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Forces & Magnetic Dipoles

B. x. q. F. F. q. m. Forces & Magnetic Dipoles. Today. Application of equation for trajectory of charged particle in a constant magnetic field: Mass Spectrometer Magnetic Force on a current-carrying wire Current Loops Magnetic Dipole Moment Torque (when in constant B field)  Motors

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Forces & Magnetic Dipoles

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  1. B x q F F . q m Forces & Magnetic Dipoles

  2. Today... • Application of equation for trajectory of charged particle in a constant magnetic field: Mass Spectrometer • Magnetic Force on a current-carrying wire • Current Loops • Magnetic Dipole Moment • Torque(when in constant B field)  Motors • Potential Energy(when in constant B field) • Nuclear Magnetic Resonance Imaging Text Reference: Chapter 28.1 and 28.3 Examples: 28.2, 28.3, and 28.6 through 28.10

  3. Moving charged particles are deflected in magnetic fields • Circular orbits • If we use a known voltage V to accelerate a particle Last Time… • Several applications of this • Thomson (1897) measures q/m ratio for “cathode rays” • All have same q/m ratio, for any material source • Electrons are a fundamental constituent of all matter! • Accelerators for particle physics • One can easily show that the time to make an orbit does not depend on the size of the orbit, or the velocity of the particle • Cyclotron

  4. Measure m/q to identify substances Mass Spectrometer • Electrostatically accelerated electrons knock electron(s) off the atom  positive ion (q=|e|) • Accelerate the ion in a known potential U=qV • Pass the ions through a known B field • Deflection depends on mass: Lighter deflects more, heavier less

  5. Electrically detect the ions which “made it through” • Change B (or V) and try again: Mass Spectrometer, cont. Applications: Paleoceanography: Determine relative abundances of isotopes (they decay at different rates  geological age) Space exploration: Determine what’s on the moon, Mars, etc. Check for spacecraft leaks. Detect chemical and biol. weapons (nerve gas, anthrax, etc.). Seehttp://www.colby.edu/chemistry/OChem/DEMOS/MassSpec.html

  6. - charged particle + charged particle Yet another example • Measuring curvature of charged particle in magnetic field is usual method for determining momentum of particle in modern experiments: e.g. B e+ e- End view: B into screen

  7. Consider a current-carrying wire in the presence of a magnetic fieldB. There will be a force on each of the charges moving in the wire. What will be the total force dFon a lengthdlof the wire? Suppose current is made up ofncharges/volume each carrying chargeqand moving with velocityvthrough a wire of cross-sectionA. • Force on each charge = • Total force = Þ • Current = Yikes! Simpler: For a straight length of wire L carrying a current I, the force on it is: Magnetic Force on a Current N S

  8. Consider loop in magnetic field as on right: If field is ^to plane of loop, the net force on loop is 0! F B x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x F F • Force on top path cancels force on bottom path(F = IBL) I F • Force on right path cancels force on left path.(F = IBL) B x • If plane of loop is not ^ to field, there will be a non-zero torque on the loop! F F . Magnetic Force on a Current Loop

  9. Preflight 13: A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. 2) What is the force on section a-b of the loop? a) zero b) out of the page c) into the page 3) What is the force on section b-c of the loop? a) zero b) out of the page c) into the page 4) What is the net force on the loop? a) zero b) out of the page c) into the page

  10. By symmetry:  ab:Fab = 0 = Fcdsince the wire is parallel toB. bc:Fbc = ILBRHR:Iis up,Bis to the right, soF points into the screen.

  11. A currentIflows in a wire which is formed in the shape of an isosceles right triangle as shown. A constant magnetic field exists in the-z direction. What is Fy, net force on the wire in the y-direction? y x x x x x x x x x x x x x x x L L x x x x x B x x x x x x x x x x L x x x x x (a) Fy < 0 x x x x x (c) Fy > 0 (b) Fy = 0 x Lecture 13, Act 1

  12. A currentIflows in a wire which is formed in the shape of an isosceles right triangle as shown. A constant magnetic field exists in the-z direction. What is Fy, net force on the wire in the y-direction? y x x x x x x x x x x x x x x x L L x x x x x B x x x x x x x x x x L x x x x x (a) Fy < 0 x x x x x (c) Fy > 0 (b) Fy = 0 x • The forces on each segment are determined by: • From symmetry,Fx = 0 • For the y-component: 45˚ F1 F2 • Therefore: F3 Lecture 13, Act 1 There is never a net force on a loop in a uniform field!

