Transcript Ch. 19

Ch. 23
Electric Potential
Chapter Overview
► Review
Work and Potential Energy
► Define Potential Difference
► Compute the Potential Difference from the Electric
Field
► Find the electric potential due to point charges
► Calculate the electric potential for a continuous
charge distribution
► Compute the Electric Field from the Potential
► Describe Properties of Equipotential Surfaces
Review of Work
►A
force of 20 N is applied at an angle of 20°
above the horizontal to a block sitting on a
frictionless horizontal surface. a) Sketch
the situation. b) If the force moves the
block 2.5 m, how much work was done?
 
W  F d
W = Fd cosθ
= 20 N x 2.5 m x cos 20°
= 47 J
Electrostatic Potential Energy
► Work
done by an electrostatic force is
► dW = F∙dr
► Electrostatic Force is a “conservative” force
► dW = -dUelectrostatic
A -2.0 µC charge is placed in a
uniform electric field of 500 N/C
a) Sketch the situation. b) What
is the work done by the electric
field on the charge if it moves 5.5
m? c) What is the change in
electrostatic potential energy of
the charge?
How would your answer for the
ΔUelec change if the charge moved
was twice as big?
(CT)
There would be no change
It would be twice as big
It would be ½ as big
It cannot be determined
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How would your answer for the
ΔUelec change if the charge moved
was half as big?
(CT)
1.
2.
3.
4.
There would be no change
It would be twice as big
It would be ½ as big
It cannot be determined
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What type of relationship is there
between the charge and the ΔUelec?
(GR)
Definition of the Potential Difference
► Since
the ΔUelec is proportional to the
charge, we define a quantity which is
independent of the charge by dividing the
potential energy by the charge.
Definition of the Potential Difference
► If
a charge goes from point A to point
B and has a change in electric potential
energy of dUelec, then the potential
difference between A and B is
dU elec
dV 
q0
Definition of the Potential Difference
► Remember
that E = F/q for a test charge
► so

 
dW
F 
dV  
   dl   E  dl
q0
q0
Definition of Potential Difference
► If
we add up the small changes in potential then
we can obtain the finite potential difference
between the points in space A and B
B
A
 
V    E  dl
Definition of the Potential Difference
► SI
Units?
► This
V
combination of units is called the volt,
1 V = 1 J/C
► volt is named after Alessandro Volta who
invented the battery
►
Ex. Two points in space are in a
region of uniform electric field of
magnitude 550 N/C with the field
pointing in the direction from one
point to the other. a) Sketch the
situation and depict the electric
field. b) Find the potential
difference between the two points.
Connection between Electrostatic
Potential Energy and Potential
Difference
► Knowing
the potential difference between
two point tells you the change in
electrostatic potential energy between those
two points
► ΔUelec
= q0ΔV = -Welec
Ex. 2.0 J of work are done when
a charge of 5.0 pC moves from
point A to point B. a) Sketch the
situation. b) Find the change in
Uelec of the charge. c) Find the
potential difference between
points A and B.
Ex. A battery maintains a
constant potential difference
between two pieces of metal of
6.0 V. A charge of -2.0 nC moves
between the plates. a) Sketch
the situation b) What is the
change of electric potential
energy for the charge.
Ex. Two parallel plates of metal are
placed in a vacuum chamber. A battery
is used to place a potential difference of
400 V between the plates. An electron
is released from rest at the plate with
the lower potential. a) Sketch the
situation. b) What will happen to the
electron and why? c) What will be the
speed of the electron once it has
crossed the gap between the plates?
Energy Conservation
► The
electrostatic force is a conservative
force, so work done by it is stored as a
potential energy
► Review:
The Work – Kinetic Energy
Theorem Wnet = ΔK where K = ½ mv2
Energy Conservation
► If
no non-conservative work is done then
mechanical energy is conserved
►E = K + U
► If
Wnc = 0 then ΔE = 0
The Electric Potential of a Point
Charge
Find the Potential difference between the
points A and B along the same radial line
outwards from the positive point charge Q
Q
A
B
V  
rb
ra
 
