Phy 213: General Physics III

1. Phy 213: General Physics III Chapter 23: Gauss’ Law Lecture Notes

2. f Electric Flux • Consider a electric field passing through a region in space. The electric flux , FE, is the scalar product of the average electric field vector and the normal area vector for the region, or: • Conceptually, the electric flux represents the flow of electric field lines (normal) through a region of space

3. Gauss’ Law • Gauss’ Law is a fundamental law of nature relating electric charge to electric flux • According to Gauss’ Law, the total electric flux through any closed (“Gaussian”) surface is equal to the enclosed charge (Qenclosed) divided by the permittivity of free space (eo): or, • Gauss’ Law can be used to determine the electric field (E) for many physical orientations (distributions) of charge • Many consequences of Gauss’ Law provide insights that are not necessarily obvious when applying Coulomb’s Law

4. Gauss’ Law for a Point Charge Consider a point charge, q • An appropriate “gaussian” surface for a point charge is a sphere, radius r, centered on the charge, since all E fields will be ∟ to the surface • The electric flux is: • Applying Gauss’ Law yields E: q r Gaussian sphere

5. Gaussian sphere q R r Spherical Symmetry: Charged Sphere • Spherical symmetry can be exploited for any spherical shape • Consider a hollow conducting sphere, radius R & charge q: • The flux through a “gaussian” sphere is: • Applying Gauss’ Law: • For r > R: • For r < R: Inside a hollow conductor, E = 0

6. r R r Gaussian sphere Spherical Symmetry: Charged Sphere (2) Consider a solid non-conducting sphere with radius R & uniform charge density, r: • Applying Gauss’ Law: • For r > R: • For r < R: Inside a uniformly charged insulator, E ≠ 0

7. Spherical Symmetry: Charged Spheres (3) Consider 2 concentric hollow conducting spheres with radii R1 & R1 and charges q1 & q2: • Applying Gauss’ Law: • For r < R1: • For R2 > r > R1: • For r > R2: {opposite charge subtracts} Gaussian sphere q1 R1 r R2 q2

8. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + l 0 r x Application: Line of Charge • Cylindrical symmetry is useful for evaluating a “line of charge” as well as charged cylindrical shapes • Consider an infinitely long line of charge, charge density l: • The flux through each of the 3 surfaces is: • Applying Gauss’ Law:

9. Gaussian “pill-box” Planar Symmetry: A Charged Sheet • A Gaussian “pill-box” is an appropriate Gaussian surface for evaluating a charged flat sheet • Consider a uniformly charged flat sheet with a surface charge density, s: • The flux through the pill-box surface is: • The electric field:

10. +q Gaussian “pill-box” -q Application: 2 Parallel Charged Plates • Two oppositely charged conducting plates, with charge density s: • Surface charges draw toward each other on the inner face of each plate • To determine the electric field within the plates, apply a Gaussian “pill-box” to one of the plates • The flux through the pill-box surface is: • The electric field is:

11. Consequences of Gauss’ Law • Electric flux is a conserved quantity for an enclosed electric charge • The electric field inside a charged solid conductor is zero • The electric field inside a charged hollow conductor or non-conductor is zero • Geometrical symmetries can make the calculation of an electric field less cumbersome even for complicated charge distributions