Chapter 23

Electric Fields
Coulomb’s Law, Equation
Mathematically,
q1 q 2
Fe  k e
r2

The SI unit of charge is the coulomb (C).

ke is called the Coulomb constant.

 ke = 8.9876 x 109 N.m2/C2 = 1/(4πeo)

 eo is the permittivity of free space.
 eo = 8.8542 x 10-12 C2 / N.m2

Section 23.3
Coulomb's Law, Notes
Remember the charges need to be in coulombs.
 e is the smallest unit of charge.
 except quarks

 e = 1.6 x 10-19 C
 So 1 C needs 6.24 x 1018 electrons or protons
Typical charges can be in the µC range.
Remember that force is a vector quantity.

Section 23.3
Particle Summary

The electron and proton are identical in the magnitude of their charge, but very
different in mass.
The proton and the neutron are similar in mass, but very different in charge.

Section 23.3
Vector Nature of Electric Forces
In vector form,
 q 1q 2
F1 2  k e 2 rˆ1 2
r
r̂12 is a unit vector directed from q1 to
q2.
The like charges produce a repulsive
force between them.

Section 23.3
Vector Nature of Electrical Forces, cont.

Electrical forces obey Newton’s Third Law.

The force on q1 is equal in magnitude and opposite in direction to the
force on q2
 
 F21  F12
With like signs for the charges, the product q1q2 is positive and the
force is repulsive.

Section 23.3
Vector Nature of Electrical Forces, 3
Two point charges are separated by a
distance r.
The unlike charges produce an
attractive force between them.
With unlike signs for the charges, the
product q1q2 is negative and the force is
attractive.
 q 1q 2 r̂12 r̂21
F1 2  k e 2 rˆ1 2
r
 q2 q1
F21  ke 2 rˆ21
r

Section 23.3
 3 significant figures
m 10
a) Number of mole: 
 107.87
m
Number of electron:  N A  47  2.62  1024

b) Number of electron added for every 109 existing electrons:
Number of electron added to q 1103
the pin  q = 1.00 mC:   6.25  1015

e 1.6  1019
6.25  1015
 2.385  2.38
2.62  10 / 10
24 9
 q 1q 2
F1 2  k e 2 rˆ1 2 q
r

 q(q)  q(q) ˆ
F  ke 2
rˆb t F  ke 2
j
r r
 9 40( 40) ˆ 6 ˆ
F  9  10 2
j   3.6× 10 jN
2000

q
 6 1.5 2  qq
F12  k e 1 2 2 rˆ12
r̂21 r̂31 r
 
F21 F31
iˆ
a) 3 2
  q1q3
q1q2 ˆ
 
F21  ke 2 i F31  ke  iˆ 
 d1  d 2 
2
d1
   q q3 
F1  F21  F31  ke q1  2 
2
 ( iˆ)  46.8iˆ N
 d1  d  d  2 
 1 2 
 
T

d FE

 
equilibrium Fext  0 Fg
d  2 L sin 
FE ke q 2  qq
tan   FE  2 F12  k e 1 2 2 rˆ12
Fg d r
 Cartesian Coordinate system

y

F
Fy

Fx

Components of a vector
O x
Fx  F cos( ) 
F  Fx iˆ  Fy ˆj
Fy  F sin( )
Multiple Charges
The resultant force on any one charge equals the vector sum of the
forces exerted by the other individual charges that are present.
 Remember to add the forces as vectors.
The resultant force on q1 is the vector sum of all the forces exerted on it
by other charges.
For example, if four charges are present, the resultant force on one of
these equals the vector sum of the forces exerted on it by each of the
other charges.
   
F1  F21  F31  F41

Section 23.3
Electrical Force with Other Forces, Example

The spheres are in equilibrium.

Since they are separated, they
exert a repulsive force on each
other.
 Charges are like charges
Model each sphere as a particle in
equilibrium.
Proceed as usual with equilibrium
problems, noting one force is an
electrical force.

