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Transcript
Chapter 27 Lecture 10 Magnetic Fields and Magnetic Forces Announcements — Stay tuned for update on TA for 11am and 12pm Discussion Sections — Exam next Thursday (will cover through Ch 26) — Ch 26 HW assigned today (due next week) Brief History of Magnetism ̣ ̣ ̣ ̣ ̣ ̣ 13th century BC: Chinese used compasses • Possibly an invention of Arabic or Indian origin 800 BC: Greeks • Discovered magnetite (Fe3O4) attracts pieces of iron 1600: William Gilbert • Expanded experiments with magnetism to a variety of materials and suggested Earth was a magnet 1819: Hans Christian Oersted • An electric current in a wire deflected a nearby compass needle: connects Electricity and Magnetism 1820’s: Faraday and Henry • A changing magnetic field creates an electric field 1820’s: Maxwell and his equations • A changing electric field produces a magnetic field. Magnetic Poles ̣ Every magnet, regardless of its shape, has two poles ➡ Called north and south poles ̣ Like poles repel • N-N or S-S ̣ Unlike poles attract • N-S ̣ The force between two poles varies as the inverse square of the distance between them Magnetic Poles ̣ A single magnetic pole has never been isolated. ̣ No matter how many times a permanent magnetic is cut in two, each piece always has a N & S pole A magnet can attract items that are not themselves magnets • Analogous to electrostatic attraction of charge objects to neutral objects via polarization • “Ferromagnets” (like Iron) can get induced to become magnets in the presence of a strong magnetic field. Magnetic Poles, cont. ● ● The name “pole” comes from the way a magnet behaves in the Earth’s magnetic field If a bar magnet is suspended so that it can move freely, it will rotate ● The magnetic north pole points toward the Earth’s north geographic pole. This implies ● Earth’s north geographic pole is a magnetic south pole ● Earth’s south geographic pole is a magnetic north pole Magnetism & Electricity First evidence of relationship between magnetism and moving charges discovered in 1820 by Hans Oersted. Current => Magnetic field. Similar experiments by Ampere, Faraday and Henry discovered that a moving magnet near a conducting loop can cause a current in the loop. Ultimately, Maxwell showed that electricity and magnetism are different manifestations of the same physical laws •“Maxwell’s Equations” for electromagnetism Magnetism & Electricity First evidence of relationship between magnetism and moving charges discovered in 1820 by Hans Oersted. Current => Magnetic field. Similar experiments by Ampere, Faraday and Henry discovered that a moving magnet near a conducting loop can cause a current in the loop. Ultimately, Maxwell showed that electricity and magnetism are different manifestations of the same physical laws •“Maxwell’s Equations” for electromagnetism Magnetic Fields ✦ ✦ ✦ Reminder: an electric field surrounds any electric charge The region of space surrounding any moving electric charge also contains a magnetic field A magnetic field also surrounds a permanent magnet Example Magnetic Fields for Common Sources Magnetic Field ! ̣ Magnetic field is a vector: B Direction is given by the direction a north pole of a compass needle points in that location ̣ Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look ̣ The compass can be used to trace the field lines ̣ The lines outside the magnet point from the North pole to the South pole ̣ ● ● Iron filings in the figures show the pattern of the magnetic field lines The direction of the field is the direction the north pole of a compass would point ● Remember N of compass would be rotated towards S of magnet Force on a Charge Moving in a Magnetic Field ! ! ! FB = qv × B Direction: Force is perpendicular to the plane defined by the velocity and the magnetic field. Force on a Charge Moving in a Magnetic Field ! ! ! FB = qv × B Magnetic force charge Magnetic field Velocity of charge Direction: Force is perpendicular to the plane defined by the velocity and the magnetic field. Direction of Force: Right-Hand Rule ! ! ! FB = qv × B cross products review Quiz At a certain instant, a proton is moving in the direction i through a magnetic field in the –k direction. What is the direction of the magnetic force exerted on the proton? ! ! ! FB = qv × B A. B. C. D. E. i –k j –j The force is zero. Quiz At a certain instant, a proton is moving in the direction i through a magnetic field in the –k direction. What is the direction of the magnetic force exerted on the proton? ! ! ! FB = qv × B A. B. C. D. E. i –k j –j The force is zero. Quiz: Magnetic Force I A positive charge enters a uniform magnetic field as shown. what is the direction of the magnetic force ? A.Left B.Right v C.Zero D.Into the page q E.Out of the page Quiz: Magnetic Force I A positive charge enters a uniform magnetic field as shown. what is the direction of the magnetic force ? A.Left B.Right v C.Zero D.Into the page q E.Out of the page The charge is moving parallel to the magnetic field, so it does not experience any magnetic force. Recall: Example Electrons are traveling at in a magnetic field: What is the magnetic force exerted on the electrons (magnitude and direction)? !" " !" F = qv × B Example Electrons are traveling at in a magnetic field: What is the magnetic force exerted on the electrons (magnitude and direction)? !" " !" F = qv × B Do we expect the force to be non-zero? Example Electrons are traveling at in a magnetic field: What is the magnetic force exerted on the electrons (magnitude and direction)? !" " !" F = qv × B Do we expect the force to be non-zero? Which direction? Example Electrons are traveling at in a magnetic field: What is the magnetic force exerted on the electrons (magnitude and direction)? !" " !" F = qv × B Do we expect the force to be non-zero? Which direction? x-comp. of velocity “feels” y-comp. of B, force in z direction y-comp. of velocity “feels” (-) x-comp. of B, force in z direction both of those forces point in -z direction Example Electrons are traveling at in a magnetic field: What is the magnetic force exerted on the electrons (magnitude and direction)? !" " !" F = qv × B ⎛ î ⎜ F = qe ⎜ 2.0 ×10 6 ⎜ 0.03 ⎜⎝ ĵ 3.0 ×10 6 −0.15 ( k̂ ⎞ ⎟ 0 ⎟ 0 ⎟⎟ ⎠ ) ( ) = qe ⎡ 0 î + 0 ĵ + ⎡⎣ −0.15 × 2.0 ×10 6 − 0.03 × 3.0 ×10 6 ⎤⎦ k̂ ⎤ ⎣ ⎦ = qe ⎡⎣ −0.3 ×10 6 − 0.09 ×10 6 ⎤⎦ k̂ =-qe ⎡⎣ 0.39 ×10 6 ⎤⎦ k̂ Differences Between E and B Fields ̣ Direction ● of force The electric force acts along the direction of the E (electric) field ~ ~ F = qE ● The magnetic force acts perpendicular to the B (magnetic) field ̣ Motion ● ● ~ F~ = q~v ⇥ B The electric force acts on a charged particle regardless of whether the particle is moving The magnetic force acts on a charged particle only when the particle is in motion Quiz A charged particle moves an infinitesimal distance, ds, under the influence of a magnetic field (and no other force). What is the work done by the magnetic field (B)? (FB below is the magnetic force.) A: q B v ds B: - FB ds C: |q| B v ds D: 0 E: None of the above because it depends on the angle between the velocity and the magnetic field. ! ! ! FB = qv × B Quiz A charged particle moves an infinitesimal distance, ds, under the influence of a magnetic field (and no other force). What is the work done by the magnetic field (B)? (FB below is the magnetic force.) A: q B v ds d~s = ~v dt dW = F~ · d~s B: - FB ds C: |q| B v ds F~B ? ~v D: 0 E: None of the above because it depends on the angle between the velocity and the magnetic field. ! ! ! FB = qv × B Work done by Magnetic Fields ! ! W = ∫ a FB • d s = b ∫ b F cos(90° )ds = 0 B a ! ! ! FB = qv × B €While electric force does work in displacing a charged particle, the magnetic force associated with a steady magnetic field does no work when a particle is displaced This is because the force is perpendicular to the displacement Work done by Magnetic Fields: KE ̣ The kinetic energy of a charged particle moving through a magnetic field cannot be altered by the magnetic field alone. ̣ When a charged particle moves with a given velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy KE = ½ m v2 v2 depends only on speed, not direction! Units of Magnetic Field ● The SI unit of magnetic field is the tesla (T) ! ! ! FB = qv × B m N =C T s ● N s N Vs T= = = 2 m C mA m A non-SI commonly used unit is a gauss (G) 1 T = 104 G €● Earth’s field is about 0.5 G (5e-5 T) but varies over the surface. ● Notation ● When vectors are perpendicular to the page, dots and crosses are used ● ● The dots represent the arrows coming out of the page The crosses represent the arrows going into the page Quiz: Magnetic Force II The figure shows a negative charge moving to the right, and a uniform magnetic field pointing into the screen. Which path does the charge follow? A.Path 1 B.Path 2 C.Path 3 D.Not enough information to determine ! ! ! FB = qv × B Quiz: Magnetic Force II The figure shows a negative charge moving to the right, and a uniform magnetic field pointing into the screen. Which path does the charge follow? A.Path 1 B.Path 2 C.Path 3 D.Not enough information to determine ! ! ! FB = qv × B q < 0, so “qV” points left Now… what path will the particle travel if we keep watching….? Charged particle in uniform magnetic field ● ● ● Consider a particle moving in an external uniform magnetic field If its initial velocity is perpendicular to the field then the force is always directed toward the center of a circular path The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle Circular Motion of a Charged Particle in B field ● Equating the magnetic and centripetal forces: mv FB = qvB = r 2 ● mv radius p depends on velocity = x mass (momentum), Solving for radius r = charge, and B-field strength qB qB ● Period T = 2πr = 2π mv = 2π m v ● Angular € speed € v qB qB v qB ω= = r m