1. EMLAB 1 Chapter 3. Gauss’ law, Divergence

2. EMLAB 2 Displacement flux : Faraday’s Experiment charged sphere (+Q) + + ++ insulator metal Two concentric conducting spheres are separated by an insulating material. The inner sphere is charged to +Q. The outer sphere is initially uncharged. The outer sphere is grounded momentarily. The charge on the outer sphere is found to be -Q.

3. EMLAB 3 Faraday concluded there was a “displacement” from the charge on the inner sphere through the inner sphere through the insulator to the outer sphere. The electric displacement (or electric flux) is equal in magnitude to the charge that produces it, independent of the insulating material and the size of the spheres. +Q -Q +Q

4. EMLAB 4 Gauss’ law D : electric displacement flux density. The electric field lines come out of positive charges and induce another charge distribution. Gauss’ law : The integral of electric flux density over a closed surface is equal to the sum of charges inside the surface.

5. EMLAB 5 Integration surface deformability If the integrand follows an inverse square law, the integration surface is deformable as long as the position of the charge remains inside or outside the surface during deformation. The result of the integration does not change after deformation.

6. EMLAB 6 Proof : Gaussian surface contains a charge The contribution of the integrand on da 1 is equal to that on da 2

7. EMLAB 7 Because S 1 and S 2 are similar, the ratio of the area S 1 to that of S 2 is where S1S1 S2S2

8. EMLAB 8

9. 9 An integration surface is deformable as long as the position of the charge remains outside the surface S. The result of the surface integral remains the same. Proof : Gaussian surface contains no charge

10. EMLAB 10 Gauss’ law Gauss’ law holds for the case with multiple charges.

11. EMLAB 11 Gauss law usefulness The integration in Gauss’ law becomes trivial when there exists a symmetry in the problem geometry. ‘D r ’ can be taken out of the integral thanks to the symmetry. Gaussian surface

12. EMLAB 12 Example 1: electric field due to spherically symmetric volume charges (r<a) (r>a) Charge enclosed in the Surface integral Gaussian surface

13. EMLAB 13 D is determined by the charges inside the Gaussian surface regardless of the charges outside the surface. Usually, the total charge on the outer surface is equal to the negative of that on the inner surface. Example 2: E-field due to an axially symmetric charge distribution

14. EMLAB 14 Gauss’ law in differential form An integral form of Gauss’ law only states that a surface integral of D over a closed surface is equal to the total charge inside. We cannot obtain the behavior of D at one specific point. To observe D at a point, an infinitesimally small integration surface is chosen for the Gauss’ law.

15. EMLAB 15 Gauss’ law on an infinitesimally small surface The coordinate of the center of the cube is (x,y,z) The differential form of Gauss’ law states that the sum of partial derivatives of D x, D y, D z with respect to x, y, z is equal to the charge density at that point. ‘Divergence’ D

16. EMLAB 16 Divergence in cylindrical coordinate Divergence in spherical coordinate

17. EMLAB 17 Boundary Conditions on the Electric Field at the Surface of a Metallic Conductor + + + + + - - - - - E = 0

18. EMLAB 18 Del operator : rectangular coordinate Divergence theorem

19. EMLAB 19 Example 3.5

20. EMLAB 20 How to obtain D using a differential form Gauss’ law Due to boundary condition