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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) • Appendix: Nuclear Magnetic Resonance Imaging • Solar Flare/Aurora Borealis Pictures

  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. 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

  10. 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:

  11. 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:

  12. 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!).

  13. 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.

  14. 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

  15. 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

  16. How can we increase the speed (rpm) of a DC motor? B 2A (a)IncreaseI (b) IncreaseB (c)Increase number of loops Lecture 13, Act 2

  17. 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 X mx=–m sin 30=–.0079 Am2 mz=mcos 30=.0136 Am2 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= 30to the xy-plane. Example: Loop in a B-Field What is themagnetic momentmof the loop? m = pr2 I = .0157 Am2 z m B q X y x

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

  19. 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

  20. 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

  21. Mass Spectrometer Force due to B on I Magnetic dipole torque potential energy Applications: dials, motors, NMR, … Nexttime:calculatingB-fieldsfrom currents Summary

  22. 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)[MRI invented by UIUC Chem. Prof. Paul Lauterbur,who shared 2003 Nobel Prize in Medicine] 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.

  23. h ≡ 6.63•10-34 J s Aligned: Anti-aligned: ? What does this have to do with Energy Difference: MRI / NMR Example μproton=1.41•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

  24. 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

  25. See it in action! Thanks to

  26. Solar Flare/Aurora Borealis links http://cfa-www.harvard.edu/press/soolar_flare.mov http://science.nasa.gov/spaceweather/aurora/gallery_01oct03_page2.html

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