Section 23.3
Electrical Force with Other Forces, Example cont.
The force diagram includes the
components of the tension, the
electrical force, and the weight.
Solve for |q|
If the charge of the spheres is not
given, you cannot determine the sign of
q, only that they both have same sign.

Section 23.3
Electric Field – Introduction
The electric force is a field force.
Field forces can act through space.
 The effect is produced even with no physical contact between
objects.
Faraday developed the concept of a field in terms of electric fields.

Test charge

FE

Section 23.4
Electric Field – Definition, cont
The electric field is defined as the electric force on the test charge
per unit charge.

The electric field vector, E , at a point in space is defined as the
electric force acting on a positive test charge, qo, placed at that
point divided by the test charge:

 F
E
qo

Section 23.4
Relationship Between F and E
 
Fe  qE
 This is valid for a point charge only.
 One of zero size
 For larger objects, the field may vary over the size of the object.
If q is positive, the force and the field are in the same direction.
If q is negative, the force and the field are in opposite directions.

Section 23.4
Electric Field, Vector Form
Remember Coulomb’s law, between the source and test charges, can be
expressed as

 qq
Fe  ke 2o rˆ
r
Then, the electric field will be

 F q
E  e  ke 2 rˆ
qo r

Section 23.4
More About Electric Field Direction
a) q is positive, the force is directed away from q.
b) The direction of the field is also away from the positive source charge.
c) q is negative, the force is directed toward q.
d) The field is also toward the negative source charge.

Section 23.4
Electric Fields from Multiple Charges
At any point P, the total electric field due to a group of source charges
equals the vector sum of the electric fields of all the charges.
 qi
E  ke  2 rˆi
i ri

Section 23.4
 A O C

d 2  d1 d1  d 2
  
E  E1  E2  0
  q2  q1 q1 q2
E1  E2 E1  ke E2  ke 2
d12 d2
q1 q2 q2
OA  x  dxx
x2  d  x  2 q1
 q2 
x d /  1
 q 
 1 
 Find the E vector at the point charge 4 C

 q1  1 ˆ 3 ˆ
E1  ke 2  i  j
q1 
d 2 2 
 q

E2  ke 22 iˆ
d
 1  q1 ˆ 3 ˆ
E  k e 2    q2  i  q1 j 
q2 d  2  2 

q2 E  198000iˆ  218238 ˆj
Electric Field – Continuous Charge Distribution, cont
Procedure:
 Divide the charge distribution into
small elements, each of which
contains Δq.
 Calculate the electric field due to
one of these elements at point P.
 Evaluate the total field by summing
the contributions of all the charge
elements.

E  Exiˆ  E y ˆj  E z kˆ

Section 23.5
Electric Field – Continuous Charge Distribution, equations
For the individual charge elements
 q
E  ke 2 rˆ
r

Because the charge distribution is continuous

 q i dq
E  ke lim
qi 0

i ri 2
ri  ke  2 rˆ
ˆ
r

Section 23.5
  
Fe  Fg  0
 

Fe   Fg   mgjˆ 
 
Fe  qE
 mgjˆ  3.8 103 kg  (9.8 m/s 2 )
E  ˆj  2070 ˆj N/C
q  18 10 C 
6

2068.88(N/C), upward
Charge Densities
Volume charge density: when a charge is distributed evenly throughout
a volume
 ρ ≡ Q / V with units C/m3 Q= ρ x V Q   dq    dV
Surface charge density: when a charge is distributed evenly over a
surface area
 σ ≡ Q / A with units C/m2 Q   dq    dA
Linear charge density: when a charge is distributed along a line
 λ ≡ Q / ℓ with units C/m Q   dq    d 

Section 23.5
Amount of Charge in a Small Volume
If the charge is nonuniformly distributed over a volume, surface, or line,
the amount of charge, dq, is given by
 For the volume: dq = ρ dV
 For the surface: dq = σ dA
 For the length element: dq = λ dℓ

