Physics
Divergence of Magnetic Field
Divergence of a magnetic field refers to the spreading out or the "source" of the magnetic field lines. It is a measure of how much the field is expanding or contracting at a given point. In mathematical terms, it is the dot product of the vector field and the del operator.
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3 Key excerpts on "Divergence of Magnetic Field"
eBook - PDFCalculus
Multivariable
- William G. McCallum, Deborah Hughes-Hallett, Daniel E. Flath, Andrew M. Gleason, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Ayse Arzu Sahin, Adam H. Spiegler, Jeff Tecosky-Feldman, Thomas W. Tucker, Aaron(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density. Figure 19.29: Vector field showing a source Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field , written div , is a scalar-valued function defined by div (, , ) = lim Volume→0 ∫ ⋅ Volume of . Here is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE DIVERGENCE OF A VECTOR FIELD 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If = 1 + 2 + 3 , then div = 1 + 2 + 3 . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div , namely ∇ ⋅ . Using ∇ = + + , we can write div = ∇⋅ = ( + + ) ⋅ ( 1 + 2 + 3 ) = 1 + 2 + 3 . Example 1 Calculate the divergence of ( ) = at the origin (a) Using the geometric definition.
eBook - PDF- Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
We say that the origin is a source. Figure 19.30 suggests flow into the origin; in this case we say that the origin is a sink. In this section we use the flux out of a closed surface surrounding a point to measure the outflow per unit volume there, also called the divergence, or flux density. Figure 19.29: Vector field showing a source Figure 19.30: Vector field showing a sink Definition of Divergence To measure the outflow per unit volume of a vector field at a point, we calculate the flux out of a small sphere centered at the point, divide by the volume enclosed by the sphere, then take the limit of this flux-to-volume ratio as the sphere contracts around the point. Geometric Definition of Divergence The divergence, or flux density, of a smooth vector field , written div , is a scalar-valued function defined by div (, , ) = lim Volume→0 ∫ ⋅ Volume of . Here is a sphere centered at (, , ), oriented outward, that contracts down to (, , ) in the limit. The limit can be computed using other shapes as well, such as the cubes in Example 2. 3 Although not all vector fields represent physically realistic fluid flows, it is useful to think of them in this way. 19.3 THE DIVERGENCE OF A VECTOR FIELD 983 In Cartesian coordinates, the divergence can also be calculated using the following formula. We show these definitions are equivalent later in the section. Cartesian Coordinate Definition of Divergence If = 1 + 2 + 3 , then div = 1 + 2 + 3 . The dot product formula gives an easy way to remember the Cartesian coordinate definition, and suggests another common notation for div , namely ∇ ⋅ . Using ∇ = + + , we can write div = ∇⋅ = ( + + ) ⋅ ( 1 + 2 + 3 ) = 1 + 2 + 3 . Example 1 Calculate the divergence of ( ) = at the origin (a) Using the geometric definition.
eBook - PDF- D. S. Jones, I. N. Sneddon, S. Ulam, M. Stark(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
Th e vector B is called the magnetic flux density. Two other vectors play a role in specifying the electromagnetic field and they are related to the lines of force which emanate from charges and currents. The vector D, which is called the electric flux density, effectively measures the number of lines of force which originate from a charge. The vector H, which is called the magnetic intensity, is such that its value on a closed curve effectively measures the current which passes through the curve. We shall assume that the vectors E, B, D and H are continuous and possess continuous derivatives at ordinary points. Their properties at the surfaces of materia bodies will have to be investigated . It will now be postulated that at ordinary points Maxwell's equation s curlE + — = 0 , (3) c u r l H -^ - = J, (4) d i v D = £ , (5) divB = 0 (6) are satisfied. The justification of this postulate lies in the fact that, firstly, in the particular cases of electrostatics, magnetostatics, the magnetic effects of steady currents and slowly varying currents, when certain terms in the equations can be neglected the phenomena predicted are in agreement with experiment. Secondly, for rapidly varying currents and for the propagation of disturbances when all terms in the equations have to be taken into account, the predictions are in accordance with experiment. It is with this second aspect that we are concerned in this book. 2* 4 THEORY OF ELECTROMAGNETISM 1.2 The equation of continuity Since the divergence of the curl of any vector vanishes identically we obtain, by taking the divergence of (4), divj = — div —— = — — -d i v D . et ct The interchange of the operators div and d/dt is permissible because we have assumed the continuity of D and its derivatives at ordinary points. A substitution from (5) uow gives d i v j + 4f = o · (7) By analogy with a corresponding equation in hydrodynamics, (7) is usually termed the equation of continuity.
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