Dr. Hugh Blanton

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ENTC 3331 RF Fundamentals

• Dr. Hugh Blanton
• ENTC 3331

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Energy Potential
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• The work done, or energy expended, in moving any
  object a vector differential distance dl under
  the influence of a force is
• Work, or energy, is measured in joules (J).

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• The differential electric potential energy dW per
  unit charge is called the differential electric
  potential (or differential voltage) dV.
• Convention potential at infinity ? 0 ? ground.

minus sign is convention
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• Electrical potential of a point charge E-field
  of a point charge, q, located at the origin?

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• Compare the electric field and potential of a
  point charge.
• Expressions are rather similar.

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• Refer to the gradient in the spherical coordinates

no dependence on q or f
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• It follows that for a point charge

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• Generalizations
• All complicated charge distributions are always
  superposition of fields due to many point
  charges.
• All these fields superimpose linearly.

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• Multiple charges
• Charge distributions

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• And importantly,
• for all distributions.
• Note

do not confuse with potential energy
Coulomb force
potential energy
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• From mechanics
• For both Coulomb and mechanical forces, the force
  is equal to the gradient of the potential energy.

potential energy
Gravitational force
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• This isnt so surprising, since
• Coulombs law
• And the gravitational force
• are so similar.

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• Forces for which holds are
  conservative.
• That is, the energy associated with the
  conservative force is conserved.

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• Determine the electric potential, V, at the
  origin for the figure.
• For multiple charges
• For all four charges

r is the origin
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• Important observation
• while

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• Find the E-field above a circular disk of radius
  a with rS constant.
• What are the potential, V, and E-field at point
  P(0,0,z) right above the circular disk?

z
y
x
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z
y
x
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• The question is, what is the field

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• Since

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• The field points along the z-axis
• This result is identical to that obtained
  earlier.
• By first calculating the potential V and then
  using , considerable geometrical
  considerations can be avoided.

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Summary

• Energy conservation requires that
• Because of energy conservation the Coulomb force
  is conservative
• Consequently,

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Stokes Theorem

• General mathematical theorem of Vector Analysis.

any vector field
any surface
closed boundary of that surface
closed path C along boundary
a surface
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• Since Stokes Theorem is of general validity, it
  can certainly be applied to the electrostatic
  fields
• and

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any closed path through

• consequence of
• energy conservation
• Coulombs force is conservative

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• Since C and S are arbitrary, this is only
  possible if

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• Recall that the curl of a gradient field is
  always zero.
• That is
• Thus

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• The curl of the electrostatic field is zero.
• The circulation of any electrostatic field is
  zero
• An electrostatic field is non-rotational
• An electrostatic field is conservative.
• All of the preceding statements are equivalent.

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• The complete set of postulates for electrostatics
  are
• and
• These are Maxwells equations of Electrostatics
• The postulates are accepted to be true for any E
  or D field as long as r is independent of time.

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Cornerstones of Electrostatics
Energy Conservation
Principle of Linear Superposition
conservative central force
generalization for multiple charges
SI units
experimental
empirical facts
Coulombs Law
field concept
charges are the sources of the field
theoretical
the field is non-rotational
?
?
Postulates of Electrostatics