1. Chapter 21
Gauss’s Law

2. Electric Field Lines
Electric field lines (convenient for visualizing electric field patterns) – lines pointing in the direction of the field vector at any point
The electric field vector is tangential to the electric field lines at each point
The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region

3. Electric Field Line Patterns
For a positive point charge the lines will radiate outward equally in all directions
For a negative point charge the lines will point inward equally in all directions

4. Electric Field Line Patterns
An electric dipole consists of two equal and opposite point charges
The number of field lines leaving the positive charge equals the number of lines terminating on the negative charge
For two equal but like point charges, at a great distance from the charges, the field would be approximately that of a single charge of 2q (bulging out of the field lines between the charges – repulsion)

5. Electric Field Line Patterns
For these two unequal and unlike point charges, at a great distance from the charges, the field would be approximately that of a single charge of +q (two lines leave the +2q charge for each line that terminates on -q)

6. Rules for Drawing Electric Field Lines
For a group of charges, the lines must begin on positive charges and end on negative charges
In the case of an excess of charge, some lines will begin or end infinitely far away
The number of lines drawn leaving a positive charge or ending on a negative charge is proportional to the magnitude of the charge
No two field lines can cross each other

7. Electric Flux
Electric flux is the product of the magnitude of the electric field and the surface area, A, perpendicular to the field
ΦE = EA

8. Electric Flux
The electric flux is proportional to the number of electric field lines penetrating some surface
The field lines may make some angle θ with the perpendicular to the surface
Then
The flux is a maximum (zero) when the surface is perpendicular (parallel) to the field

9. Electric Flux
If the field varies over the surface, Φ = EA cos θ is valid for only a small element of the area
In the more general case, look at a small area element
In general, this becomes

10. Electric Flux
The surface integral means the integral must be evaluated over the surface in question
The value of the flux depends both on the field pattern and on the surface
SI units: N.m2/C

11. Electric Flux, Closed Surface
For a closed surface, by convention, the A vectors are perpendicular to the surface at each point and point outward
(1) θ < 90o, Φ > 0
(2) θ = 90o, Φ = 0
(3) 180o > θ > 90o, Φ < 0

12. Electric Flux, Closed Surface
The net flux through the surface is proportional to the number of lines leaving the surface minus the number entering the surface

13. Electric Flux, Closed Surface
Example: flux through a cube
The field lines pass perpendicularly through two surfaces and are parallel to the other four surfaces
Side 1: Φ = – E l2
Side 2: Φ = E l2
For the other sides, Φ = 0
Therefore, Φtotal = 0

14. Chapter 21 Problem 44
What’s the flux through the hemispherical open surface of radius
R in a uniform field of magnitude E shown in the figure?

15. Chapter 21 Problem 23
The electric field on the surface of a 10-cm-diameter sphere is perpendicular to the sphere and has magnitude 47 kN/C. What’s the electric flux through the sphere?

16. Carl Friedrich Gauss
1777 – 1855
Gauss’ Law
Gauss’ Law: electric flux through any closed surface is proportional to the net charge Q inside the surface
εo = 8.85 x C2/Nm2 : permittivity of free space
The area in Φ is an imaginary Gaussian surface (does not have to coincide with the surface of a physical object)

17. Gauss’ Law
A positive point charge q is located at the center of a sphere of radius r
The magnitude of the electric field everywhere on the surface of the sphere is E = keq / r2
Asphere = 4πr2

18. Gauss’ Law
Gaussian surfaces of various shapes can surround the charge (only S1 is spherical)
The electric flux is proportional to the number of electric field lines penetrating these surfaces, and this number is the same
Thus the net flux through any closed surface surrounding a point charge q is given by q/eo and is independent of the shape of the surface

19. Gauss’ Law
If the charge is outside the closed surface of an arbitrary shape, then any field line entering the surface leaves at another point
Thus the electric flux through a closed surface that surrounds no charge is zero

20. Gauss’ Law
Since the electric field due to many charges is the vector sum of the electric fields produced by the individual charges, the flux through any closed surface can be expressed as
Although Gauss’s law can, in theory, be solved to find for any charge configuration, in practice it is limited to symmetric situations
One should choose a Gaussian surface over which the surface integral can be simplified and the electric field determined

21. Field Due to a Spherically Symmetric Charge Distribution
For r > a
For r < a

22. Field Due to a Spherically Symmetric Charge Distribution
Inside the sphere, E varies linearly with r (E → 0 as r → 0)
The field outside the sphere is equivalent to that of a point charge located at the center of the sphere

23. Electric Field of a Charged Thin Spherical Shell
The calculation of the field outside the shell is identical to that of a point charge
The electric field inside the shell is zero

24. Field Due to a Line of Charge
Select a cylindrical Gaussian surface (of radius r and length ℓ)
Electric field is constant in magnitude and perpendicular to the surface at every point on the curved part of the surface
The end view confirms the field is perpendicular to the curved surface
The field through the ends of the cylinder is 0 since the field is parallel to these surfaces

25. Field Due to a Line of Charge

26. Field Due to a Plane of Charge
The uniform field must be perpendicular to the sheet and directed either toward or away from the sheet
Use a cylindrical Gaussian surface
The flux through the ends is EA and there is no field through the curved part of the surface
Surface charge density σ = Q / A

27. Field Distance Dependences for Different Charge Distributions

28. Chapter 21 Problem 33
The electric field strength outside a charge distribution and 18 cm from its center has magnitude 55 kN/C. At 23 cm the field strength is 43 kN/C. Does the distribution have spherical or line symmetry?

29. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 1: The electric field is zero everywhere inside the conducting material
If this were not true there were an electric field inside the conductor, the free charge there would move and there would be a flow of charge – the conductor would not be in equilibrium

30. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 1: The electric field is zero everywhere inside the conducting material

31. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 1: The electric field is zero everywhere inside the conducting material

32. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 2: Any excess charge on an isolated conductor resides entirely on its surface
The electric field (and thus the flux) inside is zero whereas the Gaussian surface can be as close to the actual surface as desired, thus there can be no charge inside the surface and any net charge must reside on the surface

33. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 3: The electric field just outside a charged conductor is perpendicular to the surface and has a magnitude of σ/εo
If this was not true, the component along the surface would cause the charge to move – no equilibrium

34. Conductors in Electrostatic Equilibrium
When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium
An isolated conductor has the following properties:
Property 4: On an irregularly shaped conductor, the charge accumulates at locations where the radius of curvature of the surface is smallest
The forces from the charges at the sharp end produce a larger resultant force away from the surface.

35. Chapter 21 Problem 43
A total charge of 18 µC is applied to a thin, square metal plate 75 cm on a side. Find the electric field strength near the plate’s surface.

36. Answers to Even Numbered Problems
Chapter 21:
Problem 22
± 490 N ⋅ m2/C

37. Answers to Even Numbered Problems
Chapter 21:
Problem 26
1.81 × 103 N ⋅ m2/C

38. Answers to Even Numbered Problems
Chapter 21:
Problem 34
−2.0 × 10−4 C/m2

39. Answers to Even Numbered Problems
Chapter 21:
Problem 40
160 kN/C

40. Answers to Even Numbered Problems
Chapter 21:
Problem 58
58 nC/m2