3/22/2021 Chapter 29 Ed9 (Chapter 28 Ed10) Magnetic Fields Contents Introduction 29.1 Analysis Model: Particle in a Field (Magnetic) 29.2 Motion of a Charged Particle in a Uniform Magnetic Field 29.3 Applications Involving Charged Particles Moving in a Magnetic Field 29.4 Magnetic Force Acting on a Current-Carrying Conductor 29.5 Torque on a Current Loop in a Uniform Magnetic Field (not in the syllabus) 29.6 The Hall Effect (not in the syllabus) 1 3/22/2021 A Brief History of Magnetism 13th century BC  Chinese used a compass  Uses a magnetic needle  Probably an invention of Arabic or Indian origin 800 BC  Greeks  Discovered magnetite (Fe3O4) attracts pieces of iron Introduction A Brief History of Magnetism, cont. 1269  Pierre de Maricourt found that the direction of a needle near a spherical natural magnet formed lines that encircled the sphere .  The lines also passed through two points diametrically opposed to each other.  He called the points poles 1600  William Gilbert  Expanded experiments with magnetism to a variety of materials  Suggested the Earth itself was a large permanent magnet Introduction 2 3/22/2021 A Brief History of Magnetism, final 1750  Experimenters showed that magnetic poles exert attractive or repulsive forces on each other. 1819  Found an electric current deflected a compass needle 1820’s  Faraday and Henry  Further connections between electricity and magnetism  A changing magnetic field creates an electric field.  Maxwell  A changing electric field produces a magnetic field. Introduction Hans Christian Oersted 1777 – 1851 Discovered the relationship between electricity and magnetism An electric current in a wire deflected a nearby compass needle The first evidence of the connection between electric and magnetic phenomena Also the first to prepare pure aluminum Introduction 3 3/22/2021 Magnetic Poles Every magnet, regardless of its shape, has two poles.  Called north and south poles  Poles exert forces on one another  Similar to the way electric charges exert forces on each other  Like poles repel each other  N-N or S-S  Unlike poles attract each other  N-S Introduction Magnetic Poles, cont. The poles received their names due to 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 means the Earth’s north geographic pole is a magnetic south pole.  Similarly, the Earth’s south geographic pole is a magnetic north pole. Introduction 4 3/22/2021 Magnetic Poles, final The force between two poles varies as the inverse square of the distance between them. A single magnetic pole has never been isolated.  In other words, magnetic poles are always found in pairs.  All attempts so far to detect an isolated magnetic pole has been unsuccessful.  No matter how many times a permanent magnetic is cut in two, each piece always has a north and south pole. Introduction 29.1 Analysis Model: Particle in a Field (Magnetic) 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 magnetic substance making up a permanent magnet. Section 29.1 5 3/22/2021 Magnetic Fields, cont. A vector quantity Symbolized by 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. Section 29.1 Magnetic Field Lines, Bar Magnet Example The compass can be used to trace the field lines. The lines outside the magnet point from the North pole to the South pole. Section 29.1 6 3/22/2021 Magnetic Field Lines, Bar Magnet Iron filings are used to show the pattern of the electric field lines. The direction of the field is the direction a north pole would point. Section 29.1 Magnetic Field Lines, Opposite Poles Iron filings are used to show the pattern of the electric field lines. The direction of the field is the direction a north pole would point.  Compare to the electric field produced by an electric dipole Section 29.1 7 3/22/2021 Magnetic Field Lines, Like Poles Iron filings are used to show the pattern of the electric field lines. The direction of the field is the direction a north pole would point.  Compare to the electric field produced by like charges Section 29.1 Earth’s Magnetic Poles More proper terminology would be that a magnet has “north-seeking” and “southseeking” poles. The north-seeking pole points to the north geographic pole.  This would correspond to the Earth’s south magnetic pole. The south-seeking pole points to the south geographic pole.  This would correspond to the Earth’s north magnetic pole. The configuration of the Earth’s magnetic field is very much like the one that would be achieved by burying a gigantic bar magnet deep in the Earth’s interior. Section 29.1 8 3/22/2021 Earth’s Magnetic Field The source of the Earth’s magnetic field is likely convection currents in the Earth’s core. There is strong evidence that the magnitude of a planet’s magnetic field is related to its rate of rotation. The direction of the Earth’s magnetic field reverses periodically. Section 29.1 Definition of Magnetic Field The magnetic field at some point in space can be defined in terms of the magnetic force, F . B The magnetic force will be exerted on a charged particle moving with a velocity, v.  