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Ch28 magnetic fields
 

Ch28 magnetic fields

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    Ch28 magnetic fields Ch28 magnetic fields Document Transcript

    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Ch 28 Magnetic fields (exclude 4-crossed fields, 5-Hall effect 7-Cyclotrons) 2 lectures We are surrounded magnets and their effects. electromagnets - next chapter discusses how current in a wire creates a magnetic field Electrons have intrinsic magnetic fields (like charge, mass) Permanent magnets (ch 32) occur when these intrinsic magnetic fields have a net overall magnetic field. First we define B, the magnetic field Ch28-1/45 Definition of magnetic field B Electric field is defined in terms of force on an element of charge. There are no magnetic monopoles (i.e., isolated magnetic charges) => we must define B differently. We use magnetic force exerted on a moving charge to define B FB is zero in the same direction as B. In general Ch28-2/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 1
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Refresher on vector (or cross) products The product is perpendicular to both a and b. since sin90o = 1 If above is true, => a right-handed coordinate system Ch28-3/45 Refresher on vector (or cross) products Direction of resultant vector is determined by the right hand rule. To find direction of fingers of right hand sweep towards Vector is directed along thumb. Force vector is -in same direction as thumb for a +ve charge q -in opposite direction for a -ve charge. Ch28-4/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 2
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Defn of B (cont) Force acting on a charged particle moving with velocity through a magnetic field is always perpendicular to both and No component of in direction of motion => cannot change particle’s speed, v It accelerates particle by changing its direction. There is no force in direction of (φ=0) Ch28-5/45 Defn of B (cont) SI unit of B is tesla (T). Old unit is gauss (G) Ch28-6/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 3
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Some values of B Some approximate magnetic field values Surface neutron star 108 T Near large electromagnet 1.5 - 9 T (MRI) Near small bar magnet 10-2 T At Earth’s surface 10-4 T = 1 G In interstellar space 10-10 T Best magnetically shielded room 10-14 T Ch28-7/45 An example from particle physics Tracks of 2 electrons and positron in a bubble chamber Uniform magnetic field out of page. Event is triggered by an unseen photon (no charge) from top. Triplet production. Which way are forces? L, R, U, D electrons positron Ch28-8/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 4
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons An example from particle physics Tracks of 2 electrons and positron in a bubble chamber Uniform magnetic field out of page. Event is triggered by an unseen photon (no charge) from top. Triplet production. Which way are forces? L, R, U, D electrons Ans: to left in direction of motion since -ve charge positron Ans: to right in direction of motion since +ve charge Ch28-9/45 What is direction of magnetic force acting on moving particles: +-x, +-y, +-z, 0? Is the coordinate system right handed? Ch28-10/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 5
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons What is direction of magnetic force acting on moving particles: +-x, +-y, +-z, 0? Ans (a): +z axis Ans (b): -x axis (negative charge) Ans (c): 0 since anti-parallel. Is the coordinate system right handed? Ans: yes Ch28-11/45 Magnetic field lines Represent magnetic fields by field lines as for electric fields. 1) direction of B is tangent to field line at all points 2) field strength proportional to number of lines /unit area | lines 3) they exist inside a magnet (unlike electric field lines inside a conductor) 4) they leave a magnet from North pole and enter via South pole Ch28-12/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 6
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic poles Magnets have two poles. Referred to as a magnetic dipole. Bar, horseshoe and C magnets shown. Opposite magnetic poles attract each other and like poles repel each other Earth’s magnetic field is produced in its core. Compass is a bar magnet on a pivot. Compass North end points to North pole i.e. Earth’s “geomagnetic North Pole” is actually South pole of a magnet => field lines enter there. Ch28-13/45 Sample problem Uniform magnetic field B with magnitude 1.2 mT, is directed up throughout a lab chamber. A proton with kinetic energy 5.3 MeV enters moving horizontally from S to N. (a) What magnetic force acts on the proton as it enters? (b) What acceleration is produced by this force? Ignore Earth’s magnetic field. mp = 1.67E-27 kg. Key ideas: Use First find v from v/c about 11%, so becoming relativistic. Ch28-14/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 7
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem (cont) Direction of force? Key ideas: v is S to N, field is out of page (shown as dots in figure, x would mean into page) Right hand rule, +ve charge => force acts to right. Very small force acting on very small mass => considerable acceleration (but no change in speed!) What would happen to an electron in same situation? Ch28-15/45 Sample problem (cont) Direction of force? Key ideas: v is S to N, field is out of page (shown as dots in figure, x would mean into page) Right hand rule, +ve charge => force acts to right. Very small force acting on very small mass => considerable acceleration (but no change in speed!) What would happen to an electron in same situation? Ans: Force in opposite direction and acceleration 1836 times greater since mass much lighter (ignoring relativistic effects - which we cannot really do) Ch28-16/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 8
