Chapter 24. Electric Potential

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Chapter 24. Electric Potential

• 24.1. What is Physics?      
• 24.2. Electric Potential Energy      
• 24.3. Electric Potential      
• 24.4. Equipotential Surfaces      
• 24.5. Calculating the Potential from the
  Field      
• 24.6. Potential Due to a Point Charge      
• 24.7. Potential Due to a Group of Point
  Charges      
• 24.8. Potential Due to an Electric Dipole      
• 24.9. Potential Due to a Continuous Charge
  Distribution     
•  24.10. Calculating the Field from the
  Potential     
•  24.11. Electric Potential Energy of a System of
  Point Charges      
• 24.12. Potential of a Charged Isolated Conductor

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What is Physics?   

• Gravitational force FGm1m2/r2

• Electrostatic force FGq1q2/r2

• One thing is in common both of these forces are
  conservative

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Electric Potential Energy
Electrostatic force

• Gravitational force

Note Electric energy is one type of energy.
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Reference Point of Electric Potential Energy
The reference point can be anywhere. For
convenience, we usually set charged particles to
be infinitely separated from one another to be
zero potential energy
The potential energy U of the system at any point
f is
where W8   is the work done by the electric
field on a charged particle as that particle
moves in from infinity to point f.
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Example 1  

• A proton, located at point A in an electric
  field, has an electric potential energy of UA
  3.20 10-19 J. The proton experiences an average
  electric force of 0.80 10-9 N, directed to the
  right. The proton then moves to point B, which is
  a distance of 1.00 10-10 m to the right of
  point A. What is the electric potential energy of
  the proton at point B ?

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Electric Potential

• The electric potential V at a given point is
  the electric potential energy U of a small test
  charge q0 situated at that point divided by the
  charge itself

If we set        at infinity as our reference
potential energy,
SI Unit of Electric Potential joule/coulombvolt
(V)

• Note
• Both the electric potential energy U and the
  electric potential V are scalars.
• The electric potential energy U and the electric
  potential V are not the same. The electric
  potential energy is associated with a test
  charge, while electric potential is the property
  of the electric field and does not depend on the
  test charge.

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The Electric Potential Difference

• The electric potential difference between any
  two points i and f in an electric field.

• It is equal to the difference in potential
  energy per unit charge between the two points.
• the negative work done by the electric field on a
  unite charge as that particle moves in from point
  i to point f.

• Note
• Only the differences ?V and ?U are measurable in
  terms of the work W.
• The is ?V property of the electric field and has
  nothing to do with a test charge
• The common name for electric potential difference
  is "voltage".

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Notes Continue

• Electric field always points from higher electric
  potential to lower electric potential.
• A positive charge accelerates from a region of
  higher electric potential energy (or higher
  potential) toward a region of lower electric
  potential energy (or lower potential).
• A negative charge accelerates from a region of
  lower potential toward a region of higher
  potential.

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Conceptual Example   The Accelerations of
Positive and Negative Charges

• Three points, A, B, and C, are located along a
  horizontal line, as Figure 19.4 illustrates. A
  positive test charge is released from rest at A
  and accelerates toward B. Upon reaching B, the
  test charge continues to accelerate toward C.
  Assuming that only motion along the line is
  possible, what will a negative test charge do
  when it is released from rest at B?

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Example 2  Work, Electric Potential Energy, and
Electric Potential

• The work done by the electric force as the
  test charge (q02.0106 C) moves from A to B is
  WAB5.0105 J. (a) Find the difference,
  ?UUBUA, in the electric potential energies of
  the charge between these points. (b) Determine
  the potential difference, ?VVBVA, between the
  points.

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Example 3  Operating a Headlight

• Determine the number of particles, each carrying
  a charge of 1.601019 C (the magnitude of the
  charge on an electron), that pass between the
  terminals of a 12-V car battery when a 60.0-W
  headlight burns for one hour.

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Example 4  Electric Field and Electric Potential

• Two identical point charges (2.4109 C) are
  fixed in place, separated by 0.50 m. (see Figure
  19.32). Find the electric field and the electric
  potential at the midpoint of the line between the
  charges qA and qB.

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Equipotential Surfaces

• An equipotential surface is a surface on which
  the electric potential is the same everywhere.

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Relation of Equipotential Surfaces and the
Electric Field

1. The net electric force does no work as a charge
   moves on an equipotential surface.
2. The electric field created by any charge or group
   of charges is everywhere perpendicular to the
   associated equipotential surfaces and points in
   the direction of decreasing potential.

What will happen if the electric field E is not
perpendicular to the equipotential surface?
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Check Your Understanding 

• The drawing shows a cross-sectional view of
  two spherical equipotential surfaces and two
  electric field lines that are perpendicular to
  these surfaces. When an electron moves from point
  A to point B (against the electric field), the
  electric force does 3.21019 J of work. What
  are the electric potential differences (a) VBVA,
  (b) VCVB, and (c) VCVA?