Section 23.5
 Example – Charged Rod

 dq ˆ
 dq ˆ
 
dE  ke 2 i
dq   d    dx
x
 
E  ke  2 i
x

Section 23.5
Example – Charged Rod
 dq ˆ
 
dE  ke 2 i
x

Q
E  ke 2
a

Section 23.5
dq   d    dx symmetry

0 0
  dq ˆ  1
dq
x

dE  ke 2 iˆ E  ke  x

2
i  k 
e
ˆ
i  
 x    /2
  /2

 dq ˆ / 2  /2

 
 dq ˆ  1 
dE  ke 2 i
x x
   
E  ke  2 i  ke iˆ   
 x 0
0
dq   d    dx

P (a, b)

dq ax
dE  ke dEx  dE cos  cos  
r2 (a  x)2  b 2
dE y  dE sin 
ax
cos  
(a  x)2  b 2
Section 23.5
 dq
dE  ke 2 dEx  dE cos 
r

x dq x
r a x
2 2 2
cos   dEx  ke 2
r a  x2 a2  x2
Section 23.5
Section 23.5
 Qx Q
x>>a? E  k e 3  ke 2
x x
E=0
Ring a point charge at its center

Section 23.5
Example – Charged Disk
The disk has a radius R and a uniform
charge density σ.
Choose dq as a ring of radius r.
The ring has a surface area 2πr dr.
Integrate to find the total field.

xdq
dEx  ke
r 2
x 
2 3/2

dq   dA   2 rdr

Section 23.5
Section 23.6
Electric Field Lines, General

The density of lines through

surface A is greater than through
surface B.
The magnitude of the electric field
is greater on surface A than B.
The lines at different locations point
in different directions.
 This indicates the field is
nonuniform.

Section 23.6
Electric Field Lines – Dipole

The charges are equal and

opposite.
The number of field lines leaving
the positive charge equals the
number of lines terminating on the
negative charge.

Section 23.6
Electric Field Lines – Like Charges

The charges are equal and

positive.
The same number of lines leave
each charge since they are equal in
magnitude.
At a great distance, the field is
approximately equal to that of a
single charge of 2q.
Since there are no negative
charges available, the field lines
end infinitely far away.

Section 23.6
Electric Field Lines, Unequal Charges
The positive charge is twice the
magnitude of the negative charge.
Two lines leave the positive charge for
each line that terminates on the
negative charge.
At a great distance, the field would be
approximately the same as that due to
a single charge of +q.

Section 23.6
Motion of Particles, cont
  
Fe  qE  ma
If the field is uniform, then the acceleration is constant.
The particle under constant acceleration model can be applied to the
motion of the particle.
 The electric force causes a particle to move according to the models
of forces and motion.
If the particle has a positive charge, its acceleration is in the direction of
the field.
If the particle has a negative charge, its acceleration is in the direction
opposite the electric field.

Section 23.7
Electron in a Uniform Field, Example
The electron is projected horizontally
into a uniform electric field.
The electron undergoes a downward
acceleration.
 It is negative, so the acceleration is
opposite the direction of the field.
Its motion is parabolic while between
the plates.

Section 23.7
   
FE  Fg  n  0 Fg sin   FE  QE
  mg sin 
FE  QE E
Q
d  0.85 m
m
N proton  6.02 1023  10
18 Q2 Fg  mEarth g
F  ke 2
N electron  0.01N proton d

Q  eN electron
Chapter 24
Gauss’s Law
Electric Flux
Electric flux is the product of the
magnitude of the electric field and the
surface area, A, perpendicular to the
field.
ΦE = EA

Units: N · m2 / C

Section 24.1
Electric Flux, General Area
The electric flux is proportional to the
number of electric field lines penetrating
some surface.
The field lines may make some angle θ
with the perpendicular to the surface.
Then ΦE = EA cos θ

Section 24.1
Electric Flux, Interpreting the Equation
The flux is a maximum when the surface is perpendicular to the field.
 θ = 0°
The flux is zero when the surface is parallel to the field.
 θ = 90°
If the field varies over the surface, Φ = EA cos θ is valid for only a small
element of the area.

