Chapter 18 Electric Potential

1
Chapter 18 Electric Potential
2
Main Points of Chapter 18

• Definition of electric potential
• Calculation of electric potential of various
  charge distributions
• The relationship between electric potential and
  electric field

3
18-1 Electric Potential Energy
1. Work Done by the Electric Force
Suppose the electric field caused by a
single, stationary point charge q
b
?
O
q0
L
a
The work done by the electric force is
independent of the path, it only depends on the
initial and final positions.
4
Suppose the electric field caused by a system
of point charges.
The work done by the static electric force
depends only on the initial and final points,
not on the path. So every electric field due to a
static charge distribution is a conservative
force field.
5
The work done around a closed path
The work done by the static electrical force
around any closed path is zero.
The static electrical force is a conservative
force.
6
2. The Static Electrical Field is a Conservative
Force Field
The line integral of E around any closed path in
an electrostatic field is zero. Alternatively the
work done in taking a test charge round any
closed path in an electrostatic field is zero
The circuital law for E-fields
Using this result we can check a field whether it
is a static electrical field.
7
(1)
It can not be a static electrical field
It can not be a static electrical field
8
The differential form of the circuital law
We have for electrostatics that
9
Applying Stokes theorem to the circuital law for
E when we can make the area S infinitesimally
small
Hence the curl of any electrostatic E-field is
always zero. This is the differential form of the
circuital law.
10
(1)
It can not be a static electrical field.
(2)
It can not be a static electrical field
11
3. Electric Potential Energy
Mechanics
conservative force field
potential energy
conservative force field
Electric Potential Energy
Static electrical field
Definition We define the work done by the static
electrical force in moving a charge q0 from an
initial position to a final position as the
negative change in the electrical potential
energy of the charge q0
12
Discussion
(1)The potential energy U is a property shared
by the source charge and the test
charge.
(2) The potential energy U is related to where
the reference point is, but the
potential energy difference is independent of
the reference point.
Suppose
The potential energy U at some point is equal to
the work done by the external agent when a point
charge is brought from infinity to that point.
13

• Example Figure below shows two point charges held
  in fixed positions by forces that are not shown.
  What is the electric potential energy U of this
  system?

Q
q
d
r
Solution
14

• ACT In the figure, a proton moves from point i
  to point f in a uniform electric field directed
  as shown. (a) Does the electric field do positive
  or negative work on the proton? (b) Does the
  electric potential energy of the proton increase
  or decrease?

the electric field does negative work
the electric potential energy of the proton
increases
15
18-2 Electric Potential
1. Electric Potential
Define the electric potential at a point to be
the potential energy per unit charge at that
point.
The electric potential difference is the work per
unit charge that must be done by an external
agent to move it from initial point to final
point without changing its kinetic energy.
V is independent of the test charge
16
J/C
V (volt)
Electric Potential
SI
V/m or N/C
Electric field
Suppose
17

• ACT The units of electric potential are ________
  
• A) C/J.
• B) J/C.
• C) J.
• D) C2/m.
• E) J/kg.

• ACT Electric potential is ________
• A) measured by an accelerator.
• B) measured by a potentiometer.
• C) not measurable.
• D) measured by a ohmmeter.
• E) measured by a voltmeter

18
2. Electric Potential of a point charge
Electric Potential decreases along the electric
field lines
19
Q

A positive charge accelerates from a region of
higher electric potential (or higher potential
energy) toward a region of lower electric
potential (or lower potential energy).
A negative charge accelerates from a region of
lower potential (or higher potential energy)
toward a region of higher potential (or lower
potential energy).
20
3.The Electric Potential of Charge Distribution
For a collection of n point charges
The Superposition of Electric Potential
21
For a continuous charge distribution
22
Electric Potential vs. Electric Potential Energy
Electric potential is a property of an electric
field, regardless of whether a charged object has
been placed in that field it is measured in
joules per coulomb, or volts.
Electric potential energy is an energy of a
charged object in an external electric field (or
more precisely, an energy of the system
consisting of the object and the external
electric field) it is measured in joules.
23
4.The Potential Energy of a System of Charges
the potential energy between q1 and q2
the potential energy between q1 and q3 ,q2 and q3
24
The total potential energy of the system
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.
25
Vi is the electric potential due to all the other
charges at the location of charge qi.
26
Solution
An external agent would have to do 17 mJ of work
to disassemble the structure completely, ending
with the three charges infinitely far apart.
27

• Example How much work is done by the external
• agent when a dipole (q,l) is brought from
  infinity to rest
• in an electric field E. How about if the two
  electric
• charges are initially infinitely far apart ?

Solution
-q
q
They attract each other
28
5.The Electron-Volt
The unit of energy Electron-Volt (eV)
An electron-volt is the energy of an electron
gains when it is accelerated through a
potential difference of one volt .
Electron-volts are useful in atomic, nuclear, and
particle physics.
29

• Example The classical model of hydrogen atom in
• its normal, unexcited configuration has an
  electron
• that revolves around a proton at a distance of
  
• Determine the potential energy between the two
• particles.

Solution
The total energy of the electron
energy of ionization
30

• ACT The hydrogen atom requires an amount of
  work equal to 13.6 eV to separate the electron
  from the proton. This amount of energy is
  ________
• A) The potential energy, U, of the proton and
  electron.
• B) K - U.
• C) the kinetic energy, K, of the electron.
• D) K U.
• E) cannot be determined

31
18-3 The Potential of Charge Distribution

• If the electric field is known

• For a continuous charge distribution

32

• Example Find the electric potential at a point
  P located a distance x along the axis of a
  uniformly charged circular ring of radius R with
  total charge q.

