Download Chapter 17: Electric Potential

Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Chapter 17: Electric Potential
•Electric Potential Energy
•Electric Potential
•How are the E-field and Electric Potential related?
•Motion of Point Charges in an E-field
•Capacitors
•Dielectrics
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.1 Electric Potential Energy
Electric potential energy (Ue) is energy stored in the electric
field.
•Ue depends only on the location, not upon the path taken
to get there (conservative force).
•Ue = 0 at some reference point.
•For two point particles take Ue = 0 at r = .
•For the electric force
kq1q2
Ue 
r
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: A proton and an electron, initially separated by a
distance r, are brought closer together. How does the
potential energy of this system of charges charge?
ke2
For these two charges U e  
r
Bringing the charges closer together decreases r:.
U e  U ef  U ei  0
This is like a mass falling near the surface of the Earth;
positive work is done by the field.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued
How will the electric potential energy change if both
particles have positive (or negative) charges?
When q1 and q2 have the same algebraic sign then
Ue > 0.
This means that work must be done by an external
agent to bring the charges closer together.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What is the potential energy of a system (arrangement) of
point charges? To calculate:
Begin by placing the first charge at a place in space
far from any other charges. No work is required to
do this.
Next, bring in the remaining charges one at a time
until the desired configuration is finished.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: What is the potential energy of three point charges
arranged as a right triangle? (See text Example 17.2)
q2
q2
r12
q1
Ue  0 
r12
r23
q3
r13
kq1q2 kq1q3 kq2 q3


r12
r13
r23
r23
q3
q1
r13
kq1q2 kq1q3 kq2 q3


Ue  0 
r12
r13
r23
Are these the same?
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.2 Electric Potential
Electric potential is the electric potential energy per unit
charge.
Ue
V
qtest
Electric potential (or just potential) is a measurable scalar
quantity. Its unit is the volt (1 V = 1 J/C).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
For a point charge of charge Q:
U e kQ
V

qtest
r
When a charge q moves through a potential difference
of V, its potential energy change is Ue = q V.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: A charge Q = +1 nC is placed somewhere in space
far from other charges. Take ra = rb = rc = rd = 1.0 m and re =
rf = rg = 2.0 m.
f
b
c
e
a
Q
d
g
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(a) Compare the potential at points d and g.
Since Q>0 the potential at point d is greater than at
point g, it is closer to the charge Q.
(b) Compare the potential at points a and b.
The potential at point a is the same as at point b;
both are at the same distance from the charge Q.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(c) Place a charge of +0.50 nC at point e. What will the
change in potential (V) be if this charge is moved to point
a?
kQ 9.0 109 Nm2 /C 2 1.0 nC 
Ve 

 4.5 Volts
re
2m
kQ 9.0 109 Nm2 /C 2 1.0 nC 
Va 

 9.0 Volts
ra
1m
V = Vf – Vi = Va-Ve = +4.5 Volts
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(d) What is the change in potential energy (U) of the
+0.50 nC charge ?
Ue =qV = (+0.50 nC)(+4.5 Volts)= +2.3 nJ
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(e) How would the results of the previous questions change
if instead of a +1.0 nC charge there is a -1.0 nC charge in its
place?
(a)The potential at point d is less than the potential at
point g.
(b) Unchanged
(c) -4.5 V
(d) -2.3 nJ
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.3 The Relationship between E
and V
f
b
The circles are
called equipotentials
(surfaces of equal
potential).
c
e
a
Q
+9 V
+4.5 V
d
g
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The electric field will point in the direction of maximum
potential decrease and will also be perpendicular to the
equipotential surfaces.
f
b
c
e
a
Q
+9 V
+4.5 V
d
g
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Equipotentials
and field lines
for a dipole.
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Uniform E-field
V1
V2
V3
V4
E
Equipotential surfaces
U e
V 
  Ed
q
Where d is the distance
over which V occurs.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
If the electric field inside a conductor is zero, what is the
value of the potential?
If E=0, then V=0. The potential is constant!
What is the value of V inside the conductor? It will be
the value of V on the surface of the conductor.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.4 Moving Charges
When only electric forces act on a charge, its total
mechanical energy will be conserved.
Ei  E f
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.31): Point P is at a potential of
500.0 kV and point S is at a potential of 200.0 kV. The space
between these points is evacuated. When a charge of +2e
moves from P to S, by how much does its kinetic energy
change?
Ei  E f
Ki  U i  K f  U f
K f  K i  U i  U f  U f  U i 
 U  qV  qVs  V p 
  2e 200.0  500.0kV
 9.6 10
14
J
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.32): An electron is accelerated from
rest through a potential difference. If the electron reaches a
speed of 7.26106 m/s, what is the potential difference?
Ei  E f
0
Ki  U i  K f  U f
K f  U  qV
1
2
mv f  qV
2
2
mv f
9.1110 31 kg 7.26 106 m/s
V  

