I-3 Electric Potential

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I-3 Electric Potential
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Main Topics

• Conservative Fields.
• The Existence of the Electric Potential.
• Work done on Charge in Electrostatic Field.
• Relations of the Potential and Intensity.

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Conservative Fields

• There are special fields in the Nature in which
  the total work done when moving a particle on
  along any closed path is zero. We call them
  conservative.
• Such fields are for instance
• Gravitational - we move a massive particle
• Electrostatic - we move a charged particle

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The Existence of the Electric Potential

• From the definition of a conservative field it
  can be shown that work done by moving a charged
  particle from some point A to some other point B
  doesnt depend on the path but only on the
  difference of some scalar quality in both points.
  This quality is called the electric potential ?.

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Work Done on Charge in Electrostatic Field by an
External Agent I

• If we (as an external agent) move a charge q from
  some point A to some point B then we do work by
  definition
• W(A-gtB)q?(B)-?(A)

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Work Done on Charge in Electrostatic Field II

• Since doing positive work means increasing of the
  energy, we can define a potential energy U
• Uq?
• Then
• W(A-gtB)q?(B)-?(A) U(B)-U(A)

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Work Done on Charge in Electrostatic Field III

• In almost all situations we are interested in the
  difference of two potentials. We define this
  difference as the voltage V
• VBA ?(B)-?(A)
• Then
• W(A-gtB)q VBA

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Work Done on Charge in Electrostatic Field IV

• So we come to the general formula
• Wq?(B)-?(A)U(B)-U(A)qVBA
• Try to understand well the difference
• between the potential, the potential energy and
  the voltage!
• between the work done by the field and an
  external agent!

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The Impact of the Potential

• Since the potential exists, we can describe fully
  the electrostatic field using the scalar
  potential field ?(r) instead of the vector
  intensity field E(r).
• We need only one third of information
• Superposition means just adding numbers
• Some terms converge better

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Relations Between Potential and Intensity I

• It is convenient to describe this in terms of
  potential energy and force so we dont have to
  care about the polarity of the charge and use
  examples from the gravitation field.
• Lets have a charged particle and a force F
  acting on it.
• If the particle moves by dl the field does work
  dWF.dl .

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Relations ? versus E II

• The sign of this work depends on the projection
  of the path vector dl into the force F.
• If the field does a positive work it must be at
  the cost of lowering the potential energy of the
  particle i.e. dWgt0 means dUlt0.
• So dW F.dl -dU
• To get work for some finite path A-gtB we just
  integrate.

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Uniform ? Homogeneous Field I

• Let us move a unit charge some distance d in the
  direction of the intensity E. Since both vectors
  are parallel the work is positive and the
  integral is a simple multiplication
• Ed-?(B)-?(A)gt0
• This means ?(B)?(A)-Ed so along the field lines
  the potential ? is decreasing!

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Uniform Field II

• We can also get E from the potentials
• E-?(B)-?(A)/d?(A)-?(B)/d
• So the magnitude E is the slope of the potential
  and the vector points in the direction of
  decreasing potential.
• Roughly E depends on the change (derivative) of ?
  and ? on sum (integral) of E.

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Uniform Field III

• If we want to know the work done on a non-unity
  charge by the field or an external agent, we have
  to take into account the charge and deal with the
  potential energy. Bigger charge sees bigger
  slope in potential energy and negative charge
  sees an opposite slope!
• Roughly E depends on the change (derivative) of
  ? and ? depends on sum (integral) of E.

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The Units

• The unit of ? or V is 1 Volt.
• ? U/q gt V J/C
• E ?/d V/m
• ? kq/r V gt k Vm/C gt
• ?0CV-1m-1

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Homework 2

• The homework is selected for problem sections
  that are in the end of each chapter.
• 21 17, 19
• 22 1, 2, 4, 6, 12, 26
• 23 7, 10
• You are free to try to answer to the questions in
  the questions sections!

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Things to read

• Chapter 23-1, 23-2

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Work Done by the Field A-gtB

• Now we can divide it by the test charge q

• This is generally ho to get ? from E.