Lect 2-Gauss Law

8/12/2019 Lect 2-Gauss Law

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Electricity and Magnetism

Lecture 2 Gausss Law

Outline

Electric flux: Definition andcalculation

Gausss law and its application in

calculating electric fields

Charge on conductors


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Review

2

2

0

r

Qqke

CoulombsLaw

F

q0 q0

2r

QkeE

Multiple charges:

Calculate both force and E field by superimposition principle


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Electric Flux: Definition

3

Electric Flux, , measures how many electric field lines passing through a

given surface perpendicular to the field lines

Generally,

The perpendicular to the area A is at an angle to the field lines

It is a scalar term

cosEAAEE

Surface is face-on to electric

field:

, =0

Flux EAE

E

A

Surface is titled with an

angle to the field:

Flux cosAE

Surface is edge-on to electric

field:

, = 90Flux 0E

E

A


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4

Electric Flux, , can be defined to not only a piece of open surface, but

also a closed surface (see below). The direction of flux follows the

direction of electric field lines.

Note: Flux can be positive or negative, but it is a scalar term.

Electric Flux: Definition


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Condition of Zero Electric Flux

5

If no charge, or zeronet

charge inside a box, the net electric fluxthrough the total surface of the box is zero.


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Gauss Law

Gauss Law states that the electric flux through any

closed surface is equal to the net charge inside thesurface divided by o

6

= q2

q3

1

is the electric field at any point on the surface andqirepresents the net charge inside the surface. o= 8.85 x 10

-12C2/Nm2 is the permittivity of freespace

The area is an imaginary surface, called Gaussiansurface, but it does not have to coincide with the

surface of a physical object

= d =

In general case of non-uniform

electric field, we need integrate

Carl Friedrich

Gauss (1777

1855), German

mathematician and

physical scientist.

Contributed

significantly to manyfields, including

number theory,

algebra, statistics,

analysis, differential

geometry, geodesy,

geophysics,

electrostatics,astronomy, and

optics.


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Gauss Law

Gausss law applies to any random Gaussian surface (but it must be a

closed surface)

7

In practice, we choose high-symmetry Gaussian surfaces for convenience(see examples in next slides)

Question:

Two point charges, +qandq, arearranged as shown.

Through which closed surface(s) is

the net electric flux equal to zero?


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Applications of Gausss LawExample 1

8

To calculate the electric field, the first step is to choose the correct

Gaussian surface in a way that, either the field is the same inmagnitude, or is zero.


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9

Uniformly-charged infinite large plane.

The Gaussian surface is a cylinder passing through andperpendicular to the plane


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10

Uniformly-charged long rod.

The Gaussian surface is a cylinder surrounding the rod


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The calculation of the field outside the shell is identical to thatof a point charge

The electric field inside the shell is zero

11

224 r

QkrQE e

o

(r > b)

0E (r


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Parallel conducting plate

Assuming infinitely large parallelconducting plates, so that electric

field between plates are uniform.

Based on Gauss law, the total

electric field between the plates is

given by

The field outside the plates is zero,

because of cancelation of two

opposite electric fields by the twoplates, respectively.

12

o

E


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Charge Distribution on Conductors

Under electrostatic conditions (no charge flow), the electric field inside the conductorshould be zero. According to Gausss law, there should be no net charge inside. Any excesscharge on a conductor resides entirely on its surface.

This is an important result (see more on next slide).

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Charge Distribution on Conductors

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Summary:

Electric Flux

Flux is a measure of the flow of electric field lines

through a surface. The general form is

15

= d =

Gausss Law

The total flux through any closed surface is equal to the netcharge Qinside the surface divided by the permittivity of free

space:

can be used to calculate electric fields of highly symmetric

distributions of charge. Gausss law is an alternative to Coulombs law.

AdEAdEE

cosEAAEE

Consequence of Gausss law is that, recess charges on a

conductor reside entirely on the surface. Inside the

conductor, E = 0.

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Lect 2-Gauss Law