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. Electric Flux Density D (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).
Ds S
The total flux passing through the closed surface is d closed Ds dS
surface
3.2 Gauss’s Law 4.To a gaussian surface, the mathematical formulation of Gauss’s law DS dS charge closed Q( Qn L dL S dS d )
4. Shown in the right figure
Q D r a a (inner ) 2 r 4a Q D r b a (outer ) 2 r 4b Q D a ( a r b) 2 r 4r
3.1 Electric Flux Density
3.1 Electric Flux Density
1. Faraday’s experiment: a larger positive charge on the inner sphere induced a corresponding larger negative charge on the outer sphere. 2. Electric Flux: Q(coulombs )
S
5. The last form is usually used S
DS dS d
6. For example: placing a point charge Q at the origin of a spherical coordinate system and choose a sphere of radius a as the gaussian S.
3.2 Gauss’s Law
1. Gauss’s law: The electric flux passing through any closed surface is equal to the total charge enclosed by that surface. ---the generalization of Faraday’s 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. 3.Consider the nature of an incremental of the surface S : How to describe an incremental element of area?S S n The flux crossing S is
E
Q Q D 0 E a D a D a r a 2 r 2 r 2 r 40r 4r 4a Q dS a 2 sin d d ar D dS sin d d 4 2 Q S DS dS 0 0 4 sin d d Q
Q
7. Exercise: D3.3(P61)
3.3 Application of Gauss’s Law:some symme Determining DS if the charge distribution is known Choose a closed surface in which DS is everywhere either normal or tangential to the closed surface On that portion of the closed surface for which DS dS is not zero, DS=constant
5. If the inner sphere becomes a point charge of Q, we still have
D
6. Compared with E Q 4 0 r , we have a 2 r
Q a 2 r 4 r
D 0 E(in freespace) 7. For a general volume charge distribution: d d E a D ar r 2 2 vol 4 R vol 4R 0 8. For dielectrics, the relationship betweenD and E will be more complicated