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Circulating charged particles What does it mean if an object is moving in circle at constant velocity? => constant force acting towards centre of circle to keep particle accelerating inwards Figure: electrons circulating in a chamber with low pressure gas (to visualize). Magnetic field out of page. Electron gun G provides an initial G speed v. Force directed toward centre since | v and B. There is no effect on speed. Ch28-17/45 Circulating charged particles(cont) Determine characteristics of motion of these electrons or any charged particle in a uniform magnetic field (out of page). From Ch 6, uniform circular motion implies G a centripetal force and since B | v => Solving for r: Radius of curvature is bigger for heavy particles, bigger for higher energy particles, Which e- has smaller for higher charges and stronger fields higher energy? Ch28-18/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 9
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Circulating charged particles(cont) We can also specify periods and frequencies period (s) G frequency (1/s) angular frequency(1/s) Independent of speed of particles! (as long as non-relativistic). Faster particles go in bigger circles. For a given |q|/m: T etc are all same, independent of v r is same for same values of v Ch28-19/45 Circulating charged particles(cont) Consider looking in direction of B G (i.e. opposite of example shown) => for particles in a circular orbit, direction of +ve particles is counterclockwise, direction of -ve particles is clockwise. So far we considered particles with velocity vector | magnetic field. What happens when they are not | ? Ch28-20/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 10
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Helical paths If velocity of a charged particle has a component parallel to a (uniform) magnetic field => particle will move in a helical path about direction of magnetic field. Consider the two components of velocity in figure. The | component is used to determine radius and the || component determines pitch p (distance between adjacent turns) Ch28-21/45 Helical paths (cont) In magnetic bottle, charged particles follow field lines in a helical path. If field strong enough at ends, they can be reflected back (not shown in figure, but stronger field at ends has a component of force backwards). Earth’s magnetic field lines cause stray charged particles to be trapped in Van Allen radiation belts Bounce back and forth between N and S poles (few seconds/trip). Solar flares=> radiation disrupts reflection => electrons come down into atmosphere => aurora/northern lights (100 km high green from oxygen, pink from nitrogen - appears white when weak) Ch28-22/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 11
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Mass spectrometer Mass spectrometers are used to determine atomic mass of a substance. Element 1 is a source which uses an electric potential difference to accelerate a particle of charge q (all in vacuum). Element 2 is a uniform magnetic field B (out of page) which bends charged particle to a detector which measures radius of curvature Element 3 is solving physics equations that determine mass given V, B and x Ch28-23/45 Mass spectrometer (cont) As particle leaves source, it has roughly no kinetic energy. +ve charge accelerates going from high to low potential. At end of applied potential Using relationship (28-15) for r in a uniform magnetic field Result gives m/q in terms of what we measure. Ch28-24/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 12
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic force on a current-carrying wire A flexible wire will be bent in presence of a magnetic field. Direction of force depends on current direction (right hand rule). Inside wire consider e- with drift velocity down, or +ve charge with drift velocity up, both => force to right F = vdeB Ch28-25/45 Magnetic force on a current-carrying wire (cont) Consider a length of wire, L. All conduction electrons in this segment will drift past plane xx in time t = L/vd => in that time a charge will pass thru plane. Recall Drift velocity has dropped out of equation! Eqn applies if field B | wire. It gives the magnetic force acting on a length of wire L with current i . More generally: Ch28-26/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 13
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic force on a current-carrying wire (cont) L is a length vector, magnitude L and directed along wire in same direction as conventional current. FB is in direction of cross product because i is, by convention, in direction of +ve charge carriers. Above eqns equivalent to i.e., either pair can be used to define B. Eqns in terms of iL are usually used since measuring force on a wire is easier. If a wire is not straight, or field not uniform, force acting on wire can be found by considering wire as series of elemental components Perform an integration over actual wire to get Ch28-27/45 Question Current i in a wire passing thru a uniform magnetic field B. Resulting magnetic force is shown. Field is oriented so force is a maximum. What is direction of magnetic field? (a) +ve y axis (b) -ve z axis (c) -ve y axis Ch28-28/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 14
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Question Current i in a wire passing thru a uniform magnetic field B. Resulting magnetic force is shown. Field is oriented so force is a maximum. What is direction of magnetic field? (a) +ve y axis (b) -ve z axis (c) -ve y axis Ans: => B -ve y-axis by rhr Ch28-29/45 Sample problem A straight copper wire has a current i=28 A out of page. What are direction and magnitude of smallest magnetic field B needed to suspend the wire against gravity (linear density is 46.6 g/m). Key ideas: To be minimal force => acts directly up and B must be | L. B must be horizontal to right so force is upwards since i is out of page (and +ve). Since this must balance gravity where m is mass of wire. φ must be 90o to maximize force where m/L is the linear density. Result independent of L. This is 160 times Earth’s magnetic field. Ch28-30/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 15
