Transcript Physics 152 Walker, Chapter 20

Physics 152
Walker, Chapter 20
Electrostatic Potential Energy
Electrostatic Potential
Electric Potential and
Electric Potential Energy
 Symbol for electric potential is V
 We will first define Electric Potential Energy. Symbol is U
Scalar quantity (a magnitude, positive or negative, not a
direction) Unit is Joule (J).
 Electric Potential Energy is an energy of a charged object
in an external electric field.
 Electric Potential is the property of the electric field itself,
whether or not a charged object has been placed in it.
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Electrical Energy Terms and
Definitions
The electrostatic force is a conservative force.
Conservative because the force on a charge depends only
on the position of the charge, not its velocity or past
trajectory.
We can define an electrical potential energy U (Joules)
associated with the electrostatic force.
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Electrical Energy Terms and
Definitions (continued)
As a charge q moves parallel (in same direction) to a
constant electric field E, it experiences a force F=qE. The
work done by the electric field is, W=Fd=qEd.
(work is negative if force F and displacement d are in opposite
directions)
The change in the potential energy is just the negative of the
work done by the electric field:
DU = - W = - qEd
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Change in electric potential energy
Move the + particle opposite the direction of force = increase its potential
energy
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Question 1
W = F(xf-xi)
•
A positive charge moves
from a) to b) in the electric
field E. The work done by
the electrostatic force is:
1) Positive
2) Negative
3) zero
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(a)
(b)
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Question 2
•
A positive charge moves
from a) to b) in the electric
field E. The change in the
electrostatic potential
energy is :
1) Positive
2) Negative
3) zero
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DU = -Eq(xf-xi)
(a)
(b)
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Walker P. #1
A uniform electric field of magnitude 4.1x105 N/C points in the
positive x direction. Find the change in electric potential energy of a
+4.1 µC charge as it moves from the origin to (a) (0, 6.6 m) [ans:0], (b)
(6.6 m, 0) [ans:-11.1], and (c) (6.6 m, 6.6 m) [ans:-11.1]
E
q
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Electric Potential
It is often convenient to consider not the potential energy,
but rather the potential difference between two points.
The potential difference between points A and B, (VB -VA ),
is defined as the change in potential energy of a charge q
moved from A to B divided by that charge
U B - U A DU
DV  VB - V A 

q
q
VB - V A  - E ( x B - x A ), if E is uniform
Potential is a scalar, NOT a vector.
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Units
The potential V is measured in units of volts:
1 Volt = 1 V = 1 J /C = 1 N·m / C
With this definition of the volt, we can express the units of
the electric field as:
[E]=1 N/C = 1 V/m
Note: potential (V) potential energy (U)
Unfortunately, we use V both for the electrostatic
potential, and for its unit of measure, e.g. V(x1) = 2.5 V.
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Electric Field, Electric Potential Energy, and Work
DU = -W = -Fd
DV = DU/q = Ed
V (uniform field)
E
d
[1 N/C]=[1 V/m]
The zero of potential:
For calculating physical quantities it is the difference in potential
which has significance, not the potential itself. Therefore, we are
free to choose as having zero potential at any arbitrary point
which is convenient. Typical choices are:
• the earth
• infinity, i.e. remotely far from the charges we are studying.
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Energy Conservation
A consequence of the fact that electric force is conservative is
that the total energy of an object is conserved
(as long as nonconserative forces such as friction can be ignored)
K A  U A  KB  U B
Expressing the kinetic energy:
1 2
1 2
mvA  U A  mvB  U B
2
2
Electric potential energy is
U  qV
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Point Charges
• If we define the zero of potential to be at infinity,
then the potential at a point A which is a distance r from
a point charge q is found to have a potential given by:
q
1 q
VA  k 
r 4 0 r
10
8
Volts
(Dimensional analysis:
E = kq/r2, V has dimensions
of E times a length. r is the
only length in the problem).
Electrostatic Potential
6
4
2
q •
A
r
0
0
0.2
0.4
0.6
0.8
1
Distance (m) from 0.1nC point charge
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Many Charges and Superposition
•If we wish to know the potential at a given point in space
which results from all surrounding charges, we simply add
up the potential from each charge:
q3
q1
q2
VA  k
k
k
 ...
r1
r2
r3
•Note that because potential is a scalar, the summation is
less difficult than for the vector field E.
