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Weds., Jan. 29, 2014 PHYS 1442-004, Dr. Andrew Brandt 1
PHYS 1442 – Section 004
Lecture #5 Wednesday January 29, 2014
Dr. Andrew Brandt
CH 17
• Electric Potential due to Point Charges
• Shape of the Electric Potential
• Equi-potential Lines and Surfaces
• Electron-volt
• Capacitance
Announcements
a) Avg was 81 on HW. Good job, you should dominate that material on the
test.
b) New HW assigned on ch 17
Weds., Jan. 29, 2014 2 PHYS 1442-004, Dr. Andrew Brandt
Weds., Jan. 29, 2014 3 PHYS 1442-004, Dr. Andrew Brandt
Electric Potential and Potential Energy • The electric potential difference gives potential energy (or
the possibility to do work) based on the charge of the object.
• So what is happening in batteries or generators?
– They maintain a potential difference.
– The actual amount of energy used or transformed depends on how
much charge flows.
– How much is the potential difference maintained by a car’s
battery?
• 12Volts
– If for a given period, 5C charge flows through the headlight lamp,
what is the total energy transformed?
• Etot=5C*12V=60 What is the unit?
– If it is left on twice as long? Etot=10C*12V=120J.
C*J/C=J (Joules)
Weds., Jan. 29, 2014 4 PHYS 1442-004, Dr. Andrew Brandt
Example 17 – 2 Electrons in a TV tube: Suppose an electron in the picture tube of a
television set is accelerated from rest through a potential difference
Vba=+5000V. (a) What is the change in potential energy of the
electron? (b) What is the speed of the electron (m=9.1x10-31kg) as a
result of this acceleration? (c) Repeat for a proton (m=1.67x10-27kg)
that accelerates through a potential difference of Vba=-5000V.
• (a) What is the charge of an electron? –
• So what is the change of its potential energy?
U baqV 19 161.6 10 5000 8.0 10baeV C V J
191.6 10e C
Weds., Jan. 29, 2014 5 PHYS 1442-004, Dr. Andrew Brandt
Example 17 – 2 • (b) Speed of the electron?
– The entire potential energy of the electron is transformed into
kinetic energy. Thus the equation is
ev
K
• (c) Speed of a proton that accelerates through V=-5000V?
pv
K
210
2e em v W U ( )baU eV
16 168.0 10 8.0 10J J
2
e
K
m
167
31
2 8.0 104.2 10 /
9.1 10m s
210
2p pm v 168.0 10ba baW U e V eV J
2 ba
p
eV
m
165
27
2 8.0 109.8 10 /
1.67 10m s
Weds., Jan. 29, 2014 6 PHYS 1442-004, Dr. Andrew Brandt
Electric Potential and Electric Field • The effect of a charge distribution can be
described in terms of electric field or electric
potential.
– What kind of quantities are the electric field and the
electric potential?
• Electric Field:
• Electric Potential:
– Since electric potential is a scalar quantity, it often
can make problem solving easier.
Vector
Scalar
Weds., Jan. 29, 2014 7 PHYS 1442-004, Dr. Andrew Brandt
50V
5cm
Example 17 – 3 Uniform electric field obtained from voltage:
Two parallel plates are charged to a voltage of
50 V. If the separation between the plates is
5.0 cm, calculate the magnitude of the electric
field between them, ignoring any fringe effects.
EV
d
50
5.0
V
cm
What is the relationship between electric field and the
potential for a uniform field? ( )
/ /
W Fd Eq d
V U q W q Ed
V Ed
Solving for E 2
50
5 10
V
m1000 /V m
Weds., Jan. 29, 2014 8 PHYS 1442-004, Dr. Andrew Brandt
Electric Potential due to Point Charges • Since only the differences in potential have physical
meaning, we can choose at .
• The electrical potential V at a distance r from a single
point charge is
• So the absolute potential from a single point charge
depends only on the magnitude of the point charge
and the distance from it
V0
1
4
Q
r
0bV br
Weds., Jan. 29, 2014 9 PHYS 1442-004, Dr. Andrew Brandt
• What are the differences between the electric potential and
the electric field?