 
E  E  A

ΦE = EA cos θ

Section 24.1
Electric Flux, General
In the more general case, look at a small
area element.
 
 E  Ei Ai cos θi  Ei  A i
In general, this becomes

 E  lim
Ai 0
E i Ai
 
E  
surface
E  dA

 The surface integral means the

integral must be evaluated over the
surface in question.
In general, the value of the flux will depend
both on the field pattern and on the
surface.

Section 24.1
  
 E  E  A  EA cos θ  
 E  E  A  EA cos0
 E  1.98  106 Nm2 /C
 
 E  E  A  EA cos90  0
  
 E  E  A  EA cos10

  
 E  2.00  10 4 N/C 6.00  3.00m2 cos10
 3.54  105 Nm2 /C
  
d  E  E  dA
y 
dx dA  dx.dy kˆ
h  
dy  
E  dA  ayiˆ  bzjˆ  cxkˆ  dx.dy kˆ
 

0 x E  dA  cx dx.dy
w   w  h 
 E   E  dA   cx dx.dy    cx dx    dy 
z w
0  0 
2 2
cx h cw h
 y0 
2 0 2
 y  
d  E  E  dA
dx 
h dA  dx.dy kˆ
dy  
 
E  dA  ayiˆ  bzjˆ  cxkˆ  dx.dy kˆ 
0 x E  dA  cx dx.dy
w   w  h 
z  E  dA   cx dx.dy   0 cx dx   0 dy 
  
 
a  ax iˆ  ay jˆ  az kˆ

b  bx iˆ  by jˆ  bz kˆ
 
a  b  ax bx  ay by  az bz
 ab cos θ
Electric Flux, Closed Surface
Assume a closed surface

The vectors A i point in different
directions.
 At each point, they are
perpendicular to the surface.
 By convention, they point outward.

Section 24.1
Flux Through Closed Surface, cont.

At (1), the field lines are crossing the surface from the inside to the
outside; θ < 90o, Φ is positive.
At (2), the field lines graze surface; θ = 90o, Φ = 0
At (3), the field lines are crossing the surface from the outside to the
inside;180o > θ > 90o, Φ is negative.

Section 24.1
Flux Through Closed Surface, final
The net flux through the surface is proportional to the net number of
lines leaving the surface.
 This net number of lines is the number of lines leaving the surface
minus the number entering the surface.
If En is the component of the field perpendicular to the surface, then
 
  dA   EndA
E  E

 The integral is over a closed surface.

Section 24.1
Flux Through a Cube, Example
The field lines pass through two
surfaces perpendicularly and are
parallel to the other four surfaces.
For face 1, E = -El 2
For face 2, E = El 2
For the other sides, E = 0
Therefore, E total = 0

Section 24.1
Gauss’s Law – General
A positive point charge, q, is located at
the center of a sphere of radius r.
The magnitude of the electric field
everywhere on the surface of the
sphere is
E = keq / r2

 EdA


 dA  EAsphere
  E
q
  ke 4πr 2

r2 q
1 
ke  ε0
4πε0

Section 24.2
Gauss’s Law – General, cont.
The field lines are directed radially outward and are perpendicular to the
surface at every
 point.
E   E  dA  E  dA
This will be the net flux through the gaussian surface, the sphere of
radius r.
We know E = keq/r2 and Asphere = 4πr2,
q
 E  4πkeq 
εo

Section 24.2
Gauss’s Law – General, notes
The net flux through any closed surface surrounding a point charge, q, is
given by q/εo and is independent of the shape of that surface.
The net electric flux through a closed surface that surrounds no charge is
zero.
Since the electric field due to many charges is the vector sum of the
electric fields produced by the individual charges, the flux through any
closed surface can be expressed as
    
 
 E  dA   E1  E2   dA

Section 24.2
Gaussian Surface, Example
Closed surfaces of various shapes can
surround the charge.
 Only S1 is spherical
Verifies the net flux through any closed
surface surrounding a point charge q is
given by q/o and is independent of the
shape of the surface.

