I-5 Special Electrostatic Fields

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I-5 Special Electrostatic Fields
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Main Topics

• Electric Charge and Field in Conductors.
• The Field of the Electric Dipole.
• Behavior of E. D. in External Electric Field.
• Examples of Some Important Fields.

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A Charged Solid Conductor

• Charging a conductor means to introduce in it
  some excess charges of the same polarity.
• These charges repel themselves and since they can
  move freely as far as the surface, in
  equilibrium, they must end on a surface.
• In equilibrium there must be no forces acting on
  the charges, so the electric field inside is zero
  and also the whole solid conductor must be an
  equipotential region.

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A Hollow Conductive Shell I

• In equilibrium again
• the charges must remain on the surface.
• the field inside is zero and the whole body is an
  equipotential region.
• The above means the validity of the Gauss law.
• To proof that lets return to the Gauss law.

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The Gauss Law Revisited I

• Let us have a positive point charge Q and a
  spherical Gaussian surface of radius r centered
  on it. Let us have a radial field
• EkQ/rp
• The field lines are everywhere parallel to the
  outer normals, so the total flux is
• ?e E(r)A Qr2-p/?0
• If p?2 the flux would depend on r !

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The Gauss Law Revisited II

• The validity of the Gauss law ? p 2.
• By using a concept of the solid angle it can be
  shown that the same is valid if the charge Q is
  anywhere within the volume surrounded by the
  spherical surface.
• By using the same concept it can be shown that
  the same is actually valid for any closed
  surface.
• It is roughly because from any point within some
  volume we see any closed surface confining it
  under the solid angle of 4?.

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A Hollow Conductive Shell II

• Let first the shell be spherical. Then the charge
  density ? on its surface is constant.
• From symmetry, in the center the intensities from
  all the elementary surfaces that make the whole
  surface always compensate themselves and E 0.
• For any other point within the sphere E 0 they
  compensate themselves only if p 2.
• Again, using the concept of solid angle, it can
  be shown, the same is valid for any closed
  surface.

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A Hollow Conductive Shell III

• Conclusion The existence of a zero electric
  field within a charged conductive shell is
  equivalent to the validity of the Gauss law.
• This is the principle of
• experimental proof of the Gauss law with a very
  high precision p 2 2.7 ? 3.1 10-16.
• of shielding and grounding.

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Electric Field Near Any Conducting Surface

• Let us take a small cylinder and submerge it into
  the conductor so its axis is perpendicular to the
  surface.
• The electric field
• within the conductor is zero
• outside is perpendicular to the surface
• A non-zero flux is only through the outer cup ?
• E ?/?0.
• Beware the edges! ? is not generally constant!

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The Electric Dipole I

• An electric fields can be produced even if the
  total charge is zero by so called electric
  multipoles. Their fields are not centrosymmetric
  and decrease generally faster than the field of
  the single point charge.
• The simplest one is the electric dipole. It is
  the combination of two charges of the same
  absolute value but different sign Q and Q
  separated by some distance l. We define the
  dipole moment
• p Ql .

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The Electric Dipole II

• Electric dipoles (multipoles) are important
  because they are responsible for all the
  electrical behavior of neutral matter.
• The components of material (molecules, domains)
  can be polar or their dipole moment can be
  induced.

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Behavior of the Electric Dipole in External
Electric Fields

• In uniform electric fields the dipoles are
  subjected to a torque which is trying to turn
  their dipole moments in the direction of the
  field lines
• In non-uniform electric fields the dipoles are
  also dragged.

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Some Examples

• The field of homogeneously charged sphere
• Parallel uniformly charged planes
• Electrostatic xerox copier

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Homework

• Now, you should be able to solve all the problems
  due Monday!

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Things to read

• Repeat the chapters 21, 22, 23 !
• Try to see the physicist Bible
• The Feynman Lectures on Physics

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The Solid Angle I

• Let us have a spherical surface of radius r. From
  its center we see an element of the surface da
  under a solid angle d?

The whole surface we see under
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The Solid Angle II

• If there is a point charge Q in the center the
  elementary flux through da is

Since the last fraction is d?, the total flux is

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Intensities near more curved surfaces are
stronger!

• Lets have a large and a small conductive spheres
  R, r connected by a long conductor and lets
  charge them. Charge is distributed between them
  to Q, q so that the system is equipotential

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Potential of Electric Dipole I

• Let us have a charge Q at the origin and a Q in
  l. What is the potential in r? We use the
  superposition principle and the gradient

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Potential of Electric Dipole II

• The first two terms cancel

• The potential has axial symmetry with the dipole
  in the axis and axial anti-symmetry perpendicular
  to it. It decreases with 1/r2!

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Electric Dipole - The Torque

• Let us have a uniform field with intensity E.
  Forces on both charges contribute simultaneously
  to the torque

• The general relation is a cross product

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Electric Dipole - The Drag

• Let us have a non-uniform field with intensity E
  with the dipole parallel to a field line (-Q in
  the origin).

• Generally

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The vector or cross product I

• Let ca.b
• Definition (components)

The magnitude c
Is the surface of a parallelepiped made by a,b.
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The vector or cross product II
The vector c is perpendicular to the plane made
by the vectors a and b and they have to form a
right-turning system.
?ijk 1 (even permutation), -1 (odd), 0 (eq.)