W02D2 Gauss

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W02D2Gausss Law
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From Last ClassElectric Field Using Coulomb and
Integrating

• Dipole E falls off like 1/r3
• Spherical charge E falls off like 1/r2
• Line of charge E falls off like 1/r
• (infinite)
• Plane of charge E uniform on
• (infinite) either side of plane

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Announcements
Math Review Week Three Tuesday from 9-11 pm in
26-152Vector Calculus PS 2 due Week Three
Tuesday at 9 pm in boxes outside 32-082 or
26-152 W02D3 Reading Assignment Course Notes
Chapter Course Notes Sections 3.6, 3.7, 3.10
Make sure your clicker is registered
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Outline

• Electric Flux
• Gausss Law
• Calculating Electric Fields using Gausss Law

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Gausss Law

• The first Maxwell Equation!
• A very useful computational technique to find the
  electric field when the source has enough
  symmetry.

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Gausss Law The Idea
The total flux of field lines penetrating any
of these closed surfaces is the same and depends
only on the amount of charge inside
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Gausss Law The Equation
Electric flux (the surface integral of E
over closed surface S) is proportional to charge
enclosed by the volume enclosed by S
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Electric Flux
Case I E is a uniform vector field
perpendicular to planar surface S of area A
Our Goal Always reduce problem to finding a
surface where we can take E out of integral and
get simply EArea
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Electric Flux
Case II E is uniform vector field directed at
angle to planar surface S of area A
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Concept Question Flux
The electric flux through the planar surface
below (positive unit normal to left) is

1. positive.
2. negative.
3. zero.
4. Not well defined.

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Concept Question Answer Flux
Answer 2. The flux is negative.

• The field lines go from left to right, opposite
  the assigned normal direction. Hence the flux is
  negative.

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Open and Closed Surfaces
A rectangle is an open surface it does NOT
contain a volume A sphere is a closed surface
it DOES contain a volume
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Area Element Closed Surface
Case III not uniform, surface curved For
closed surface, is normal to surface and
points outward ( from inside to outside)
if points out
if points in
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Group Problem Electric Flux Sphere
Consider a point-like charged object with charge
Q located at the origin. What is the electric
flux on a spherical surface (Gaussian surface) of
radius r ?
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Arbitrary Gaussian Surfaces
True for all surfaces such as S1, S2 or S3 Why?
As area gets bigger E gets smaller
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Gausss Law
Note Integral must be over closed surface
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Concept Question Flux thru Sphere
The total flux through the below spherical
surface is

1. positive (net outward flux).
2. negative (net inward flux).
3. zero.
4. Not well defined.

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Concept Question Answer Flux thru Sphere
Answer 3. The total flux is zero

• We know this from Gausss Law
• No enclosed charge ? no net flux.
• Flux in on left cancelled by flux out on right

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Concept Question Gausss Law

• The grass seeds figure shows the electric field
  of three charges with charges 1, 1, and -1,
  The Gaussian surface in the figure is a sphere
  containing two of the charges. The electric flux
  through the spherical Gaussian surface is
• Positive
• Negative
• Zero
• Impossible to determine without more
  information.

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Concept Question Answer Gausss Law
Answer 3 Zero. The field lines around the two
charged objects inside the Gaussian surface are
the field lines associated with a dipole, so the
charge enclosed in the Gaussian surface is zero.
Therefore the electric flux on the surface is
zero. Note that the electric field E is clearly
NOT zero on the surface of the sphere. It is
only the INTEGRAL over the spherical surface of E
dotted into dA that is zero.
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Virtual ExperimentGausss Law Applet

• Bring up the Gausss Law Applet and answer the
  experiment survey questions
• http//web.mit.edu/viz/EM/visualizations/electrost
  atics/flux/closedSurfaces/closed.htm

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Choosing Gaussian Surface In Doing Problems
True for all closed surfaces Useful (to calculate
electric field ) for some closed surfaces for
some problems with lots of symmetry.

• Desired E Perpendicular to surface and uniform
  on surface.
• Flux is EA or -EA.
• Other E Parallel to surface. Flux is zero

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Symmetry Gaussian Surfaces
Desired E perpendicular to surface and constant
on surface. So Gausss Law useful to calculate
electric field from highly symmetric sources
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Using Gausss Law to do Problems

1. Based on the source, identify regions in which to
   calculate electric field.
2. Choose Gaussian surface S Symmetry
3. Calculate
4. Calculate qenc, charge enclosed by surface S
5. Apply Gausss Law to calculate electric field

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ExamplesSpherical SymmetryCylindrical
SymmetryPlanar Symmetry
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Group Problem Gauss Spherical Symmetry
Q uniformly distributed throughout
non-conducting solid sphere of radius a. Find
everywhere.
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Concept Question Spherical Shell
We just saw that in a solid sphere of charge the
electric field grows linearly with distance.
Inside the charged spherical shell at right (rlta)
what does the electric field do?
Q

1. Zero
2. Uniform but Non-Zero
3. Still grows linearly
4. Some other functional form (use Gauss Law)
5. Cant determine with Gauss Law

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Concept Question Answer Flux thru Sphere
Answer 1. Zero
Q

• Spherical symmetry
• ? Use Gauss Law with spherical surface.
• Any surface inside shell contains no charge
• ? No flux
• E 0!

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Demonstration Field Inside Spherical Shell
(Grass Seeds)
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Worked Example Planar Symmetry
Consider an infinite thin slab with uniform
positive charge density . Find a vector
expression for the direction and magnitude of the
electric field outside the slab. Make sure you
show your Gaussian closed surface.
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Gauss Planar Symmetry
Symmetry is Planar Use Gaussian Pillbox
Note A is arbitrary (its size and shape) and
should divide out
Gaussian Pillbox
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Gauss Planar Symmetry
Total charge enclosed
NOTE No flux through side of cylinder, only
endcaps

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Concept Question Superposition
Three infinite sheets of charge are shown above.
The sheet in the middle is negatively charged
with charge per unit area , and the other
two sheets are positively charged with charge per
unit area . Which set of arrows (and zeros)
best describes the electric field?
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Concept Question Answer Superposition
Answer 2 . The fields of each of the plates are
shown in the different regions along with their
sum.
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Group Problem Cylindrical Symmetry
An infinitely long rod has a uniform positive
linear charge density .Find the direction
and magnitude of the electric field outside the
rod. Clearly show your choice of Gaussian closed
surface.
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Electric Fields

• Dipole E falls off like 1/r3
• Spherical charge E falls off like 1/r2
• Line of charge E falls off like 1/r
• (infinite)
• Plane of charge E uniform on
• (infinite) either side of plane