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Chapter 22
Gauss’s Law
Introduction: Our approach
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Look at why to study Gauss’s Law
Give attention to Electric Flux
Grasp Gauss’s Law
Grasp when and how to use it
Grasp making use of symmetry
Master calculating Electric Fields through
Gauss’s Law
Why study Gauss’s Law?
• To get experience with mathematical
elegance
• To make powerful use of symmetry to get
quantitative results
• To gain familiarity with Electric Field
behavior and gain qualitative clarity
Electric Flux
(on whiteboards)
• What is electric flux (in words)?
• Write an expression for the electric flux
through a planar area in a region where
the electric field is uniform.
• What is the general mathematical
expression for electric flux?
• Exercises
– Question
– from ILT: Flux through a cube (next)
On E-flux through a cube
3. A cube has sides of length L. On one face of the
cube, the electric field is uniform with magnitude E
and has a direction pointing directly into the cube.
The total charge on and within the cube is zero.
Which statements are necessarily true?
1. The electric field on the opposite face of the box
has magnitude E and points directly out of the face.
2. The total flux through the remaining five faces is
out of the box and equal to EL^2.
3. At least one other face of the cube must have an
inward flux.
4. None of the previous statements must be true.
Gauss’s Law
(on whiteboards)
• What (in words) two things does Gauss’s
Law say are proportional?
• What is the general mathematical
expression of Gauss’s Law?
• Make a sketch illustrating Gauss’s Law.
To solve for E directly, one must
• (1) be able to determine from the
symmetry of the charge distribution what
direction E points and on what variables E
depends on so that one can
• (2) create a Gaussian surface on which
E·dA is a constant. Once such a
Gaussian surface has been created, one
can then
• (3) solve for E by pulling it out of the
integral.
– Rachel Pepper, et. al. PERC Proceedings 2010
When there’s sufficient symmetry…
• Identify the position at which you want to
determine the E-field from a specific
charge distribution.
• Make a judicious “Gaussian surface”
through that point, so that there and
elsewhere E·dA = E dA, E a constant, and
possibly elsewhere E·dA = 0.
• Use symmetry of charge distribution to find
such a surface. (What symmetries….?)
Using symmetry…
• From a superposition standpoint (We’ve
been introduced to it in applications of
Coulomb’s Law; e.g. cancellation.)
• Now from a geometric standpoint (very
powerful – e.g. reductio ad absurdum
argument)
• Examples
Compound use of Gauss’s Law
• Extending beyond the direct calculation of
E is sometimes possible using
superposition
• Examples
Sample Gauss’ Law problems
• Let’s make one up in our groups and
share with the class
• Work out as groups/class
the end
• review these lecture notes to refresh your
memory of what we did, what you learned
during our classes, and what is left to
understand on chapter 22 subject matter
• Consider a constant E-field, a flat surface
of area A, and a number of field lines, N,
through the surface. (Make a sketch.)
• The E-flux is proportional singly to which
of E, A, or N?
• What other products or quotients of these
quantities is it proportional to? And why?
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