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
What is Physics?
• Gravitational force: F=Gm1m2/r2
•
Electrostatic force: F=Gq1q2/r2
• One thing is in common: both of these forces are
conservative
Electric Potential Energy
Gravitational force
Electrostatic force
Note: Electric energy is one type of energy.
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 W∞ is the work done by the electric field on a charged
particle as that particle moves in from infinity to point f.
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 ?
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/coulomb=volt (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.
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 ΔV and ΔU are measurable in terms of
the work W.
•The is ΔV property of the electric field and has nothing to do
with a test charge
•The common name for electric potential difference is
"voltage".
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.
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?
Example 2 Work, Electric Potential Energy, and Electric
Potential
The work done by the electric force as the test
charge (q0=+2.0×10–6 C) moves from A to B is
WAB=+5.0×10–5 J. (a) Find the difference, ΔU=UB–UA,
in the electric potential energies of the charge
between these points. (b) Determine the potential
difference, ΔV=VB–VA, between the points.
Example 3 Operating a Headlight
Determine the number of particles, each carrying a
charge of 1.60×10–19 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.
Example 4 Electric Field and Electric
Potential
Two identical point charges (+2.4×10–9 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.
Equipotential Surfaces
An equipotential surface is a
surface on which the electric
potential is the same
everywhere.
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?
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 +3.2×10–19
J of work. What are the electric potential
differences (a) VB–VA, (b) VC–VB, and (c) VC–VA?
Calculating the Potential from the Field
Calculating the Field from the Potential
The potential gradient gives the component of the
electric field along the displacement Δs
V
Es  
s
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
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.
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.
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 conductor—whether on the
surface or inside—come to the same
potential. This is true even if the
conductor has an internal cavity and even
if that cavity contains a net charge.
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.
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 ΔV 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 ΔV by moving the positive test charge q0 from
i to f along the path icf shown in Fig. 24-5 b.
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.
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:
Vtot=V1+V2+∙∙∙+VN
Potential Due to an Electric Dipole
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 V=0 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)?
Potential Due to a Continuous Charge
Distribution
Line of Charge
In Fig. 24-12 a, a thin
nonconducting rod of length L
has a positive charge of
uniform linear density λ . 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.
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 σ on one surface. Here
we derive an expression for
V(z), the electric potential at
any point on the central axis.
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.
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 d=12 cm and that
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.
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.
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.

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