 E  dl
but for a point charge
 kQ
E  2 rˆ
r
and in this case dl  r̂dr, so
rb
rb
kQ
kQ
V    2 rˆ  rˆdr    2 dr
ra
ra
r
r
Carrying out the integral we get
rb
kQ 
kQ kQ
V 



r  ra
rb
ra
Suppose the points A and B were at the
same radii but in different directions as
shown. Would the answer to the
potential difference change?
B
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Yes
No
Depends on the
exact position
Q
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The Electric Potential of a Point
Charge
► The
potential difference is given by the
difference of
kQ kQ
V 

rb
ra
► The
answer depends only on the radius of
each point and not the direction
The Electric Potential of a Point
Charge
► We
can conclude, if a point charge of charge Q is
a distance r from a point P, then the electric
potential at the point P is given by
Q
V k
r
At what distance from the
charge Q is the electric
potential 0?
(TPS)
r = 0 (at the charge)
r = ∞ (very far from the
charge
The electric potential is
never 0
Cannot be determined
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At what distance from the charge
Q is the electric potential 0?
(TPS)
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r = 0 (at the charge)
r = ∞ (very far from the
charge
The electric potential is
never 0
Cannot be determined
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The Electric Potential of a Point
Charge
► We
can conclude, if a point charge of charge Q is
a distance r from a point P, then the electric
potential at the point P is given by
Q
V k
r
► The
electric potential of a point charge has built
in that the potential is 0 when r = ∞
Ex. A point P is 5.0 cm away
from a small charged object with
charge 2.0 μC. a) Sketch the
situation. b) Find the electric
potential at the point P c) What
work would be done to place a 2.0 μC at the point P from very
far away. d) What does the
work?
Superposition
► The
electric potential due to several charges
is simply the sum of the potentials of the
individual charges
► The electric potential is a scalar. No need to
take into account direction, but do need to
include signs in calculation
Ex. a) Find the electric potential at the point P. b)
How much work would it require to bring a 7.5 pC
charge to point P from very far away? c) Does the
field do work or is work done on the field?
-5.0 pc
3.0 PC
The Potential of a Continuous
Distribution of Charge
► Suppose
charge
we a continuous distribution of
dqi
++++++++
++++++
++++++
► We
ri
P
can divide the object into small pieces
that we treat like point objects
The Potential of a Continuous
Distribution of Charge
► To
find the total potential at P, we add up all
the little pieces
dq
V  k
r
Ex. Find the potential on the axis
of a uniform positively charged
ring of charge at a point P a
distance x from the center of the
ring.
The Potential of a Continuous
Distribution of Charge
► Each
small piece of charge dqi contributes a
small piece to the potential dVi given by
dqi
dVi  k
r
The Potential of a Continuous
Distribution of Charge
► To
find the total potential at the point P
dq
V  k
r
Determining the Potential from the
Field
► We
can in principle find the potential for any
distribution of charge using
dq
V  k
r
► However
if we know the electric field, it can
be easier to use the definition of the
potential difference to find the potential
 
V    E  dl
Ex. A very long, insulating, solid
cylinder of radius is uniformly
charged with a positive charge. a)
Sketch the cylinder and discuss what
the symmetry tells you about the Efield. b) Use Gauss’s Law to find
the Electric Field for r < R and R > r.
c) Use the electric field to find the
potential for r < R and r > R
Determining E from V
► If
you know the potential in a region of space,
you can determine the electric field
► Since
 
V    E  dl
► We
can find E from V by taking a derivative
► In 1D we can write
dV
Ex  
dx
Ex. A potential along the x-axis is
given by V(x) = kQ/|x|. Find the
electric field as a function of x. If Q is
positive, what is the direction of the
electric field in the +x and –x
directions?
Ex. Two parallel metal plates are
separated by 2.0 mm. The plates
are connected to the opposite
terminals of a 6.0 V battery. a)
Sketch the situation. b) Find the
average electric field in the region
between the plates. c) Indicate the
direction of the electric field in the
region between the plates
A region of space has a constant
positive electric potential. What
can you say about the electric field
in that region?
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2.
3.
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It is constant and positive
It is constant and negative
It is 0
Cannot be determined
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A region of space has an electric
field of o. What can you say about
the electric potential in that
region?
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It is constant and positive
It is constant and negative
It is 0
Cannot be determined
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Equipotential Surfaces
►A
surface at a constant electric potential is
an equipotential surface
► Electric field lines are perpendicular to
equipotential surfaces and point form higher
ot lower potential
► Two conductors in contact at electrostatic
equilibrium will be equipotential
► Electric Field is higher near more sharply
curved points on an equipotential surface
The figure shows a set of equipotential surfaces measured by a
student. Find the average electric field
and indicate the direction of the
electric field in each region.
Equipotential Surfaces
► An
equipotential surface is one in which the
potential difference between any two points
on the surface is 0
► Is
a conductor in electrostatic equilibrium an
equipotential surface?
Dielectric Beakdown