewett, Physics for Scientists and Engineers 9th Ed. (2014), Cengage Learning (Brooks and Cole), Fig. 29.26 (pp. 890). Fig. 7.3.16: A. Ajaja, “The Hall Effect,” 2000, web. Accessed 06 January 2016. G.P.II L07P3
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