Assume (for now) there are no gravitational or electric fields present. Section 29.1 9 3/22/2021 Properties of a Force on a Charge Moving in a Magnetic Field The magnitude FB of the magnetic force exerted on the particle is proportional to the charge, q, and to the speed, v, of the particle. When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero. When the particle’s velocity vector makes any angle q 0 with the field, the force acts in a direction perpendicular to the plane formed by the velocity and the field. The magnetic force exerted on a positive charge is in the direction opposite the direction of the magnetic force exerted on a negative charge moving in the same direction. The magnitude of the magnetic force is proportional to sin q, where q is the angle the particle’s velocity makes with the direction of the magnetic field. Section 29.1 More About Direction Section 29.1 10 3/22/2021 Force on a Charge Moving in a Magnetic Field, Formula The properties can be summarized in a vector equation: FB  qv  B  FB is the magnetic force  q is the charge  v is the velocity of the moving charge  B is the magnetic field Section 29.1 Direction: Right-Hand Rule #1 This rule is based on the right-hand rule for the cross product. Your thumb is in the direction of the force if q is positive. The force is in the opposite direction of your thumb if q is negative. Section 29.1 11 3/22/2021 Direction: Right-Hand Rule #2 Alternative to Rule #1 The force on a positive charge extends outward from the palm. The advantage of this rule is that the force on the charge is in the direction you would push on something with your hand. The force on a negative charge is in the opposite direction. Section 29.1 More About Magnitude of F The magnitude of the magnetic force on a charged particle is FB = |q| v B sin q  q is the smaller angle between v and B  FB is zero when the field and velocity are parallel or antiparallel  q = 0 or 180o  FB is a maximum when the field and velocity are perpendicular  q = 90o Section 29.1 12 3/22/2021 Differences Between Electric and Magnetic Fields Direction of force  The electric force acts along the direction of the electric field.  The magnetic force acts perpendicular to the magnetic field. Motion  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. Work  The 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 of its point of application. Section 29.1 Work in Fields, cont. 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. Section 29.1 13 3/22/2021 Units of Magnetic Field The SI unit of magnetic field is the tesla (T). T Wb N N   2 m C ( m / s ) A  m  Wb is a weber A non-SI commonly used unit is a gauss (G).  1 T = 104 G Section 29.1 14 3/22/2021 15 3/22/2021 29.2 Motion of a Charged Particle in a Uniform Magnetic Field Notation Notes 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. The same notation applies to other vectors. Section 29.2 Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field with its velocity perpendicular to the field. The force is always directed toward the center of the circular path. The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle. Section 29.2 16 3/22/2021 Force on a Charged Particle Use the particle under a net force and a particle in uniform circular motion models. Equating the magnetic and centripetal forces: 2 FB  qvB  mv r Solving for r: mv r qB  r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field. Section 29.2 More About Motion of Charged Particle The angular speed of the particle is ω v qB  r m  The angular speed, w, is also referred to as the cyclotron frequency. The period of the motion is T 2πr 2π 2πm   v ω qB Section 29.2 17 3/22/2021 Motion of a Particle, General If a charged particle moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helix. Same equations apply, with v replaced by v   v y2  v z2 Section 29.2 Bending of an Electron Beam Electrons are accelerated from rest through a potential difference. The electrons then enter a uniform magnetic field that is perpendicular to the velocity vector. The electrons travel in a curved path. Conservation of energy will give v Other parameters can be found Section 29.2 18 3/22/2021 Particle in a Nonuniform Magnetic Field The motion is complex. For example, the particles can oscillate back and forth between two positions. This configuration is known as a magnetic bottle. Section 29.2 Van Allen Radiation Belts The Van Allen radiation belts consist of charged particles surrounding the Earth in doughnut-shaped regions. The particles are trapped by the Earth’s nonuniform magnetic field. The particles spiral from pole to pole.  