I-2 Gauss’ Law

1. I-2 Gauss’ Law

2. Main Topics • The Electric Flux. • The Gauss’ Law. • The Charge Density. • Use the G. L. to calculate the field of a • A Point Charge • An Infinite Uniformly Charged Wire • An Infinite Uniformly Charged Plane • Two Infinite Charged Planes

3. The Electric Flux • The electric flux is defined as It represents amount of electric intensity which flows perpendicularly through a surface, characterized by its outer normal vector . The surface must be so small that can be considered constant there. • Let’s revisit the scalar product.

4. The Gauss’ Law I • Total electric flux through a closed surface is equal to the net charge contained in the volume surrounded by the surface divided by the permitivity of vacuum . • It is equivalent to the statement that field lines begin in positive charges and end in negative charges.

5. The Gauss’ Law II • Field lines can both begin or end in the infinity. • G. L. is roughly because the decrease of intensity the with r2 in the flux is compensated by the increase with r2 of surface of the sphere. • The scalar product takes care of the mutualorientation of the surface and the intensity.

6. The Gauss’ Law III • If there is no charge in the volume each field line which enters it must also leave it. • If there is a positive charge in the volume then more lines leave it than enter it. • If there is a negative charge in the volume then more lines enter it than leave it. • Positive charges are sources and negative are sinks of the field. • Infinity can be either source or sink of the field.

7. The Gauss’ Law IV • Gauss’ law can be taken as the basis of electrostatics as well as Coulomb’s law. It is actually more general! • Gauss’ law is useful: • for theoretical purposes • in cases of a special symmetry

8. The Charge Density • In real situations we often do not deal with point charges but rather with chargedbodies with macroscopic dimensions. • Then it is usually convenient to define the charge density i.e. charge per unit volume or surface or length, according to the symmetry of the problem. • Since charge density may depend on the position, its use makes sense mainly if the bodies are uniformly charged e.g. conductors in equilibrium.

9. A Point Charge I • As a Gaussian surface we choose a spherical surface centered on the charge. • Intensity is perpendicular to the spherical surface in every point and so parallel (or antiparallel) to its normal. • At the same time E is constant on the surface, so :

10. A Point Charge II • So we get the same expression for the intensity as from the Coulomb’s Law : • Here we also see where from the “strange term” appears in the Coulomb’s Law!

11. An Infinite Uniformly Charged Wire I • Conductive wire (in equilibrium) must be charged uniformly so we can define thelengthchargedensity as charge per unit length: • Both Q and Lcan be infinite, yet have a finite ratio. • The wire is axis of the symmetry of the problem.

12. An Infinite Uniformly Charged Wire II • Intensity lies in planes perpendicular to the wire and it is radial. • As a Gaussian surface we choose a cylindricalsurface (of some length L) centered on the wire. • Intensity is perpendicular to the surface in every point and so parallel to its normal. • At the same time E is constanteverywhere on this surface.

13. Infinite Wire III • Flux through the flat caps is zero since here the intensity is perpendicular to the normal. • So :

14. Infinite Wire IV • By making one dimension infinite the intensity decreases ~ 1/r instead of 1/r2 which was the case of a point charge! • Again, we can obtain the same result using the Coulomb’s law and the superposition principle but it is “a little” more difficult!

15. An Infinite Charged Conductive Plane I • If the charging is uniform, we can define the surfacechargedensity : • Again both Q and A can be infinite yet reach a finite ratio, which is the charge per unit surface. • From the symmetry the intensity must be everywhere perpendicular to the surface.

16. Infinite Plane II • As a Gaussian surface we can take e.g. a cylinder whose axis is perpendicular to the plane. It should be cut in halves by the plane. • Nonzero flux will flow only through both flat cups (with some magnitude A) since is perpendicular to them.

17. Infinite Plane III • This time doesn’t change with the distance from the plane. Such a field is called homogeneous or uniform! • Note that both magnitude and direction of the vectors must be the same if the vectorfield should be uniform.

18. Quiz: Two Parallel Planes • Two large parallel planes are d apart. One is charged with a charge density , the other with -. Let Eb be the intensity between and Eooutside of the planes. What is true? • A) Eb= 0, Eo=/0 • B) Eb= /0, Eo=0 • C) Eb= /0, Eo=/20

19. Homework • The one from yesterday is due tomorrow! • The next one will be assigned tomorrow.

20. Things to read • This lecture covers: Giancoli: Chapter 22 • Advance reading : Giancoli : Chapter 23-1, 23-2

21. The scalar or dot product Let Definition I. (components) • Definition II. (projection) Can you proof their equivalence? ^

22. Gauss’ Law • The exact definition: • In cases of a special symmetry we can find Gaussian surface on which the magnitude E is constant andis everywhere parallel to the surface normal. Then simply: ^

23. Infinite Wire by C.L.– die hard! Only radial component Er of is non-zero • We have to substitute all variables using  and integrate from 0 to : • “Quiz”: What was easier? ^