PHY2049: Chapter 281Magnetic Field and WorkÎMagnetic force is alwaysperpendicular to velocity Therefore B field does no work! Why? BecauseÎConsequences Kinetic energy does not change Speed does not change Only direction changes Particle moves in a circle (if )()0KFxFvt∆=⋅∆=⋅∆=rrrrvB⊥rrPHY2049: Chapter 282x 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 x x x x x xx x x x x x x x x x x x x xTrajectory in a Constant Magnetic FieldÎA charge q enters B field with velocity v perpendicular to B. What path will q follow? Force is always ⊥ velocity and ⊥ B Path will be a circle. F is the centripetal force needed to keep the charge in its circular orbit. Let’s calculate radius RFFvRvBqFvPHY2049: Chapter 283x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x xCircular Motion of Positive ParticleBqFv2mvqvBR=mvRqB=PHY2049: Chapter 284Magnetic Force ÎTwo particles of the same charge enter a magnetic field with the same speed. Which one has the bigger mass? A B Both masses are equal Cannot tell without more infox x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xABmvRqB=Bigger mass means larger inertia, less acceleration, thus bigger radiusPHY2049: Chapter 285Work and Energy ÎA charged particle enters a uniform magnetic field. What happens to the kinetic energy of the particle? (1) it increases (2) it decreases (3) it stays the same (4) it depends on the direction of the velocity (5) it depends on the direction of the magnetic fieldMagnetic field does no work, so K is constantPHY2049: Chapter 286Cosmic Ray ExampleÎProtons with energy 1 MeV move ⊥ earth B field of 0.5 gauss or 5 × 10-5T. Find radius & frequency of orbit.2122KKmv vm=⇒=2mv mKReB eB==()( )619 1327K 10 1.6 10 =1.6 10 J1.67 10 kgm−−−=× ×=×()122/ 2vveBfTR mveB mπππ== = =760Hzf=2900mR=Frequency is independent of v!PHY2049: Chapter 287Helical Motion in B FieldÎVelocity of particle has 2 components (parallel to B and perp. to B) Only v⊥= v sinφ contributes to circular motion v||= v cosφ is unchangedÎSo the particle moves in a helical path v||is the constant velocity along the B field v⊥is the velocity around the circlevv v⊥=+rr rmvRqB⊥=Bvφv⊥v||PHY2049: Chapter 288Helical Motion in Earth’s B FieldParticles moving along field lines cause Aurora Borealis and Australis:http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.htmlPHY2049: Chapter 289Mass SpectrometerÎOriginally developed by physicists, now an important tool in chemistry, biology, environmental studies, forensics, pharmaceutics, etc.ÎSample is vaporized, broken into fragments of molecules, which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B (sometimes V) allows ions of different masses to reach the detector.PHY2049: Chapter 2810Mass Spectrometer (simplified)ÎSample is vaporized, broken into fragments of molecules, which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B (sometimes V) allows ions of different masses to reach the detector. Spectrometer determines mass from B (sometimes from V)DVBrqm2)(2=2rdetectorPHY2049: Chapter 2811Torque on Current LoopÎ Rectangular current loop in uniform magnetic field (lengths a & b) Forces in left & right branches are 0 Force in top branch is into plane Force in bottom branch is out of planeÎ Equal forces give net torque! Bottom side up, top side down (RHR) Rotates around horizontal axisÎ µ = NiA ⇒ “magnetic dipole moment” Assuming N turns τ = µB, true for any shape!!Î If plane tilted angle θ to B field τ = µBsinθ θ is angle between normal and BBabab()Fd iBa b iBab iBAτ== = =Plane normal is ⊥ B(θ = 90°)PHY2049: Chapter 2812Magnetic Dipole MomentPHY2049: Chapter 2813Torque ExampleÎA 3-turn circular loop of radius 3 cm carries 5A current in a B field of 2.5 T. Loop is tilted 30° to B field.ÎÎÎRotation is alwaysin direction to align µ with B field30°()2223 3 5 3.14 0.03 0.0339A mirµπ==××× = ⋅sin 30 0.0339 2.5 0.5 0.042 N mBτµ==××=⋅PHY2049: Chapter 2814Magnetic Force ÎA rectangular current loop is in a uniform magnetic field. What direction is the net force on the loop? (a) + x (b) + y (c) zero (d) – x (e) – y BxzyForces cancel onopposite sides of loopPHY2049: Chapter 2815Electromagnetic Flowmeter¾ Moving ions in the blood are deflected by magnetic force¾ Positive ions deflected down, negative ions deflected up¾ This separation of charge creates an electric field E pointing up¾ E field creates potential difference V = Ed between the electrodes¾ The velocity of blood flow is measured by v = E/BEPHY2049: Chapter 2816Hall Effect: Do + or – Charges Carry Current?Î + charges moving counter-clockwise experience upward forceÎ Upper plate at higherpotentialÎ – charges moving clockwise experience upward forceÎ Upper plate at lowerpotentialVery quickly, equilibrium between electrostatic & magnetic forces is established and potential difference stops growing:up driftFqvB=Hdown inducedwVFqE q==Hdrift"Hall Voltage"VvBw==¾ This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges¾ Hall effect is used to measure moderate to moderately high B (10-4 T –3 T)¾ It is also used to measure the speed of computer hard drivePHY2049: Chapter 2817FAQ on Magnetic Field and WorkÎ Magnetic force does no work. But an electric motor (=a current loop in B) does work. Where does this work come from? Magnetic force does no work on a moving charge Magnetic torque on a current loop does work: ∆W=τ∆θÎ There is no net force, only torque, on a current loop (=magnetic dipole moment)
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