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Physics 1402.001 - Week Eight The Magnetic Field - Chapter 26 Magnetic Forces on Currents Magnetic Energy Hall Effect Torques on Current Loops Magnetic Fields Office Hour • My office hour this Tuesday March 7, from 3:30 – 4:30 PM, will be moved to Wednesday March 8, at the same time. • No Office Hours during Spring Break Magnetic Fields • Magnets occur naturally in Nature Magnetic Fields • Magnets occur naturally in Nature • Magnets ALWAYS come with BOTH North and South poles No Magnetic Single Charges Force Exerted by Magnetic Fields Magnetic Fields • Magnets occur naturally in Nature • Magnets always come with BOTH North and South poles • Unlike Electric Charges, ALL magnets are Dipoles • There are no Magnetic Monopoles Gauss’ Law for Magnetism • Magnetic flux Φ = Magnets - How Do They Work? ∫B•n dA • There are no net converging or diverging magnetic field lines • Therefore Φ = ∫B•n dA = 0 Force Exerted by Magnetic Fields • Moving electric charges create magnetic fields (More on this in Chapter 27) • If moving charges are in presence of external magnetic field, these two magnetic fields can interact • Net effect is that a charge moving in an external magnetic field feels an additional Force! Force Exerted by Magnetic Fields Force Exerted by Magnetic Fields Force Exerted by Magnetic Fields • F = q v x B • Newton = (Coul)(meter/sec)B • 1 Tesla = Newton/Coul-m/sec • 1Tesla = Newton/Amp-m = 104 Gauss Force on Moving Proton Force on Moving Proton Force on Moving Proton Clicker Question No. One • Force = q v x B q = + 1.6 x 10-19 Coul • V = 107 m/sec j |B| = 0.6 G = 6 x 10-5 T • B = By j + Bz k = (Bcosθ)j + (-Bsinθ)k • (j x j ) = 0 (j x –k) = - i • F = (1.6 x 10-19 Coul)(107 m/sec)(-6 x 10-5 T sin70°)i • F = - 9 x 10-17 N i The left diagram shows a positively charged particle is moving with velocity v in a magnetic field B. Using the arrows in the right diagram, what is the direction of the magnetic force on the particle? Clicker Question No. One Clicker Question No. Two The left diagram shows a positively charged particle is moving with velocity v in a magnetic field B. Using the arrows in the right diagram, what is the direction of the magnetic force on the particle? The left diagram shows a force F on a negatively charged particle moving a magnetic field B. Using the arrows in the right diagram, what is the direction of the velocity of the particle? Clicker Question No. Two Force on Current-Carrying Wire The left diagram shows a force F on a negatively charged particle moving a magnetic field B. Using the arrows in the right diagram, what is the direction of the velocity of the particle? I = n q Vol/time = n q A L/time = n q A vd Force on Current-Carrying Wire Force on Current-Carrying Wire B = Bx i + By j = Bcos30°i + Bsin30° j L = 3 mm = 3 x 10-3 m I = 3.0 A B = 0.02 T F = (q vd x B)n A L I = n q A vd F = (3 A)(3 x 10-3 m)(0.02T)(sin30°) (i x j) F=ILxB dF = I dL x B F = 9 x 10-5 N k Physics 1402.001 - Week Eight Force on Curved Wire The Magnetic Field - Chapter 26 Magnetic Forces on Currents Magnetic Energy Hall Effect Torques on Current Loops Force on Curved Wire Point Charge Moving in Uniform Magnetic Field • F = qvB = ma = mv2/R • R = mv/qB • (2πR) = v τ = (2πmv/qB) • τ = 2πmv/vqB = 2πm/qB • ω = 2πf = 2π/τ = qB/m Point Charge Moving in Uniform Magnetic Field Clicker Question No. Three A particle with charge q and mass m is moving with speed v in the +x direction enters a magnetic field of strength B pointing in the +y direction. The work done by the magnetic force on the particle as it travels one semi-circle is A. B. C. D. E. πmqvB πmv2 πqvB zero πmv/qB Clicker Question No. Three A particle with charge q and mass m is moving with speed v in the +x direction enters a magnetic field of strength B pointing in the +y direction. The