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Torque on a current loop (electric motors) A current loop in a uniform magnetic field forms a simple motor. As shown, at ends i is parallel or anti-parallel to B => no force acting. On sides forces are in opposite directions => rotate the current loop. Trick is to reverse current flow when loop is vertical (commutator does so). Ch28-31/45 Torque on a current loop (electric motors) (cont) Define n, normal vector | plane of loop. Direction of n defined by another right-hand rule with fingers curling along direction of current. Left hand figure is not like previous one since field is into page. If loop in (a) is in plane of page => net force on loop is 0 since everything is equal and opposite. But once at an angle wrt a horizontal axis there is a net force/torque. Ch28-32/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 16
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Torque on a current loop (electric motors) (cont) Looking from side 2 end of loop along it (side 4 is behind side 2) . θ is the arbitrary angle between n and B. Side 2: L is in direction of i, magnitude b. Angle with B is 90 - θ. Using θ 90 - θ n rhr tells us F2 is out of page. θ Side 4: 90 - θ B L is in opposite direction => force is equal and opposite direction L => no net force in or out of page. Ch28-33/45 Torque on a current loop (electric motors) (cont) Sides 1 & 3: L is | to B so F1 and F3 both have magnitude = iaB. In opposite directions => no net movement up or down. Forces are not on same line of action => net torque which tends to rotate loop so that n is aligned with B. Projection of force F1 | wire is F1cos(90-θ) =F1sinθ θ 90 - θ n zero when θ is 0 (aligned). Torque has a moment arm θ B so magnitude of torque from F1 & F3 is b/2 sinθ Ch28-34/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 17
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Torque on a current loop (electric motors) (cont) Suppose single loop is replaced by a coil with N loops or turns. A=ab, area of coil. (NiA) are all characteristics of coil. This eqn holds for all flat coils, no matter the shape, as long as B uniform. In an electric motor the current must be reversed when θ is 0 (via commutator). Ch28-35/45 Magnetic dipole moment A current carrying coil is said to be a magnetic dipole just like a bar magnet since a magnetic field generates a torque on it. We assign a magnetic dipole moment µ to the coil. µ has same direction as normal to coil (same rhr as before for that direction) and magnitude N is # turns in coil, i the current in wire and A area enclosed. Rewrite eqn for τ as where θ is angle between µ and B. Generalize to which is very much like Ch28-36/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 18
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Magnetic dipole moment (cont) A magnetic dipole in a magnetic field => magnetic potential energy that depends on orientation. in field. For electric dipoles we showed In analogy, for magnetic case we have Aligned µ and B => lowest energy (-µ B) => state torque wants to achieve. Higher energy when anti-parallel (+µ B) U=+µB U=-µB Must work against torque to get there. Ch28-37/45 Magnetic dipole moment (cont) Work done on dipole by magnetic field as rotated is: If an external agent acts on dipole during change in orientation, work done on dipole by applied torque is the -ve of work done by magnetic field on dipole, i.e. The first eqn suggests another unit for µ, viz [J/T] recall Ch28-38/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 19
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Some magnetic dipole moments There are many magnetic dipoles aside from the current carrying loop. Some examples magnetic dipole moments Small bar magnet 5 J/T Earth 8.0x1022 J/T proton 1.4x10-26 J/T electron 9.3x10-24 J/T Note: electron has a much larger magnetic dipole moment than proton. Ch28-39/45 Question Consider above 4 orientations of µ, a magnetic dipole moment in magnetic field B. Rank them, largest first, according to (a) magnitude of torque on dipole (b) potential energy of dipole. Ch28-40/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 20
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Question Consider above 4 orientations of µ, a magnetic dipole moment in magnetic field B. Rank them, largest first, according to (a) magnitude of torque on dipole Ans: => all the same (b) potential energy of dipole. Ans: => 1&4 same& highest since cos(180-θ) is same and negative. Or because they required moving against torque which wants them to align with B. Ch28-41/45 Sample problem Circular coil has 250 turns, A=2.52E-4 m2, i=100µA. Coil at rest in B with magnitude B=0.85 T and µ aligned with B. What is the direction of the current in the coil? Ans: rhr fingers follow current & thumb points to µ => current down screen on closest side. How much work would an external agent have to do via a torque to rotate to 90o? Key idea: applied work = change in potential due to change in orientation. Ch28-42/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 21
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Following not used in class Ch28-43/45 Sample problem: galvanometer A galvanometer measures current flow by measuring torque exerted by a uniform magnetic field on a current-carrying coil. Example: Coil is 1.2 x 1.2cm2, 250 turns, can rotate about an axis into page. Uniform magnetic field of 0.23 T as shown so field | normal vector of coil (along pointer) and parallel plane of coil. A spring counterbalances torque so for a given i, pointer is steady with an angular deflection dφ. Greater current => greater deflection. If 100 µA causes 28o deflection, what is torsional spring constant κ ( )? Ch28-44/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 22
    • Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem: galvanometer Which way does the current flow? Ans: up left, down right What is torsional spring constant κ ( )? Key idea: spring’s torque is balanced by magnetic field torque when there is a constant current. where A=area of coil, φ is the angular deflection (28deg) and θ = 90 since field always | to normal to coil. Digital ammeters and voltmeters do not use a moving coil. Ch28-45/45Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 23