  13. Note: if loop^B, sinq = 0Þt = 0 maximumtoccurs when loopparallel toB B x q w . Þ Þ t = AIBsinq F = IBL where A = wL = area of loop Calculation of Torque • Suppose the coil has widthw(the side we see) and lengthL(into the screen). The torque is given by:

  14. Current increased • μ=I • Area increases • Torque fromBincreases • Angle of needle increases Current decreased • μdecreases • Torque fromBdecreases • Angle of needle decreases Applications: Galvanometers(≡Dial Meters) We have seen that a magnet can exert a torque on a loop of current – aligns the loop’s “dipole moment” with the field. • In this picture the loop (and hence the needle) wants to rotate clockwise • The spring produces a torque in the opposite direction • The needle will sit at its equilibrium position This is how almost all dial meters work—voltmeters, ammeters, speedometers, RPMs, etc.

  15. B Free rotation of spindle Motors Slightly tip the loop Restoring force from the magnetic torque Oscillations Now turn the current off, just as the loop’s μ is aligned with B Loop “coasts” around until itsμ is ~antialigned withB Turn current back on Magnetic torque gives another kick to the loop Continuous rotation in steady state

  16. VS I t Motors, continued • Even better • Have the current change directions every half rotation • Torque acts the entire time • Two ways to change current in loop: • Use a fixed voltage, but change the circuit (e.g., break connection every half cycle •  DC motors • 2. Keep the current fixed, oscillate the source voltage • AC motors

  17. What should we do to increase the speed (rpm) of a DC motor? B 2A (a)IncreaseI (a)IncreaseI (b) IncreaseB (b) IncreaseB (c)Increase number of loops (c)Increase number of loops 2B Lecture 13, Act 2 • What should we do to increase the speed of an AC motor?

  18. What should we do to increase the speed (rpm) of a DC motor? B 2A (a)IncreaseI (b) IncreaseB (c)Increase number of loops Lecture 13, Act 2 All of the above, though for variable speed motors, it is always the current that is adjusted.

  19. What should we do to increase the speed (rpm) of an AC motor? B 2B (a)IncreaseI (b) IncreaseB (c)Increase number of loops Lecture 13, Act 2 Again, all of the above! BUT, while the answers above are still true, the main change is to vary the drive frequency—this must match the rotation rate.

  20. direction:^to plane of the loop in the direction the thumb of right hand points if fingers curl in the direction of current. m = AI magnitude: B x q . q • Torque on loop can then be rewritten as: Þ t = AIBsinq • Note: if loop consists of N turns,m= NAI Magnetic Dipole Moment • We can define the magnetic dipole moment of a current loop as follows:

  21. μ N = • You can think of a magnetic dipole moment as a bar magnet: Bar Magnet Analogy • In a magnetic field they both experience a torque trying to line them up with the field • As you increase I of the loop  stronger bar magnet • N loops  N bar magnets • We will see next lecture that such a current loop does produce magnetic fields, similar to a bar magnet. In fact, atomic scale current loops were once thought to completely explain magnetic materials (in some sense they still are!).

  22. E B +q x q . . q -q (per turn) Electric Dipole Analogy

  23. Preflight 13: A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. 6) What is the net torque on the loop? a) zero b) up c) down d) out of the page e) into the page

  24. The torque due toFcbis up, sincer(from the center of the loop) is to the right, andFcbis into the page. Same goes forFda. mpoints out of the page (curl your fingers in the direction of the current around the loop, and your thumb gives the direction of m). Use the RHR to find the direction of t to be up.

  25. Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. B x q . q • Define a potential energy U (with zero at position of max torque) corresponding to this work. Þ Therefore, Þ Þ Potential Energy of Dipole

  26. m x B B B m m t = mB x x X negative work positive work Potential Energy of Dipole t = 0 U = -mB t = 0 U = mB U = 0

  27. Preflight 13: Two current carrying loops are oriented in a uniform magnetic field. The loops are nearly identical, except the direction of current is reversed. 8) What direction is the torque on loop 1? a) clockwise b) counter-clockwise c) zero 9) How does the torque on the two loops compare? a) τ1 > τ2 b) τ1 = τ2 c) τ1 < τ2 10) Which loop occupies a potential energy minimum, and is therefore stable? a) loop 1 b) loop 2 c) the same

  28. Loop 1: U1 = +mB Loop 2: U2 = -mBU2is a minimum. Loop 1: m points to the left, so the angle between mand B is equal to 180º, hence t = 0. Loop 2: m points to the right, so the angle between mand B is equal to 0º, hence t = 0.

