1 Physics for Scientists and Engineers, 3rd editionLecture PowerPoint Physics for Scientists and Engineers, 3rd edition Fishbane Gasiorowicz Thornton © 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.
2 The Production and Properties of Magnetic FieldsChapter 29 The Production and Properties of Magnetic Fields
3 29-1 Ampère’s Law Experimental observation: two parallel current-carrying wires exert forces on each other Assumption: current creates a magnetic field Field can be mapped out Field makes circles around wire, direction given by right-hand rule
4 29-1 Ampère’s Law Field around a current-carrying wire:
5 29-1 Ampère’s Law We find that: where(29-2) (29-3) In choosing a value for µ0, the magnitude of the coulomb has been determined. This is the way the coulomb is defined.
6 29-1 Ampère’s Law In addition, we have the magnetic field of a long straight wire: (29-5) We can do a line integral around the wire, following the magnetic field lines: (29-7)
7 29-1 Ampère’s Law In this case, there is a current through the surface whose edge is defined by the path. Ampère’s Law is the generalization of this, valid for any current and path: (29-10)
8 29-1 Ampère’s Law Using Ampère’s Law to Find the Magnetic FieldThis can be done, but only in cases with a high degree of symmetry, such as a current-carrying wire
9 29-2 Gauss’ Law for MagnetismNo magnetic monopoles (single magnetic charge) have ever been observed Magnetic field lines must be continuous The magnetic “charge” inside any closed surface must always be zero.
10 29-2 Gauss’ Law for MagnetismDefinition of magnetic flux: (29-11) Therefore, for any closed surface: (29-12) This is Gauss’ law for magnetism.
11 29-2 Gauss’ Law for MagnetismField lines for a bar magnet: Similar to those for a magnetic dipole But no magnetic charges, so all field lines are closed loops
12 29-3 Solenoids A solenoid is a length of wire coiled into a cylinderIt produces a nearly uniform magnetic field within the cylinder parallel to its axis The magnetic field outside the solenoid is very small
13 Using Ampère’s law to find the magnetic field in a solenoid:29-3 Solenoids Using Ampère’s law to find the magnetic field in a solenoid: The dot product of B and ds is zero everywhere except along leg 4, where it is
14 29-3 Solenoids This gives for the magnetic field:(29-15) Here, n is the number of turns per unit length.
15 29-3 Solenoids A toroidal solenoidNo end effects, as the ends are connected But field is not constant across cross section of torus
16 29-3 Solenoids A toroidal solenoidHowever, if radius of cross section r0 is small compared to radius of torus R, magnetic field can be assumed to be constant.
17 29-4 The Biot–Savart Law Want to be able to calculate magnetic fields of current distributions that can’t be done with Ampère’s law Can use the magnetic field of a straight section of wire; this gives the Biot–Savart law: (29-19)
18 29-4 The Biot–Savart Law Magnetic DipolesMagnetic field of a current loop is that of a magnetic dipole Dipole strength is characterized by dipole moment µ For a circular current loop: For a loop of any shape: (29-25) (29-27)
19 29-5 The Maxwell Displacement CurrentAmpère’s law has a flaw when currents are varying One closed line can be the edge of an infinite number of surfaces As long as currents through all surfaces are the same, no problem But if current is varying this may not be true
20 29-5 The Maxwell Displacement CurrentFor example, a capacitor charging up: Surface 1 has a current going through it but Surface 2 does not But: electric flux is changing through Surface 2
21 29-5 The Maxwell Displacement CurrentThis can be fixed by adding a term called the displacement current Displacement current is zero unless there is a changing electric field (29-30) Generalized form of Ampère’s law: (29-31)
22 Magnetic field of a long straight wire carrying current I:Summary of Chapter 29 Magnetic field of a long straight wire carrying current I: (29-5) Ampère’s law: (29-10) Gauss’ law for magnetism: (29-12)
23 Summary of Chapter 29, cont.Magnetic field of a solenoid: (29-15) Biot–Savart law: (29-19) Current loops form magnetic dipoles with magnetic moment µ: (29-27)
24 Summary of Chapter 29, cont.If currents are not constant, Ampère’s law needs modification: (29-31)