Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These are the questions we will address. Chapter Goal: To learn how to calculate and use the magnetic field.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix B. Parabola C. Circle D. Ellipse E. HyperbolaCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix B. Parabola C. Circle D. Ellipse E. HyperbolaCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The magnetic field of a straight, current-carrying wire is A. parallel to the wire. B. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The magnetic field of a straight, current-carrying wire is A. parallel to the wire. B. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. http://solar.gmu.edu/teaching/2008_CSI769/solar_magnetic_field.jpg The Earth and Sun are magneticCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Connections to current A magnetic field can be sensed with a magnetic material (compass) and is associated with a current. The earth’s field results from large scale internal currents. The field of a permanent magnet results from atomic scale currents.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The field appears only if there is current. It is associated with moving charge.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Source of the Magnetic Field: Moving Charges The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: where r is the distance from the charge and θ is the angle between v and r. (Valid if v<<c.) The Biot-Savart law can be written in terms of the cross product asCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The field of an element of circuit dB =µo4πIds ׈ r r2 dI=Ids dB r The average B field dB due to the moving charge in an element of circuit of vector length ds carrying current I follows from superposing the fields of the moving charges:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The field of straight wire 11 All current elements produce B out of page x a r = x2+ a2r ! dB =µo4πIds ׈ r r2=µo4πIr2sinθ=µo4πIr2ar=µo4πIax2+ a2( )3 / 2 B =µoIa4πdxx2+ a2( )3 / 2=−∞∞∫µoI2πaAdd them all up:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 33.4 The magnetic field strength near a heater wireCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 33.4 The magnetic field strength near a heater wire Twice B(Earth).Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Magnetic field lines close upon themselves – they circulate rather them emanate from charges. Although a current loop can appear like a dipole pair of charges, there is no magnetic charge. Magnetic field linesCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Magnetic Dipoles The magnetic dipole moment of a current loop enclosing an area A is defined as The SI units of the magnetic dipole moment are A m2. The on-axis field of a magnetic dipole isCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 33.7 The field of a magnetic dipoleCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. EXAMPLE 33.7 The field of a magnetic dipoleCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The superposition of the fields of a stack of loops is a field like that of a bar magnet. The bar magnetic field results from alignment of many atomic scale electronic currents/magnetic dipoles. Magnetic materialsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Line integrals Magnetic field lines close upon themselves – they circulate rather them emanate from poles. The line integral of B around a closed loop is a measure of the strength in circulation.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Ampère’s law Whenever total current Ithrough passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is given by Ampère’s law:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Ampère’s law example • Could have used Ampere’s law to calculate B r I B(r) B • ds∫ = Bds =∫ B ds∫ = B2πr =µoI ⇒ B =µoI2πrCircular path Surface bounded by path B||ds B constant on path path length = 2πrCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The strength of the uniform magnetic field inside a solenoid is where n = N/l is the number of turns per unit length.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Gauss’s law for magnetism • Net magnetic flux through any closed surface is always zero B • dA∫= 0No magnetic ‘charge’, so right-hand side=0 in the case of magnet field. E • dA∫=QenclosedεoCompare to Gauss’ law for electric fieldCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. General laws of electromagnetism Gauss’ law Ampere’s law B • dA∫= 0 E • dA∫=Qenclosedεo B • ds∫=µoIMagnetostatics Electrostatics E • ds∫= ?0 • Integral of E-field around closed loop is is the change in electric potential going around = zero. xCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Magnetic Force on a Moving Charge The magnetic force on a charge q as it moves through a magnetic field B with
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