work done by the magnetic force on the particle as it travels one semi-circle is A. B. C. D. E. πmqvB πmv2 πqvB zero πmv/qB Clicker Question No. Four Clicker Question No. Four • Three particles travel through a region of space where the magnetic field is out of the page, as shown in the figure. The electric charge of each of the three particles is, respectively, • • • • • A) 1 is neutral, 2 is negative, and 3 is positive. B) 1 is neutral, 2 is positive, and 3 is negative. C) 1 is positive, 2 is neutral, and 3 is negative. D) 1 is positive, 2 is negative, and 3 is neutral. E) 1 is negative, 2 is neutral, and 3 is positive. Point Charge Moving in Uniform Magnetic Field • Three particles travel through a region of space where the magnetic field is out of the page, as shown in the figure. The electric charge of each of the three particles is, respectively, • • • • • A) 1 is neutral, 2 is negative, and 3 is positive. B) 1 is neutral, 2 is positive, and 3 is negative. C) 1 is positive, 2 is neutral, and 3 is negative. D) 1 is positive, 2 is negative, and 3 is neutral. E) 1 is negative, 2 is neutral, and 3 is positive. Electron-Positron Pairs Electron-Positron Pairs Electron-Positron Pairs Point Charge Moving in Uniform Magnetic Field Magnetic Bottle Radiation Belts Aurora Borealis Clicker Question No. Five Electrons travel at an initial velocity v0. They pass through a set of deflection plates, between which there exists an electric field which deflects them upwards toward point b. In which direction should a magnetic field be applied so that the electrons land undeflected at a? Clicker Question No. Five Electrons travel at an initial velocity v0. They pass through a set of deflection plates, between which there exists an electric field which deflects them upwards toward point b. In which direction should a magnetic field be applied so that the electrons land undeflected at a? Velocity Selector Velocity Selector J.J. Thomson’s Measurement of e/m • F = - qE + qv x B • If F = 0 • qE = q v x B = qvB • v = E/B Velocity Selector Point Charge Moving in Uniform Magnetic Field Mass Spectrometer Mass Spectrometer q|ΔV| = (1/2) mv2 ω = 2πf = qB/m R = mv/qB v = qBR/m v2 = q2B2R2/m2 (1/2) mv2 = (1/2) m (q2B2R2/m2) = q|ΔV| m/q = (1/2)B2R2/|ΔV| Al Nier – University of Minnesota Al Nier – University of Minnesota Mass Spectrometer Mass Spectrometer Torques on Current Loops due to Magnetic Fields Torques on Current Loops due to Magnetic Fields Use the right hand rule to define orientation of current loop, represented by normal vector n F1 = F2 = I L x B = IaB Torques on Current Loops due to Magnetic Fields F1 and F2 form a “couple” and “every couple has its moment!” (But it’s just a lot of torque!) Torques on Current Loops due to Magnetic Fields Calculate torque τ about point P (so F1 does not contribute to torque) τ = F2bsinθ = IaBbsinθ = IB (ab)sinθ µ = IANn τ = µ x Β Torques on Current Loops due to Magnetic Fields Torques on Current Loops due to Magnetic Fields Torques on Current Loops due to Magnetic Fields Potential Energy of Magnetic Dipole Spinning Charged Disc • dW = - τ dθ = - µB sinθ dθ = - dU • U = ∫ µB sinθ dθ = - µB cosθ + Uo • If U = 0 when θ = 90° • U = - µB cosθ = - µ • B The Hall Effect The Hall Effect • Magnetic force FM = q vd B • Balanced by electric force FE = q EH • EH = vd B • For a width w, VH = EH w = vd B w • Hall Voltage VH = vd B w The Hall Effect • Current is deflected by external magnetic field. The Hall Effect • I = n |q | vd A – check units • Area A = w t • Charges pile up at sides of material, building up an electric field. I = n |q | vd w t • vd w = I/n |q| t = I/ n e t • VH = vd w B = I B/n e t • When EH = VH/width the field is strong enough to prevent one more charge from being deflected The Quantum Hall Effect • n = I B/VH e t • Measure I, B and VH – determine n!