Presentation on theme: "Sources of Magnetic Field"— Presentation transcript:
1 Sources of Magnetic Field Chapter 30Sources of Magnetic Field
2 IntroductionThis chapter will focus on the sources of magnetic fields: moving charges.We’ll look at the field created by a current carrying conductor, as well as other symmetrical current distributions.We’ll look at the force between two current-carrying conductors.We’ll finish by looking at the processes that result in materials being naturally magnetic.
3 30.1 The Biot-Savart LawShortly after Oersted discovers that a compass needle is deflected by a nearby current-carrying conductor (1819) Jean-Baptiste Biot and Felix Savart begin quantitative experimentation.Their experimental results have given a mathematical expression for the magnetic field at some point in space in terms of the current that causes it.
4 30.1We will summarize results of their experiment concerning the magnetic field dB at a point P associated with a length element ds on a wire carrying current I.
5 30.1 The results: The vector dB is perpendicular to both ds and r. The magnitude of dB is inversely proportional to r2, where r is the distance from ds to point P.The magnitude of dB is proportional to the current and to the magnitude of ds.The magnitude of dB is proportional to sin θ, where θ is the angle between ds and r.
6 30.1The mathematical expression summarizng these observations is known as the Biot-Savart Law:Remember μo is the permeability of free spaceAgain, note that dB is only the field created by a single element of the conductor and to find the total magnetic field B at point P, we must integrate.
7 30.1Integration givesWe must be careful with this integration as in does involved the vector cross product.For direction of the B field, the right hand rule is used.Point thumb in direction of I, curling fingers show the direction of B.
8 30.1Quick Quiz p. 928Example 30.1Resulting Equations 30.2, 30.3
9 30.2 Magnetic Force between Two Parallel Conductors If we have two current carrying wires, the B field caused by one current will exert a force on the other.If the conductors are parallelto each other, then the Bfield is perpendicular to thecurrent.
10 30.2 The force on wire one from B-Field two is And assuming long wires ( << a)So the Force on wire one is given
11 30.2 Often we make use of the Force per unit length By applying the right hand rule, we can see that two wires carrying current in the same direction will attract each other.Currents in the opposite direction will result in the conductors repelling.
12 30.2 This is how the fundamental unit of the Ampere is defined. When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x N/m, the current in each wire is 1 A.
13 30.2 From this the quantity of the Coulomb is defined. When a conductor carries a steady current of 1 A, the quantity of charge that flows through the cross section in 1 sec is 1 C.Quick Quizzes p. 933
14 30.2Ex: A long wire carries a current of 80 A. How much current must a second parallel wire carry if it is located 20 cm below the first wire such that it will not fall due to the force of gravity? Assume the lower wire has a linear density of 0.12 g/cm and a length of 1 m.
15 30.3 Ampere’s LawWe know that a current will create a magnetic field in a circular path around a conductor.
16 30.3From symmetry we can assume that for a given circular path, where the conductor passes perpendicularly through the center, the magnitude of B is the same.We also know that B varies proportionally with current I, and inverse proportionally with the distance from it, a.
17 30.3If we look at the product of B and length element ds, and sum this around the circular path, this is called a line integral.B and ds are parallel to each other.And B is constant at radius r
18 30.3 So the line integral goes as follows. But r and a are the same value, so the circumference cancels, giving Ampere’s Law
19 30.3Ampere’s Law- the line integral of B.ds around any closed path, equals μoI where I is the net steady current passing through any surface bound by the closed path.Ampere’s Law describes the creation of magnetic fields and will have similar application to Gauss’s Law, for applications of high symmetry.
22 30.3 Example 30.6 “Infinite Current Sheet” Js is the linear current densityalong the z axis in the picturegiven.
23 30.4 Magnetic Field of Solenoid A solenoid is a long wire wound in a helix. It can create a reasonably uniform magnetic field in its interior.If the turns are tight togetherand the solenoid has a finitelength, it closely resembles themagnetic field of a bar magnet.
27 30.5 Magnetic FluxMagnetic Flux is similar to Electric flux in that it describes the amount of electric field lines penetrating a surface.Consider and arbitrary object and element of surface area, dA.The flux through theelement is B.dA
28 30.5The total magnetic flux through the surface is is sum of the flux through each surface element.If the field is uniform at an angle θ, to the area vector then
29 30.5So if the field lines run parallel to the surface, θ = 90o and the flux is zero.If the field lines are perpendicular to the surface, then θ = 0o and the flux is a maximumn value.
30 30.5 If the field is not uniform, and integration is often performed. Example 30.8 p. 941
31 30.6 Gauss’s Law for Magnetism Different from Gauss’s Law for Electric Fields.Electric FieldsThe net electric flux through a closed surface depends on the net charge inside (Qin)Magnetic FieldsThe net magnetic flux through any closed surface is zero.This is because all field lines are closed loops.They do not originate/terminate on discrete charges.