  29. A circular loop of radiusRcarriescurrentI as shown in the diagram. A constantmagnetic fieldBexists in the+xdirection.Initially the loop is in the x-y plane. The coil will rotate to which of the following positions? y B R 3A b I a x 3B Whatis the potential energyU0of the loop in its initial position? (c)neither (b)U0 is maximum (a)U0 is minimum Lecture 13, Act 3 y y w w (b) (a) (c)It will not rotate b a b a z z

  30. A circular loop ofradius R carriescurrent Ias shown in the diagram. A constantmagnetic field Bexists in the+x direction. Initially the loop is in the x-y plane. The coil will rotate to which of the following positions? y B R 3A b I a x y y w w b a b a z z • The coil will rotate if the torque on it is non-zero: • The magnetic momentm is in+z direction. • Therefore the torquet is in the+y direction. • Therefore the loop will rotate as shown in (b). Lecture 13, Act 3 (b) (a) (c)It will not rotate

  31. A circular loop ofradius Rcarriescurrent Ias shown in the diagram. A constantmagnetic field Bexists in the+x direction. Initially the loop is in the x-y plane. What is the potential energy U0 of the loop in its initial position? y B R b I a x • The potential energy of the loop is given by: • In its initial position, the loop’s magnetic moment vector points in the +z direction, so initial potential energy is ZERO • This does NOT mean that the potential energy is a minimum!!! • When the loop is in the y-z plane and its magnetic moment points in the same direction as the field, its potential energy is NEGATIVE and is in fact the minimum. • Since U0 is not minimum, the coil will rotate, converting potential energy to kinetic energy! 3B (c)neither (b)U0 is maximum (a)U0 is minimum Lecture 13, Act 3

  32. B y x x x x x x x x x x x x x x x x x x x x x x x x x x x x I The direction ofmis perpendicular to the plane of the loop as in the figure. Find thexandzcomponents ofm: x z z m X mx=–m sin45=–.0111 Am2 B mz=mcos45=.0111 Am2 q X y x A circular loop has radiusR = 5 cmand carries currentI = 2 Ain the counterclockwise direction. A magnetic fieldB =0.5 T exists in the negative z-direction. The loop is at an angleq= 45to the xy-plane. Example: Loop in a B-Field What is themagnetic momentmof the loop? m = pr2 I = .0157 Am2

  33. Mass Spectrometer Force due to B on I Magnetic dipole torque potential energy Applications: dials, motors, NMR, … Nexttime:calculatingB-fieldsfrom currents Summary Reading assignment: Chapter 29.1 through 29.3

  34. In an external B-field: • Classically: there will be torques unless is aligned along B or against it. • QM: The spin is always ~aligned along B or against it Aligned: Anti-aligned: Energy Difference: MRI (Magnetic Resonance Imaging) ≡ NMR (Nuclear Magnetic Resonance) A single proton (like the one in every hydrogen atom) has a charge (+|e|) and an intrinsic angular momentum (“spin”). If we (naively) imagine the charge circulating in a loop  magnetic dipole moment μ.

  35. h ≡ 6.6•10-34 J s Aligned: Anti-aligned: ? What does this have to do with Energy Difference: MRI / NMR Example μproton=1.36•10-26 Am2 B = 1 Tesla (=104 Gauss) (note: this is a big field!) In QM, you will learn that photon energy = frequency • Planck’s constant

  36. B Small B low freq. Bigger B high freq. Signal at the right frequency only from this slice! • If we “bathe” the protons in radio waves at this frequency, the protons can flip back and forth. • If we detect this flipping  hydrogen! • The presence of other molecules can partially shield the applied B, thus changing the resonant frequency (“chemical shift”). • Looking at what the resonant frequency is  what molecules are nearby. • Finally, because , if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution. MRI / NMR continued

  37. See it in action! Thanks to

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