Section 24.2
Gaussian Surface, Example 2
The charge is outside the closed
surface with an arbitrary shape.
Any field line entering the surface
leaves at another point.
Verifies the electric flux through a
closed surface that surrounds no
charge is zero.

Section 24.2
Gauss’s Law – Final
  q
The mathematical form of Gauss’s law states 
E   E  d A  in
εo

 qin is the net charge inside the surface.


E represents the electric field at any point on the surface.

 E is the total electric field and may have contributions from
charges both inside and outside of the surface.

1
ε0 
4πke

Section 24.2
 E  15Nm2 /C
qin
 E  15Nm /C=
2

ε0
qin  1.33  10 10 C
 E  0

q
 hemisphere   hemisphere   flat  0
2ε0
q
 flat 
2ε0
Applying Gauss’s Law

Although Gauss’s law can, in theory, be solved to find E for any charge
configuration, in practice it is limited to symmetric situations (3 cases)

To use Gauss’s law, you want to choose a gaussian surface over which
the surface integral can be simplified and the electric field determined.
Take advantage of symmetry.
Remember, the gaussian surface is a surface you choose, it does not
have to coincide with a real surface.

Section 24.3
 P

Q
 What kind of surface we can choose so that
E is constant on this surface or part of this
surface?

We can find a Gaussian surface S over

which the electric field has constant
magnitude.
 Conditions for a Gaussian Surface
Try to choose a surface that satisfies one or more of these conditions:
 The value of the electric field can be argued from symmetry to be
constant over the surface / part of the surface.
 
 The dot product of E  dA can be expressed as a simple algebraic
product EdA because E and dA are parallel.
 
 The dot product is 0 because E and dA are perpendicular.
 The field is zero over the portion of the surface.
If the charge distribution does not have sufficient symmetry such that a
gaussian surface that satisfies these conditions can be found, Gauss’
law is not useful for determining the electric field for that charge
distribution.

Section 24.3
Field Due to a Spherically Symmetric Charge Distribution
Select a sphere as the gaussian
surface.
For r >a

  qin

E   E  d A  EdA 
εo
qin
 dA  εo
 E
dA  4πr 2

Q Q
E  k e 2
4πεo r 2 r

Section 24.3
Spherically Symmetric, cont.
Select a sphere as the gaussian
surface, r < a.
qin < Q
qin = (4/3πr3) with =Q/ (4/3πa3)

  qin

E   E  d A  EdA 
εo
qin Q
E 2
 k e 3
r
4πεo r a

Section 24.3
Spherically Symmetric Distribution, final
Inside the sphere, E varies linearly with r
 E → 0 as r → 0
The field outside the sphere is equivalent
to that of a point charge located at the
center of the sphere.

Section 24.3
 qin
E 
εo
E
 E _ side 
6
Field at a Distance from a Line of Charge (Density )
Select a cylindrical charge distribution .
 The cylinder has a radius of r and a
length of ℓ.

E is constant in magnitude and
perpendicular to the surface at every
point on the curved part of the surface.
Use Gauss’s law to find the field.
  qin 1

E   E  d A   EdA 
εo
ke 
4πε0
λ
E  2πr   
εo
λ λ
E  2ke
2πεo r r
Section 24.3
 qin qin  ρV
E 
εo a Voutside  πR 2h
 total  E.2πr .h
Vinside  πr 2 h
ρr
Einside 
2ε0
ρR 2 h
Eoutside 
2ε0 r

λ
E
2πεo r
Field at a point (inside and outside) of
a solid charged cylinder (Density )
Field Due to a Line of Charge, cont.
The end view confirms the field is
perpendicular to the curved surface.
The field through the ends of the
cylinder is 0 since the field is parallel to
these surfaces.