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Calculating the Potential from the Field
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Calculating the Field from the Potential

• The potential gradient gives the component of
  the electric field along the displacement ?s

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Check Your Understanding 

• The sketch below shows cross sections of
  equipotential surfaces between two charged
  conductors that are shown in solid black. Various
  points on the equipotential surfaces near the
  conductors are labeled A, B, C, ..., I. At which
  of the labeled points will the electric field
  have the greatest magnitude

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Example 4  

• The metal contacts of an electric wall socket
  are about 1.0 cm apart and are maintained at a
  potential difference of 120 V. What is the
  average electric field strength between the
  contacts? What is the direction of the electric
  field if the left contact is the higher
  potential? The lower potential? Treat the
  potential difference between the contacts as
  being constant in time.

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Sample Problem
The electric potential at any point on the
central axis of a uniformly charged disk is given
by Eq. 24-37 ,




                                                       


                                                                             

Starting with this expression, derive an
expression for the electric field at any point on
the axis of the disk.
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Potential of a Charged Isolated Conductor

• An excess charge placed on an isolated
  conductor will distribute itself on the surface
  of that conductor so that all points of the
  conductorwhether on the surface or insidecome
  to the same potential. This is true even if the
  conductor has an internal cavity and even if that
  cavity contains a net charge.

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Isolated Conductor in an External Electric Field

• The free conduction electrons distribute
  themselves on the surface in such a way that the
  electric field they produce at interior points
  cancels the external electric field that would
  otherwise be there.
• The electron distribution causes the net
  electric field at all points on the surface to be
  perpendicular to the surface.

                                                                                                

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Sample Problem

• (a) Figure 24-5 a shows two points i and f in a
  uniform electric field E . The points lie on the
  same electric field line (not shown) and are
  separated by a distance d. Find the potential
  difference ?V by moving a positive test charge
  q0 from i to f along the path shown, which is
  parallel to the field direction. (b) Now find the
  potential difference ?V by moving the positive
  test charge q0 from i to f along the path icf
  shown in Fig. 24-5 b.

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Potential Due to a Point Charge


A zero reference potential is at infinity
A positively charged particle produces a positive electric potential. A negatively charged particle produces a negative electric potential.                                       

    
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Potential Due to a Group of Point Charges
The potential at a point due to any number of
point charges can be found by simply finding the
potential at the point due to each alone and
adding the potentials VtotV1V2VN
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Potential Due to an Electric Dipole

                                                                                                                                                                
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Sample Problem

• (a) In Fig. 24-9 a, 12 electrons (of charge -e)
  are equally spaced and fixed around a circle of
  radius R. Relative to V0 at infinity, what are
  the electric potential and electric field at the
  center C of the circle due to these electrons?
• (b) If the electrons are moved along the circle
  until they are nonuniformly spaced over a 120
  arc (Fig. 24-9 b), what then is the potential at
  C? How does the electric field at C change (if at
  all)?

                                                                                                                                           
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Potential Due to a Continuous Charge Distribution
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Line of Charge

• In Fig. 24-12 a, a thin nonconducting rod of
  length L has a positive charge of uniform linear
  density ? . Let us determine the electric
  potential V due to the rod at point P, a
  perpendicular distance d from the left end of the
  rod.

                                                                               
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Charged Disk

• In Section 22.7 , we calculated the magnitude
  of the electric field at points on the central
  axis of a plastic disk of radius R that has a
  uniform charge density s on one surface. Here we
  derive an expression for V(z), the electric
  potential at any point on the central axis.

                                                                         

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Electric Potential Energy of a System of Point
Charges

• The electric potential energy of a system of
  fixed point charges is equal to the work that
  must be done by an external agent to assemble the
  system, bringing each charge in from an infinite
  distance.

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Sample Problem

• Figure 24-16 shows three point charges held in
  fixed positions by forces that are not shown.
  What is the electric potential energy U of this
  system of charges? Assume that d12 cm and that

                                                                     

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Conceptual Questions

• 1. The drawing shows three possibilities for the
  potentials at two points, A and B. In each case,
  the same positive charge is moved from A to B. In
  which case, if any, is the most work done on the
  positive charge by the electric force? Account
  for your answer.

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• 2. The electric field at a single location is
  zero. Does this fact necessarily mean that the
  electric potential at the same place is zero? Use
  a spot on the line between two identical point
  charges as an example to support your reasoning.

3. The potential is constant throughout a given
region of space. Is the electric field zero or
nonzero in this region? Explain.
4. In a region of space where the electric field
is constant everywhere, is the potential constant
everywhere? Account for your answer.
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• 5. A positive test charge is placed in an
  electric field. In what direction should the
  charge be moved relative to the field, such that
  the charge experiences a constant electric
  potential? Explain.

6. The potential at a point in space has a
certain value, which is not zero. Is the electric
potential energy the same for every charge that
is placed at that point? Give your reasoning.