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Forces & Magnetic Dipoles B x q F F q . m m AI mB U m B 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 Last Time… • Moving charged particles are deflected in magnetic fields F q v B mv • Circular orbits R qB • If we use a known voltage V to accelerate a particle q 2V 2 1 2 2 qV 2 mv m R B • 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 Mass Spectrometer • Measure m/q to identify substances m R2 B2 q 2V 1. Electrostatically accelerated electrons knock electron(s) off the atom positive ion (q=|e|) 2. Accelerate the ion in a known potential U=qV 3. Pass the ions through a known B field – Deflection depends on mass: Lighter deflects more, heavier less Mass Spectrometer, cont. 4. Electrically detect the ions which “made it through” 5. Change B (or V) and try again: 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.). See http://www.colby.edu/chemistry/OChem/DEMOS/MassSpec.html Yet another example • Measuring curvature of charged particle in magnetic field is usual method for determining momentum of particle in modern experiments: e.g. e+ mv 2mK R qB qB - charged particle + charged particle B e- End view: B into screen Magnetic Force on a Current • Consider a current-carrying wire in the N presence of a magnetic field B. • There will be a force on each of the charges moving in the wire. What will be the total force dF on a length dl of the wire? • Suppose current is made up of n charges/volume each carrying charge q and moving with velocity v through a wire of cross-section A. • Force on each charge = qv B • Total force = dF nA(dl )qv B • Current = dq nAv(dt )q I nAvq dt dt Yikes! Simpler: For a straight length of wire L carrying a current I, the force on it is: dF Idl B F IL B S Magnetic Force on a Current Loop • Consider loop in magnetic field as on right: If field is ^ to plane of loop, the net force on loop is 0! – Force on top path cancels force on bottom path (F = IBL) x x Fx x x x x x x x x x x x x – Force on right path cancels force on left path. (F = IBL) • If plane of loop is not ^ to field, there will be a non-zero torque on the loop! x x x x x B x x x x F x B x F 1 F x x x x x x x x x xI F F . Lecture 13, Act 1 • A current I flows 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? (a) Fy < 0 (b) Fy = 0 y B (c) Fy > 0 x x x x x x x x x x x x x x x Lx x x x2L x x x x x x x x Lx x x x x x x x x x x x x x Calculation of Torque • Suppose the coil has width w (the side we see) and length L (into the screen). The torque is given by: τ r F w τ 2 F sin q 2 F = IBL where AIB sinq B x q w F F . r A = wL = area of loop F • Note: if loop ^ B, sinq = 0 = 0 maximum occurs when loop parallel to B r ×F Magnetic Dipole Moment • We can define the magnetic dipole moment of a current loop as follows: magnitude: m AI q direction: ^ to plane of the loop in the direction the thumb of right hand points if fingers curl in the direction of current. B x F q F m • Torque on loop can then be rewritten as: τ μ B AIB sinq • Note: if loop consists of N turns, m = NAI . Bar Magnet Analogy • You can think of a magnetic dipole moment as a bar magnet: μ N = – 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!). 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 Current increased μ = I • Area increases Torque from B increases Angle of needle increases Current decreased μ decreases Torque from B decreases Angle of needle decreases This is how almost all dial meters work—voltmeters, ammeters, speedometers, RPMs, etc. Motors Free rotation of spindle B 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 its μ is ~antialigned with B Turn current back on Magnetic torque gives another kick to the loop Continuous rotation in steady state Motors, continued Even better Have the current change directions every half rotation Torque acts the entire time Two ways to change current in loop: 1. 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 VS I t 2 Lecture 13, Act 2 • How can we increase the speed (rpm) of a DC 2A motor? (a) Increase I (b) Increase B B (c) Increase number of loops Example: Loop in a B-Field A circular loop has radius R = 5 cm and carries current I = 2 A in the counterclockwise direction. A magnetic field B =0.5 T exists in the negative z-direction. The loop is at an angle q = 30 to the xy-plane. B y x x x x x x x z x x x x x x xI x x x x x x x What is the magnetic moment m of the loop? x x x x x x x m = p r2 I = .0157 Am2 The direction of m is perpendicular to the plane of the loop as in the figure. Find the x and z components of m : x z m mx = –m sin 30 = –.0079 Am2 X B mz = m cos 30 = .0136 Am2 q yX x Electric Dipole Analogy +q F E q p F τ r F F qE B x . -q F q F . m τ r F F IL B (per turn) p 2qa μ NAI τ p E τ μ B Potential Energy of Dipole • Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. q F • Define a potential energy U (with zero at position of max torque) corresponding to this work. θ U τdθ 90 B x F q . m θ Therefore, U μB cos θ θ 90 U μB sin θdθ 90 U μB cos θ U μ B Potential Energy of Dipole m B m x B x m x =0 = mB X =0 U = -mB U=0 U = mB negative work positive work B Summary • Mass Spectrometer • Force due to B on I dF I dl B • Magnetic dipole m AIN – torque mB – potential energy U m B zero defined at q 90 • Applications: dials, motors, NMR, … • Next time: calculating B-fields from currents 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. In an external B-field: – Classically: there will be torques unless m is aligned along B or against it. – QM: The spin is always ~aligned along B or against it Aligned: U1 m B Anti-aligned: U 2 m B Energy Difference: U U 2 U1 2m B MRI / NMR Example Aligned: U1 m B Anti-aligned: U 2 m B Energy Difference: U U 2 U1 2m B μproton = 1.41•10-26 Am2 B = 1 Tesla (=104 Gauss) U 2m B 2.82 10 J 26 (note: this is a big field!) In QM, you will learn that photon energy = frequency • Planck’s constant h ≡ 6.63•10-34 J s 2.82 1026 J f 42.5 MHz -34 6.63 10 J s What does this have to do with ? MRI / NMR continued 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 f U B, if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution. B Small B low freq. Bigger B high freq. Signal at the right frequency only from this slice! See it in action! Thanks to Solar Flare/Aurora Borealis links http://cfa-www.harvard.edu/press/soolar_flare.mov http://science.nasa.gov/spaceweather/aurora/gallery_01oct03_page2.html