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# Phy 213: General Physics III - PowerPoint PPT Presentation

Phy 213: General Physics III. Chapter 23: Gauss’ Law Lecture Notes. f. Electric Flux. Consider a electric field passing through a region in space. The electric flux , F E , is the scalar product of the average electric field vector and the normal area vector for the region, or:

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### Phy 213: General Physics III

Chapter 23: Gauss’ Law

Lecture Notes

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

• 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

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

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

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

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

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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:

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:

+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:

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