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1、3.1 Electric Flux Density,2. Electric Flux:,3. Electric Flux Density (coulombs/square meter):direction (the direction of the flux lines at that point) and magnitude (the number of flux lines crossing a surface normal to the lines divided by the S. area).,4. Shown in the right figure,1. Faradays expe
2、riment: a larger positive charge on the inner sphere induced a corresponding larger negative charge on the outer sphere.,3.1 Electric Flux Density,5. If the inner sphere becomes a point charge of Q, we still have,6. Compared with , we have,7. For a general volume charge distribution:,8. For dielectr
3、ics, the relationship between and will be more complicated,3.1 Electric Flux Density,9. Example 3.1: find D in the region about a uniform line charge of 8nC/m lying along the z axis in free space.,10. Exercise: D3.1, D3.2,3.2 Gausss Law,1. Gausss law: The electric flux passing through any closed sur
4、face is equal to the total charge enclosed by that surface. -the generalization of Faradays experiment,2.A cloud of point charges(total charge Q) are shown in the following Figure. There is some value DS at every point on the surface.,How to describe an incremental element of area?,The flux crossing
5、 is,The total flux passing through the closed surface is,3.Consider the nature of an incremental of the surface :,3.2 Gausss Law,4.To a gaussian surface, the mathematical formulation of Gausss law,5. The last form is usually used,6. For example: placing a point charge Q at the origin of a spherical
6、coordinate system and choose a sphere of radius a as the gaussian S.,7. Exercise: D3.3(P61),3.3 Application of Gausss Law:some symmetrical charge.,1. Determining if the charge distribution is known,Choose a closed surface in which is everywhere either normal or tangential to the closed surface,On th
7、at portion of the closed surface for which is not zero, =constant,3.3 Application of Gausss Law,3. A second example: the uniform line charge distribution lying along the z axis and extending from - to +,Only the radial component of D is present,So we can choose a cylindrical surface to which is ever
8、ywhere normal,4. If symmetry does not exist, we cannot use Gausss Law to obtain a solution.,The radial component is a function of only,3.3 Application of Gausss Law,5. A coaxial cable, the inner of radius a and the outer radius b, a charge distribution of on the outer surface of the inner conductor,
9、Choose a right circular cylindrical of length L and radius and , so we have,3.3 Application of Gausss Law,6. Example 3.2: Select a 50-cm length of coaxial cable having an inner radius of 1mm, and an outer radius of 4mm. The space between conductors is assumed to be filled with air. The total charge
10、on the inner conductor is 30nC. We wish to know the charge density on each conductor, and E and D vector fields.,7. Exercise: D3.5(P66),3.4 Application of Gausss Law:Differential Volume Element,1. Apply Gausss Law to the problem without any symmetry.,Choose a very small closed surface: D is almost c
11、onstant and the small change in D can be represented by the first two terms of Taylors-series expansion for D.,2. The aim is to receive some information about the way D varies in the region of our small surface.,3. Consider any point, shown in the following Figure. Choose the small rectangular box a
12、s the closed surface,3.4 Application of Gausss Law:Differential Volume Element,3. Consider any point, shown in the following Figure. Choose the small rectangular box as the closed surface,3.4 Application of Gausss Law:Differential Volume Element,By the same process, we can obtain,These results may b
13、e collected to yield,4. Apply Gausss law to the closed surface surrounding the volume element and have an approximate result,3.4 Application of Gausss Law:Differential Volume Element,5. Example 3.3: find an approximate value for the total charge enclosed in an incremental volume of 10-9 m3 located a
14、t the origin, if,6. Exercise: D3.6(P70),3.5 Divergence,1. This equation can be written as,Or as a limit:,2. The last term is the volume charge density, hence,3. The methods could have been used on any vector A,3.5 Divergence,4. The divergence of the vector flux density A is : the outflow of flux fro
15、m a small closed surface per unit volume as the volume shrinks to zero.,5. A positive divergence indicates a source; a negative divergence indicates a sink.,6. In cartesian coordinate system:,7. Note:Divergence is performed on a vector; but the result is a scalar.,3.5 Divergence,8. Example 3.4: Find
16、 div D at the origin if,9. Exercise: D3.7(P73),3.6 Maxwells First Equation (Electrostatics),1. The first is the definition of divergence; The second is the result of applying the definition to a differential volume element.,2. This is the first of Maxwells equations as they apply to electrostatics a
17、nd steady magnetic fields. It states that the electric flux per unit volume leaving a vanishingly small volume unit is exactly equal to the volume charge density there.,3. Maxwells first equation is described as the differential-equation form of Gausss law and Gausss law is recognized as the integra
18、l form of Maxwells first equation.,3.6 Maxwells First Equation (Electrostatics),4. Consider the divergence of D in the region about a point charge Q located at the origin.,5. Exercise: D3.8(P74),3.7 The vector operator and the divergence theorem,1. Define the del operator as a vector operator,2. The
19、 use of is much more prevalent than that of,3. Divergence theorem: The integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector throughout the volume enclosed by the closed surface,3.7 The vector operator and the divergence theorem,4. The divergence theorem is true for any vector field,5. Give an example to specify the divergence theorem.,6. Example 3.5: Evaluate both sides of the divergence theorem for the field and the rectangular parellelepiped formed by the planes x=0 and 1, y=0 and 2, z=0 and 3.,7. D3.9(P78),