•If we have a continuous distribution of charge, we use
techniques of integral calculus to calculate V(x,y,z).
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Potential and Work
For any group of charges, we can calculate the work
done by the electrostatic force as the charges are
brought together from infinity.
The potential energy associated with a two charge
system:
q1q2
U k
r
•q
2
•
q1
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Walker P. #33
The three charges are held in place in the figure below, where L = 1.25 m.
(a) Find the electric potential at point P [ans:76.9 kV]
(b) Suppose that a fourth charge, with a charge of 6.11 mC and a mass of
4.71 g, is released from rest at point P. What is the speed of the fourth
charge when it has moved infinitely far away from the other three charges?
[ans:14.1 m/s]
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The Electron Volt (eV)
It is often convenient to work with a unit of energy
called the electron volt.
One electron volt is defined as the amount of energy
an electron (with charge e) gains when accelerated
through a potential difference of 1 V:
1 eV = (1.6 · 10-19 C)V= 1.6 · 10-19 J
A Battery is an electron pump.
A battery (1.5 V), each electron pumped through the
battery from + to - is given a potential energy of
1.5eV.
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Equipotential Surfaces
A (real or imaginary) surface in space for which the
potential is the same everywhere is called an
equipotential surface.
• The electric field at every point on an equipotential
surface is perpendicular to the surface.
• Equipotential surfaces are like contour lines on a
topographic map.
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Electric Field Lines and Equipotential Surfaces
for two point charges
(Electric field lines and Equipotential surfaces are always mutually
perpendicular).
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Capacitance
A capacitor is device that stores the energy
associated with a configuration of charges.
In general, a capacitor consists of 2 conductors,
one with a charge +Q and the other with a charge
–Q (on the surfaces). Any geometry is a capacitor
-
+
++ -Q + - Q
+
-
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• The capacitance C is defined as the ratio of the
magnitude of the charge on either conductor to the
magnitude of the potential difference between the
conductors:
Q
C
V
For parallel plate C = A 0 /d. (C does not depend on Q or
V)
[V = Ed, E=Q/(A 0), V = Qd / (A 0)]
The unit of capacitance is the Farad (F): 1 F = 1 C/V
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The Parallel-plate Capacitor
A common type of capacitor is the parallel-plate
capacitor, made up simply of two flat plates of area A
separated by a distance d. Its capacitance is given
by:
A
C  0
d
where 0 is a constant called the permittivity of free
space.
0
=8.8510-12
C2
/
Nm2
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k
1
40
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Dielectrics
A dielectric is an insulating material in which the individual
molecules polarize in proportion to the strength of an external
electric field.
This reduces the electric field inside the dielectric by a factor k,
called the dielectric constant.
For fixed charge Q on plates
E
E0
and V 
V0
k
k
Capacitance is
increased by k.
C  kC0
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Dielectric Strength
• Dielectrics are insulators: charges are not free to
move (beyond molecular distances)
• Above a critical electric field strength, however,
the electrostatic forces polarizing the molecules
are so strong that electrons are torn free and
charge flows.
• This is called Dielectric Breakdown, and can
disturb the mechanical structure of the material
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Dielectric Properties
of common materials
Material
Dielectric
Constant: k
Dielectric
Strength (V/m)
Vacuum
1
2.5·1018
Air (lightening)
1.00059
(k-1) Density
3.0·106
Teflon
2.1
60 ·106
Paper
3.7
16 ·106
Mica
5.4
100 ·106
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Energy Stored in a Capacitor
Recall that work is required to move charges about or
“charge” the capacitor. The work required to charge a
capacitor with a charge q to a voltage V is:
1
E  QV
2
So this must correspond to the energy stored in the capacitor.
Because Q=CV, this can be rewritten:
2
1
2 Q
E  CV 
2
2C
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Walker P. #50
(a) What plate area is required if an air-filled, parallel-plate capacitor
with a plate separation of 2.8 mm is to have a capacitance of 26 pF?
[ans:0.00822 m2]
(b) What is the maximum voltage that can be applied to this capacitor
without causing dielectric breakdown? [ans:8.4 kV]
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