Section 24.3
Field Due to a Plane of Charge

E must be perpendicular to the
plane and must have the same
magnitude at all points equidistant
from the plane.
Choose a small cylinder whose axis
is perpendicular to the plane for the
gaussian surface.

E is parallel to the curved surface
and there is no contribution to the
surface area from this curved part
of the cylinder.
The flux through each end of the
cylinder is EA and so the total flux
is 2EA.
Section 24.3
Field Due to a Plane of Charge, final
The total charge in the surface is σA.
Applying Gauss’s law:
σA σ
 E  2E A  and E 
εo 2εo
Note, this does not depend on r.
Therefore, the field is uniform everywhere.

Section 24.3
Properties of a Conductor in Electrostatic Equilibrium
When there is no net motion of charge within a conductor, the conductor
is said to be in electrostatic equilibrium.
The electric field is zero everywhere inside the conductor.
 Whether the conductor is solid or hollow
If the conductor is isolated and carries a charge, the charge resides on
its surface.
The electric field at a point just outside a charged conductor is
perpendicular to the surface and has a magnitude of σ/εo.
  is the surface charge density at that point.
On an irregularly shaped conductor, the surface charge density is
greatest at locations where the radius of curvature is the smallest.

Section 24.4
Property 1: Fieldinside = 0

Consider a conducting slab in an

external field.
If the field inside the conductor
were not zero, free electrons in the
conductor would experience an
electrical force.
These electrons would accelerate.
These electrons would not be in
equilibrium.
Therefore, there cannot be a field
inside the conductor.

Section 24.4
Property 1: Fieldinside = 0, cont.
Before the external field is applied, free electrons are distributed
throughout the conductor.
When the external field is applied, the electrons redistribute until the
magnitude of the internal field equals the magnitude of the external field.
There is a net field of zero inside the conductor.
This redistribution takes about 10-16 s and can be considered
instantaneous.
If the conductor is hollow, the electric field inside the conductor is also
zero.
 Either the points in the conductor or in the cavity within the conductor
can be considered.

Section 24.4
Property 2: Charge Resides on the Surface
Choose a gaussian surface inside but
close to the actual surface.
The electric field inside is zero (property
1).
There is no net flux through the
gaussian surface.
Because the gaussian surface can be
as close to the actual surface as
desired, there can be no charge inside
the surface.

Section 24.4
Property 2: Charge Resides on the Surface, cont.
Since no net charge can be inside the surface, any net charge must
reside on the surface.
Gauss’s law does not indicate the distribution of these charges, only that
it must be on the surface of the conductor.

Section 24.4
Property 3: Field’s Magnitude and Direction

Choose a cylinder as the gaussian

surface.
The field must be perpendicular to
the surface.
 If there were a parallel
component to E , charges
would experience a force and
accelerate along the surface
and it would not be in
equilibrium.

Section 24.4
Property 3: Field’s Magnitude and Direction, cont.
The net flux through the gaussian surface is through only the flat face
outside the conductor.
 The field here is perpendicular to the surface.
Applying Gauss’s law

σA σ
 E  EA  and E 
εo εo

Section 24.4
Sphere and Shell Example
Conceptualize
 Similar to the sphere example
 Now a charged sphere is
surrounded by a shell
 Note charges
Categorize
 System has spherical symmetry
 Gauss’ Law can be applied

Section 24.4
Sphere and Shell Example, cont.
Analyze
 Construct a Gaussian sphere
between the surface of the solid
sphere and the inner surface of the
shell.
 Region 2
 a<r<b
 Charge inside the surface is +Q

 The electric field lines must be

directed radially outward and be
constant in magnitude on the
Gaussian surface.