– Electric potential
• Electric potential energy per unit charge
• Inversely proportional to the distance
• Simply add the potential from each of the charges to obtain the total potential
from multiple charges, since potential is a scalar quantity
– Electric field
• Electric force per unit charge
• Inversely proportional to the square of the distance
• Need vector sums to obtain the total field from multiple charges
• Potential for a positive charge is large near the charge and
decreases to 0 at large distances.
• Potential for the negative charge is small (large magnitude but
negative) near the charge and increases with distance to 0
Properties of the Electric Potential
2
0
1
4
QE
r
0
1
4
QV
r
Weds., Jan. 29, 2014 10 PHYS 1442-004, Dr. Andrew Brandt
Shape of the Electric Potential • So, what does the electric potential look like as a function of
distance?
– What is the formula for the potential by a single charge?
V0
1
4
Q
r
Positive Charge Negative Charge
A uniformly charged sphere would have the same potential as a single point charge.
What does this mean? Uniformly charged sphere behaves like all the charge is on the single point in the center.
Weds., Jan. 29, 2014 11 PHYS 1442-004, Dr. Andrew Brandt
Since we obtain
Example 17.5 Work to bring two positive charges close together: What
minimum work is required by an external force to bring a
charge q=3.00 μC from a great distance away ( ) to a
point 0.500 m from a charge Q=20.0 μC?
What is the work done by the electric field in terms of potential
energy and potential?
W
0.500 ,b ar m r
W
In other words, the external force must input work of +1.08J to bring the charge
3.00 C from infinity to 0.500m from the 20.0 C charge.
baqV04 b a
q Q Q
r r
0
04 b
q Q
r04 b
q Q
r
9 2 2 6 68.99 10 3.00 10 20.00 101.08
0.500
N m C C CJ
m
r
Weds., Jan. 29, 2014 12 PHYS 1442-004, Dr. Andrew Brandt
Since we obtain
More on Example 17-5 Work to bring two positive charges close together: What
minimum work is required by an external force to bring a
charge q=3.00 μC from a great distance away ( ) to a
point 0.500 m from a charge Q=20.0 μC?
What is the work done by the electric field in terms of potential
energy and potential?
W
0.500 ,b ar m r
W
In other words, the external force must input work of +1.08J to bring the charge
3.00 C from infinity to 0.500m from the 20.0 C charge.
baqV04 b a
q Q Q
r r
0
04 b
q Q
r04 b
q Q
r
9 2 2 6 68.99 10 3.00 10 20.00 101.08
0.500
N m C C CJ
m
r
Weds., Jan. 29, 2014 13 PHYS 1442-004, Dr. Andrew Brandt
Electrostatic Potential Energy: Two charges
• What is the electrostatic potential energy of a configuration of
charges? (Choose V=0 at r=
– If there are no other charges around, a single point charge Q1 in
isolation has no potential energy and feels no electric force
• If a second point charge Q2 is to a distance r12 from Q1 ,the
potential at the position of Q2 is
• The potential energy of the two charges relative to V=0 at r=
is
-- This is the work that needs to be done by an external force to bring Q2
from infinity to a distance r12 from Q1.
– It is also a negative of the work needed to separate them to infinity.
V 1
0 12
1
4
Q
r
2U Q V1 2
0 12
1
4
Q Q
r
Weds., Jan. 29, 2014 14 PHYS 1442-004, Dr. Andrew Brandt
Electrostatic Potential Energy: Three Charges • So what do we do for three charges?
• Work is needed to bring all three charges together
– There is no work needed to bring Q1 to a certain place without
the presence of any other charge
– The work needed to bring Q2 to a distance to Q1 is
– The work need to bring Q3 to a distance to Q1 and Q2 is
• So the total electrostatic potential of the three charge
system is
12U
3U
U
1 2
0 12
1
4
Q Q
r
13U1 3
0 13
1
4
Q Q
r
2 3
0 23
1
4
Q Q
r23U
12 13 23U U U1 3 2 31 2
0 12 13 23
1 0 at
4
Q Q Q QQ QV r
r r r
Weds., Jan. 29, 2014 15 PHYS 1442-004, Dr. Andrew Brandt
Equi-potential Surfaces • Electric potential can be visualized using equipotential lines in
2-D or equipotential surfaces in 3-D
• Any two points on equipotential surfaces (lines) have the same
potential
• What does this mean in terms of the potential difference?
