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Unformatted text preview: Chapter 17 Magnetic Field: Causes 137 17 Magnetic Field: Causes This chapter is about magnetism but lets think back to our introduction to charge for a moment. We talked about the electric field before saying much about what caused it. We said the electric field exerts a force on a particle that has charge. Later we found out that charged particles play not only the role of victim to the electric field but, that charged particles cause electric fields to exist. Now we have been talking about the magnetic field. We have said that the magnetic field exerts a torque on a particle that has magnetic dipole moment. You might guess that a particle that has magnetic dipole moment would cause a magnetic field. Youd be right! A particle that has the physical property known as magnetic dipole moment causes a magnetic field to exist in the region of space around it. A magnetic field can be caused to exist by a particle having magnetic dipole moment or a distribution of particles having magnetic dipole moment. The magnetic field at point P , an empty point in space in the vicinity of a particle that has a magnetic dipole moment, due to that particle-with-magnetic-dipole-moment, is given by 3 o ) ( 3 4 r B h h h = r r (17-1) where A m T 10 4 7 o = is a universal constant which goes by the name of the magnetic permeability of free space. This value is to be taken as exact. (Do not treat the 4 as a value known to only one significant digit.) B h is the magnetic field vector at point P , where P is an empty point in space a distance r away from the particle-with-magnetic-dipole-moment that is causing B h . h is the magnetic dipole moment of the particle that is causing the magnetic field. r is a unit vector in the direction from the particle, toward point P . Defining r h to be the position vector of point P relative to the location of the particle-with-magnetic-dipole- moment, r r r = h so r r r h = . r is the distance that point P is from the particle-with-magnetic-dipole-moment. A particle-with-magnetic-dipole-moment is called a magnetic dipole. Note that the magnetic field due to a magnetic dipole dies off like 3 1 r . Chapter 17 Magnetic Field: Causes 138 Example A particle is at the origin of a Cartesian coordinate system. The magnetic dipole moment of the particle is j 2 m A 1 . . Find the magnetic field vector, due to the particle, at (3 . cm, 4 . cm). Solution Im going to start with a diagram of the configuration. Note that I do not know the direction of B in advance, so, I have drawn B h on the diagram in a fairly arbitrary direction. I did want to put B h on there to make it more evident that we are dealing with the magnetic field at point P, caused by the particle at the origin. Also, I intentionally drew B h in a direction other than that of r h , to avoid conveying the false impression that B h is necessarily in the direction of r h . (At some points, it is, but those points are the...
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