May result in auroras Section 29.2 19 3/22/2021 20 3/22/2021 21 3/22/2021 29.3 Applications Involving Charged Particles Moving in a Magnetic Field Charged Particles Moving in Electric and Magnetic Fields In many applications, charged particles will move in the presence of both magnetic and electric fields. In that case, the total force is the sum of the forces due to the individual fields.  The total force is called the Lorentz force. In general: F  qE  qv  B Section 29.3 22 3/22/2021 Velocity Selector Used when all the particles need to move with the same velocity. A uniform electric field is perpendicular to a uniform magnetic field. When the force due to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight line. This occurs for velocities of value. v=E/B Section 29.3 Velocity Selector, cont. Only those particles with the given speed will pass through the two fields undeflected. The magnetic force exerted on particles moving at a speed greater than this is stronger than the electric field and the particles will be deflected to the left. Those moving more slowly will be deflected to the right. Section 29.3 23 3/22/2021 Mass Spectrometer A mass spectrometer separates ions according to their mass-to-charge ratio. In one design, a beam of ions passes through a velocity selector and enters a second magnetic field. After entering the second magnetic field, the ions move in a semicircle of radius r before striking a detector at P. If the ions are positively charged, they deflect to the left. If the ions are negatively charged, they deflect to the right. Section 29.3 Mass Spectrometer, cont. The mass to charge (m/q) ratio can be determined by measuring the radius of curvature and knowing the magnetic and electric field magnitudes. m rBo rBoB   q v E In practice, you can measure the masses of various isotopes of a given atom, with all the ions carrying the same charge.  The mass ratios can be determined even if the charge is unknown. Section 29.3 24 3/22/2021 Thomson’s e/m Experiment Electrons are accelerated from the cathode. They are deflected by electric and magnetic fields. The beam of electrons strikes a fluorescent screen. e/m was measured Section 29.3 Cyclotron A cyclotron is a device that can accelerate charged particles to very high speeds. The energetic particles produced are used to bombard atomic nuclei and thereby produce reactions. These reactions can be analyzed by researchers. Section 29.3 25 3/22/2021 Cyclotron, cont. D1 and D2 are called dees because of their shape. A high frequency alternating potential is applied to the dees. A uniform magnetic field is perpendicular to them. A positive ion is released near the center and moves in a semicircular path. Section 29.3 Cyclotron, final The potential difference is adjusted so that the polarity of the dees is reversed in the same time interval as the particle travels around one dee. This ensures the kinetic energy of the particle increases each trip. The cyclotron’s operation is based on the fact that T is independent of the speed of the particles and of the radius of their path. K 1 q 2B2R 2 mv 2  2 2m When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play. Most accelerators currently used in research are synchrotrons. Section 29.3 26 3/22/2021 29.4 Magnetic Force Acting on a Current-Carrying Conductor Magnetic Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field.  The current is a collection of many charged particles in motion. The direction of the force is given by the right-hand rule. Section 29.4 Force on a Wire In this case, there is no current, so there is no force. Therefore, the wire remains vertical. Section 29.4 27 3/22/2021 Force on a Wire, 2 The magnetic field is into the page The current is up the page The force is to the left The wire deflects to the left Section 29.4 Force on a Wire, 3 The magnetic field is into the page The current is down the page The force is to the right The wire deflects to the right 28 3/22/2021 Force on a Wire, equation The magnetic force is exerted on each moving charge in the wire.  F  qvd B The total force is the product of the force on one charge and the number of charges.  F   qvd  B  nAL Section 29.4 Force on a Wire, Equation cont. In terms of the current, this becomes FB  IL  B  I is the current.  L is a vector that points in the direction of the current.  Its magnitude is the length L of the segment.  B is the magnetic field. Section 29.4 29 3/22/2021 Force on a Wire, Arbitrary Shape Consider a small segment of the wire, ds The force exerted on this segment is dFB  I ds  B The total force is b FB  I  ds  B a Section 29.4 30 3/22/2021 31 3/22/2021 The following are the recommended problems, at the end of chapter. You should try to work on these by your own as part the preparation for the tests. Some of these questions will be discussed during the problem-solving class. (Note: these are the minimum number of problems you should work on). Chapter 29: 2, 8, 11, 13, 19, 25, 33, 37, 39, 46, 48 Also complete the WebAssign Homework 7 32
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