– The potential difference between the two points on an equipotential
surface is 0.
• How about the potential energy difference?
– Also 0.
• What does this mean in terms of the work to move a charge
along the surface between these two points?
– No work is necessary to move a charge between these two points.
Weds., Jan. 29, 2014 16 PHYS 1442-004, Dr. Andrew Brandt
Equi-potential Surfaces • An equipotential surface (line) must be perpendicular to the electric field. Why?
– If there are any parallel components to the electric field, it would
require work to move a charge along the surface.
• Since the equipotential surface (line) is perpendicular to the electric field, we can draw
these surfaces or lines easily.
• There can be no electric field inside a conductor in static case, thus the entire volume of
a conductor must be at the same potential.
• So the electric field must be perpendicular to the conductor surface.
Point
charges Parallel
Plate Just like a topographic map
Weds., Jan. 29, 2014 17 PHYS 1442-004, Dr. Andrew Brandt
Electrostatic Potential Energy: electron Volt
• What is the unit of electrostatic potential energy?
– Joules
• Joules is a very large unit in dealing with electrons, atoms or
molecules
• For convenience a new unit, electron volt (eV), is defined
– 1 eV is defined as the energy acquired by a particle carrying the
charge equal to that of an electron (q=e) when it moves across a
potential difference of 1V.
– How many Joules is 1 eV then?
• eV however is not a standard SI unit. You must convert the
energy to Joules for computations.
1eV 191.6 10 1C V 191.6 10 J
Weds., Jan. 29, 2014 18 PHYS 1442-004, Dr. Andrew Brandt
Capacitors (or Condensers) • What is a capacitor?
– A device that can store electric charge without letting the charge flow
• What does it consist of?
– Usually consists of two oppositely charged conducting objects (plates or
sheets) placed near each other without touching
– Why can’t they touch each other?
• The charges will neutralize each other
• Can you give some examples?
– Camera flash, surge protectors, computer keyboard, binary circuits…
• How is a capacitor different than a battery?
– Battery provides potential difference by storing energy (usually chemical
energy) while the capacitor stores charge but very little energy.
19
Capacitors • A simple capacitor consists of a pair of parallel plates
of area A separated by a distance d.
– A cylindrical capacitors are essentially parallel plates
wrapped around as a cylinder.
• Symbols for a capacitor and a battery:
– Capacitor -||-
– Battery (+) -|l- (-) Circuit
Diagram
Weds., Jan. 29, 2014 PHYS 1442-004, Dr. Andrew Brandt
Weds., Jan. 29, 2014 20 PHYS 1442-004, Dr. Andrew Brandt
• What do you think will happen if a battery is connected
(voltage is applied) to a capacitor?
– The capacitor gets charged quickly, one plate positive and the other
negative with an equal amount. of charge
• Each battery terminal, the wires and the plates are
conductors. What does this mean?
– All conductors are at the same potential.
– the full battery voltage is applied across the capacitor plates.
• So for a given capacitor, the amount of charge stored in the
capacitor is proportional to the potential difference Vba
between the plates. How would you write this formula?
– C is a proportionality constant, called capacitance of the device.
– What is the unit?
Capacitors
baQ CV
C/V or Farad (F)
C is a property of a capacitor so does not depend on Q or V.
Normally use F or pF.
Weds., Jan. 29, 2014 21 PHYS 1442-004, Dr. Andrew Brandt
Determination of Capacitance • C can be determined analytically for capacitors w/ simple
geometry and air in between.
• Let’s consider a parallel plate capacitor.
– Plates have area A each and separated by d.
• d is smaller than the length, so E is uniform.
– For parallel plates E= 0, where is the surface charge density.
• E and V are related
• So from the formula:
– What do you notice?
baV Ed0
0ba
AQ QC
V Qd A d
C only depends on the area
(A) and the separation (d) of
the plates and the permittivity
of the medium between them.