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.
• Uniform field.

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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 by
  definition work
• W(A-gtB) ? q?(B)-?(A)

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Work Done on Charge in Electrostatic Field II

• Since doing positive work on some particle means
  increasing its energy, we can define a potential
  energy U
• Uq?
• This definition clearly matches the above
• 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
• VAB ?(B)-?(A)
• Then
• W(A-gtB)q VAB

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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)qVAB
• Try to understand well the difference
• between the potential ?, the potential energy U
  and the voltage V!
• between the work done by the field W and done by
  an external agent W - W (skiing)!

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The Impact of the Potential

• Since the potential exists, we can describe the
  electrostatic field fully using the scalar
  potential field instead of the vector intensity
  field
• 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 study this relation first in
  terms of potential energy and force so we dont
  have to care about the polarity of the charge and
  we can use examples from the gravitation field.
• Lets have a charged particle in a field which
  acts by a force on it.
• If the particle moves by the field does work
  W on it

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Relations of ? versus II

• The sign of this work depends on the projection
  of the path vector into the vector of force
  .
• If it has the same direction as the force, the
  field does a positive work. Such a shift can take
  place without some external agent acting. But the
  it must be done at the cost of lowering the
  potential energy of the particle
• Further we can talk only about shifts in the
  direction of the force or those opposite to it.

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Relations of ? versus III

• When shifting the particle into the direction of
  the force the (positive) work is done by the
  field and when shifting it into the opposite
  direction the work is done by an external agent
• in this case the potential energy of the particle
  increases
• the field can return this energy later
• thats why we call it potential energy

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Relations of ? versus IV

• The work done by the field for a certain path
  A-gtB we can get by integration
• Finally, after dividing by the charge we get the
  relation between the intensity and the potential,
  we were looking for

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Relations of ? versus V

• Lets have a particle with a unit positive
  charge. Force acting on it is numerically equal
  to the intensity and its potential energy is
  numerically equal to the potential in the
  particular point.
• But we have to understand that
• the intensity and the potential are properties of
  the field
• the force and the potential energy are the
  properties of the particular particle in the
  field and their dimensions differ by the factor Q
  C.

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Relations of ? versus VI

• Lets shift a unit charge (1C) in the direction
  of the intensity by . Then we have
• So ?(B) ?(A) - Edl and the potential
  decreases in the direction of the intensity and
  also along the field lines.

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Relations of ? versus VII

• So when moving along a field line, we can get the
  intensity as a change of the potential
• We see that the potential is connected to the
  integral properties of the intensity while the
  intensity on the other hand to the derivative
  properties of the potential.

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Uniform ? Homogeneous Field I

• The simplest electrostatic field is the uniform
  or homogeneous field whose intensities are
  constant vectors (they have the same magnitudes
  and directions) in every point.
• In a uniform field we can illustrate the
  properties derived in the easiest way.
• The potential changes only in the direction of
  the intensities. And it is the only important
  direction.
• The field lines are all parallel lines.

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Uniform ? Homogeneous Field II

• Everything derived above is valid now, even for a
  shift of any distance d along a field line
• The intensity can be understood as a slope of the
  change of potential along a field line.

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Homogeneous Field III

• If we want to find the work necessary to shift a
  charge from one point into another one, we have
  to find what is the projection into the direction
  of a field lines and we have to take into account
  what charge we particularly shift.
• Large charge feels steeper slope of its potential
  energy than a small one.
• Negative charge feels the decrease of potential
  of the field as an increase of its potential
  energy.

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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 from 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

• This lecture covers
• Chapter 23-1, 23-2
• Advance reading
• Chapter 21-10, 23-5, 23-8