2q
2  1.60 10 19 C
 150 Volts
Note: the electron moves





2
from low V to high V.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.5 Capacitors
A capacitor is a device that stores electric potential energy
by storing separated positive and negative charges. Work
must be done to separate the charges.
+
+
-
-
+
+
+
+
+
-
-
-
-
-
Parallel plate
capacitor
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
For a parallel plate capacitor:
EQ
E  V
 Q  V
Written as an equality: Q = CV, where the proportionality
constant C is called the capacitance.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What is the capacitance for a parallel plate capacitor?

Q
V  Ed  d 
d
0
0 A
0 A
Q 
d
where C 
V  CV
0 A
d
.
Note: C depends only on constants and geometrical factors.
The unit of capacitance is the farad (F). 1 F = 1 C2/J = 1 C/V
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.42): A parallel plate capacitor
has a capacitance of 1.20 nF. There is a charge of
magnitude 0.800 C on each plate.
(a) What is the potential difference between the plates?
Q  CV
Q 0.800 C
V  
 667 Volts
C
1.20 nF
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) If the plate separation is doubled, while the charge is
kept constant, what will happen to the potential difference?
Q Qd
V  
C 0 A
V  d
If d is doubled so is the potential difference.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.86): A parallel plate capacitor has a
charge of 0.020 C on each plate with a potential difference
of 240 volts. The parallel plates are separated by 0.40 mm of
air.
(a) What is the capacitance of this capacitor?
Q
0.020 C
C

 8.3 10 11 F  83 pF
V 240 Volts
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) What is the area of a single plate?
C
A
0 A
d
Cd
0

83 pF0.40 mm 
8.85 10 12 C 2 / Nm2
 0.0038 m 2  38 cm 2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.6 Dielectrics
As more and more charge is placed on capacitor plates,
there will come a point when the E-field becomes strong
enough to begin to break down the material (medium)
between the capacitor plates.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
To increase the capacitance, a dielectric can be placed
between the capacitor plates.
C   C0
where C0 
0 A
d
and  is the dielectric constant.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.55): A capacitor can be made from
two sheets of aluminum foil separated by a sheet of waxed
paper. If the sheets of aluminum are 0.3 m by 0.4 m and the
waxed paper, of slightly larger dimensions, is of thickness
0.030 mm and has  = 2.5, what is the capacitance of this
capacitor?
C0 
0 A
d
8.85 10 12 Nm2 /C 2 0.40 * 0.30m 2

0.030 10-3 m
8
 3.54 10 F




and C   C0  2.5 3.54 10 8 F  8.85 10 8 F.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§17.7 Energy Stored in a Capacitor
A capacitor will store energy equivalent to the amount of
work that it takes to separate the charges.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The energy stored in the electric field between the plates is:
1
U  QV
2
1
2
 C V 
2
Q2

2C
}
These are found by
using Q=CV and
the first relationship.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 17.63): A parallel plate capacitor is
composed of two square plates, 10.0 cm on a side, separated
by an air gap of 0.75 mm.
(a) What is the charge on this capacitor when the potential
difference is 150 volts?
Q  CV 
0 A
d
8
V  1.77 10 C
(b) What energy is stored in this capacitor?
1
U  QV  1.33 10 6 J
2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Summary
•Electric Potential Energy
•Electric Potential
•The Relationship Between E and V
•Motion of Point Charges (conservation of energy)
•Parallel Plate Capacitors (capacitance, dielectrics, energy
storage)
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