ENE 325 Electromagnetic Fields and Waves

1. ENE 325Electromagnetic Fields and Waves Lecture 3 Gauss’s law and applications, Divergence, and Point Form of Gauss’s law

2. Review (1) • Coulomb’s law • Coulomb’s force • electric field intensity or V/m

3. Review (2) • Electric field intensity in different charge configurations • infinite line charge • ring charge • surface charge

4. Outline • Gauss’s law and applications • Divergence and point form of Gauss’s law

5. Gauss’s law and applications • “The net electric flux through any closed surface is equal to the total charge enclosed by that surface”. • If we completely enclose a charge, then the net flux passing through the enclosing surface must be equal to the charge enclosed, Qenc.

6. Gauss’s law and applications • The integral form of Gauss’s law: • Gauss’s law is useful in finding the fields for problems that have a high degree of symmetry by following these steps: • Determine what variables influence and what components of are present. • Select an enclosing surface, Gaussian surface, whose surface vector is directed outward from the enclosed volume and is everywhere either tangential to or normal to

7. Gauss’s law and applications • The enclosing surface must be selected in order for to be constant and to be able to pull it out of the integral.

8. Ex1 Determine from a charge Q located at the origin by using Gauss’s law. 1. 2. Select a Gaussian surface 3.Drat a fixed distance is constant and normal to a Gaussian surface, can be pulled out from the integral.

9. Ex2 Find at any point P (, , z) from an infinite length line of charge density L on the z-axis. 1. From symmetry, 2. Select a Gaussian surface with radius  and length h. 3. D at a fixed distance is constant and normal to a Gaussian surface, can be pulled out from the integral. ant and normal to a Gaussian surface, can be pulled out from the integral.

10. Ex3 A parallel plate capacitor has surface charge +S located underneath a top plate and surface charge -S located on a bottom plate. Use Gauss’s law to find and between plates.

11. Ex4 Determine electric flux density for a coaxial cable.

12. Ex5 A point charge of 0.25 C is located at r = 0 and uniform surface charge densities are located as follows: 2 mC/m2 at r = 1 cm and -0.6 mC/m2 at r = 1.8 cm. Calculate at • r = 0.5 cm • r = 1.5 cm

13. c) r = 2.5 cm

14. Divergence and Point form of Gauss’s law(1) • Divergence of a vector field at a particular point in space is a spatial derivative of the field indicating to what degree the field emanates from the point. Divergence is a scalar quantity that implies whether the point source contains a source or a sink of field. where = volume differential element

15. Divergence and Point form of Gauss’s law(2) or we can write in derivative form as Del operator: It is apparent that therefore we can write a differential or a point form of Gauss’s law as

16. Divergence and Point form of Gauss’s law(3) For a cylindrical coordinate: For a spherical coordinate:

17. Physical example The plunger moves up and down indicating net movement of molecules out and in, respectively. • positive indicates a source of flux. (positive charge) • negative indicates a sink of flux. (negative charge) An integral form of Gauss’s law can also be written as

18. Ex6Let . Determine

19. Ex7Let C/m2 for a radius r = 0 to r = 3 m in a cylindrical coordinate system and for r > 3 m. Determine a charge density at each location.

20. Ex8 Let in a cylindrical coordinate system. Determine both terms of the divergence theorem for a volume enclosed by r = 1 m, r = 2 m, z = 0 m, and z = 10 m.