3.1 Laplace

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3.1 Laplaces Equation
Common situation Conductors in the system,
which are a at given potential V or which carry
a fixed amount of charge Q.
The surface charge distribution is not known.
We want to know the field in regions, where there
is no charge.
Reformulate the problem.
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Boundary conditions. (e.g. over a surface
Vconst.)
Important in various branches of
physics gravitation, magnetism, heat
transportation, soap bubbles (surface tension)
fluid dynamics
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One dimension
Boundary conditions
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V has no local minima or maxima.
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Two Dimensions
Partial differential equation. To determine the
solution you must fix V on the boundary
boundary condition.
Rubber membrane Soap film
V has no local minima or maxima inside the
boundary.
A ball will roll to the boundary and out.
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Three Dimensions
Partial differential equation. To determine the
solution you must fix V on the boundary, which
is a surface, boundary condition.
V has no local minima or maxima inside the
boundary.
Earnshaws Theorem A charged particle cannot be
held in a stable equilibrium by electrostatic
forces alone.
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First Uniqueness Theorem
The solution to Laplaces equation in some
volume V is uniquely determined if V is
specified on the boundary surface S.

• The potential in a volume V is uniquely
  determined if
• the charge density in the region, and
• the values of the potential on all boundaries are
  specified.

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Second Uniqueness Theorem
In a volume surrounded by conductors and
containing a specified charge density, the
electrical field is uniquely determined if the
charge on each conductor is given.
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Image Charges
What is V above the plane?
Boundary conditions
There is only one solution.
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The region zlt0 does not matter. There, V0.
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Induced surface charge
Force exerted by the image charge
Force on q
Different from W of 2 charges!!
Energy
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Example 3.2
Find the potential outside the conducting
grounded sphere.
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