Section 24.4
Sphere and Shell Example, 3
Analyze, cont.
 The electric field for each area can be calculated.
Q
E1  ke 3
r (for r  a )
a
Q
E2  ke 2 (for a  r  b )
r
E3  0 (for b  r  c )
Q
E 4  k e (for r  c )
r2

Section 24.4
Sphere and Shell Example
Finalize
 Check the net charge.
 Think about other possible combinations.
 What if the sphere were conducting instead of insulating?

Section 24.4
a) 15 Nm2/C

b) 1.33x10-10 C

c) No. the
direction of all
electric field is
not same
  Q1 
E1  ke 3 r1
 8a
 Q2 
E2  ke 3 r2
 a
r2



r1


Etotal


 E1
E2
 Q
-Q
5 10 15
Q
4 8 12 16
Chapter 25
Electric Potential
25.1 Electrical Potential Energy
When a test charge is placed in an electric field, it experiences a force.
 
 Fe  qoE
 The force is conservative.
If the test charge is moved in the field by some external agent, the work
done
 by the field is the negative of the work done by the external agent.
ds
is an infinitesimal displacement vector that is oriented tangent to a
path through space.
 The path may be straight or curved and the integral performed along
this path is called either a path integral or a line integral.

Section 25.1
Electric Potential Energy, cont
The work done within the charge-field system by the electric field on the
charge is    
F  ds  qoE  ds

As this work is done by the field, the

 potential energy of the charge-field
system is changed by ΔU = qoE  ds
For a finite displacement of the charge from A to B, the change in
potential energy of the system is
B  
U  UB  U A  qo  E  ds
A

Because the force is conservative, the line integral does not depend on
the path taken by the charge.

Section 25.1
Electric Potential

The potential energy per unit charge, U/qo, is the electric potential.
 The potential is characteristic of the field only.
 The potential energy is characteristic of the charge-field system.
 The potential is independent of the value of qo.
 The potential has a value at every point in an electric field.
The electric potential is
U
V
qo

Section 25.1
Electric Potential, cont.
The potential is a scalar quantity.
 Since energy is a scalar
As a charged particle moves in an electric field, it will experience a
change in potential.
U B  
V     E  ds
qo A

The infinitesimal displacement is interpreted as the displacement

between two points in space rather than the displacement of a point
charge.

Section 25.1
Electric Potential, final
The difference in potential is the meaningful quantity.
We often take the value of the potential to be zero at some convenient
point in the field.
Electric potential is a scalar characteristic of an electric field,
independent of any charges that may be placed in the field.
The potential difference between two points exists solely because of a
source charge and depends on the source charge distribution.
 For a potential energy to exist, there must be a system of two or
more charges.
 The potential energy belongs to the system and changes only if a
charge is moved relative to the rest of the system.

Section 25.1
Work and Electric Potential
Assume a charge moves in an electric field without any change in its
kinetic energy.
The work performed on the charge is
W = ΔU = q ΔV
Units:1 V ≡ 1 J/C
 V is a volt.
 It takes one joule of work to move a 1-coulomb charge through a
potential difference of 1 volt.
In addition, 1 N/C = 1 V/m
 This indicates we can interpret the electric field as a measure of the
rate of change of the electric potential with respect to position.

Section 25.1
Voltage
Electric potential is described by many terms.
The most common term is voltage.
A voltage applied to a device or across a device is the same as the
potential difference across the device.
 The voltage is not something that moves through a device.

Section 25.1
25.2 Potential Difference in a Uniform Field
The equations for electric potential between two points A and B can be
simplified if the electric field is uniform:
B   B
VB  VA  V    E  ds  E  ds  Ed
A A

The displacement points from A to B and is parallel to the field lines.

The negative sign indicates that the electric potential at point B is lower
than at point A.
 Electric field lines always point in the direction of decreasing electric
potential.