Solution
33

• ACT (a) In Fig. a , what is the potential at
  point P due to charge Q at distance R from P? Set
  V 0 at infinity. (b) In Fig. b , the same
  charge Q has been spread uniformly over a
  circular arc of radius R and central angle 40.
  What is the potential at point P, the center of
  curvature of the arc? (c) In Fig. c , the same
  charge Q has been spread uniformly over a circle
  of radius R. What is the potential at point P,
  the center of the circle? (d) Rank the three
  situations according to the magnitude of the
  electric field that is set up at P, greatest
  first.

(d) a,b,c
34

• Example Find the potential of a spherical shell
  (radius R) with total charge q uniformly
  distributed

Solution
It is easy to use Gauss law to find the electric
field
Where r gt R
Where r lt R
35

• Act Two concentric uniformly charged sphere
  shells
• of radii R1 and R2 respectively carry charges
  q1 and
• q2 respectively as shown. Find the potential.

36

• Example Find the potential of a uniformly
  charged sphere with total charge q and radius R

Solution
It is easy to use Gauss law to find the electric
field
Where r gt R
Where r lt R
Follow-up Please solve this problem by the
superposition of electric potential!
37
Solution
38

• Example Find the electric potential at a
  distance r from a very long line of charge with
  liner charge density ?.

Solution
The electric field of a line charge is given by
If we let
A reference point at infinity is not suitable for
this field!
39
Solution
The electric field is
Choose Vx00
40
If I choose Vx-a/20
41

• Act There is a small cavity on an uniformly
  charged
• sphere shell (R,s ). The radius of the cavity
  is h as shown.
• Find the electric field and potential at the
  center.

o
42

• Example Find the potential energy of a uniformly
  charged spherical shell with total charge Q and
  radius R

Solution
The electric potential energy of a system 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.
Suppose we have brought charge q in and they are
uniformly distributed.
To bring an additional charge dq ,we must do work
43

• Example Find the potential energy of a uniformly
  charged spherical shell with total charge Q and
  radius R

Alternative Solution
Vi is the electric potential due to all the other
charges at the location of charge qi.
44
18-4 Equipotentials
An equipotential surface is a three-dimensional
surface on which the electric potential is the
same at every point.
The equipotential surfaces are drawn such that
the potential difference between adjacent
surface is constant.
45
(No Transcript)
46
(1) No work is done by the electric force when a
charge moves on an equipotential .
Because potential energy does not change
(2) Field lines is everywhere perpendicular to
equipotential surfaces.
47
(3) In regions where the field is stronger, the
equipotential surfaces are closer, in regions
where the field is weaker, the equipotential
surfaces are farther apart.
Because the potential differences between
adjacent surfaces are equal.
(4) The direction of the electric field is always
from regions with higher values of the electric
potential to the regions with lower values of the
electric potential.
48
18-5 Determining Fields from Potentials
Consider a small displacement between two
equipotential surfaces
The rate of change of V with distance is
greatest in the direction perpendicular to the
equipotential at that point.
We define
A vector that points in the direction of the
greatest change in a scalar function and has a
magnitude equal to the derivative of that
function with respect to the distance in that
direction is called the gradient of the function
49
In Cartesian Coordinates
The electric field lines point in the direction
of the greatest rate of decrease with respect to
distance in the potential function.
50
For any infinitesimal displacement
We can write
51
In vector notation
The electric field always points in the direction
of maximum potential decrease.
We have
52
The relationship between electric potential and
electric field

• To find the potential from the field integrate
  along
• 
  the path
• To find the field from the potential take
  gradient

If we know either the potential or the electric
field over some region of space, we can use one
to calculate the other.
53

• ACT Figure below gives the electric potential V
  as a function of x. (a) Rank the five regions
  according to the magnitude of the x component of
  the electric field within them, greatest first.
  What is the direction of the field along the x
  axis in (b) region 2 and (c) region 4?

(a) 2, 4 and then a tie of 1, 3 and 5 (where E0 )
(b) negative x direction
(c) positive x direction
54

• Example Suppose we know
  ,
• find the electric field at point(2,3,0)

Solution
55

• Example Use the expression of electric potential
  of a point charge to find its electric field .

Solution
We know in the radial direction the rate of
change of V is greatest
56

• Example Find the electric field at a distance x
  along the axis of a uniformly charged disk of
  radius R and charge Q

Solution
Calculate the electric potential first
dV
57
(No Transcript)
58

• Example Calculate the electric potential and
  electric
• field of the electric dipole.
  

Solution
59
Discussion
60

• ACT If the potential at a distance d from a
  dipole is Vo for distances large compared to the
  length of the dipole, then the potential at twice
  the distance from the dipole is ________

(a) Vo
(b) Vo/2
(c) Vo/4
61

• Act Which is V(x)? Which is E(x)?

E
V
x
x
1
2
3
4
5
6
1
2
3
4
5
6
V
E
x
x
1
2
3
4
5
6
1
2
3
4
5
6
62
Solution
63
E
Xa
(2)
x
a
2a
3a
0
-V0/a
-2V0/a
X2a
64
Summary of Chapter 18
Electric potential
Relationship of field and potential
Electric field is perpendicular to equipotential
surface