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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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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
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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
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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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Applications of Gausss LawExample 2
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Uniformly-charged infinite large plane.
The Gaussian surface is a cylinder passing through andperpendicular to the plane
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Applications of Gausss LawExample 3
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Uniformly-charged long rod.
The Gaussian surface is a cylinder surrounding the rod
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Applications of Gausss LawExample 4
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.