Section 25.2
Energy and the Direction of Electric Field
When the electric field is directed
downward, point B is at a lower
potential than point A.
When a positive test charge moves
from A to B, the charge-field system
loses potential energy.
Electric field lines always point in the
direction of decreasing electric
potential.

Section 25.2
More About Directions
A system consisting of a positive charge and an electric field loses
electric potential energy when the charge moves in the direction of the
field.
 An electric field does work on a positive charge when the charge
moves in the direction of the electric field.
The charged particle gains kinetic energy and the potential energy of the
charge-field system decreases by an equal amount.
 Another example of Conservation of Energy

Section 25.2
Directions, cont.
If qo is negative, then ΔU is positive.
A system consisting of a negative charge and an electric field gains
potential energy when the charge moves in the direction of the field.
 In order for a negative charge to move in the direction of the field, an
external agent must do positive work on the charge.

Section 25.2
Equipotentials
Point B is at a lower potential than point
A.
Points B and C are at the same
potential.
 All points in a plane perpendicular
to a uniform electric field are at the
same electric potential.
The name equipotential surface is
given to any surface consisting of a
continuous distribution of points having
the same electric potential.

Section 25.2
Charged Particle in a Uniform Field, Example
A positive charge is released from rest
and moves in the direction of the
electric field.
The change in potential is negative.
The change in potential energy is
negative.
The force and acceleration are in the
direction of the field.
Conservation of Energy can be used to
find its speed.

Section 25.2
25.3 Potential and Point Charges
An isolated positive point charge
produces a field directed radially
outward.
The potential difference between points
A and B will be
 1 1
VB  VA  k eq   
 rB rA 

Section 25.3
Potential and Point Charges, cont.
The electric potential is independent of the path between points A and B.
It is customary to choose a reference potential of V = 0 at rA = ∞.
Then the potential due to a point charge at some point r is
q
V  ke
r

Section 25.3
Electric Potential with Multiple Charges
The electric potential due to several point charges is the sum of the
potentials due to each individual charge.
 This is another example of the superposition principle.
 The sum is the algebraic sum
qi
V  ke 
i ri
 V = 0 at r = ∞

Section 25.3
Potential Energy of Multiple Charges

q1q2
The potential energy of the system is U  ke .
r12

If the two charges are the same sign, U is positive and work must be done to
bring the charges together.
If the two charges have opposite signs, U is negative and work is done to keep
the charges apart.

Section 25.3
U with Multiple Charges, final

If there are more than two charges,

then find U for each pair of charges
and add them.
For three charges:
 q1q2 q1q3 q2q3 
U  ke    
r
 12 r13 r23 

 The result is independent of the

order of the charges.

Section 25.3
Section 25.3
Section 25.3
25.4 Finding E From V
Assume, to start, that the field has only an x component.
dV
Ex  
dx
Similar statements would apply to the y and z components.
Equipotential surfaces must always be perpendicular to the electric field
lines passing through them.

Section 25.4
E and V for an Infinite Sheet of Charge

The equipotential lines are the

dashed blue lines.
The electric field lines are the
brown lines.
The equipotential lines are
everywhere perpendicular to the
field lines.

Section 25.4
E and V for a Point Charge

The equipotential lines are the

dashed blue lines.
The electric field lines are the
brown lines.
The electric field is radial.
Er = - dV / dr
The equipotential lines are
everywhere perpendicular to the
field lines.

Section 25.4
E and V for a Dipole

The equipotential lines are the

dashed blue lines.
The electric field lines are the
brown lines.
The equipotential lines are
everywhere perpendicular to the
field lines.

Section 25.4
Electric Field from Potential, General
In general, the electric potential is a function of all three dimensions.
Given V (x, y, z) you can find Ex, Ey and Ez as partial derivatives:

V V V
Ex   Ey   Ez  
x y z

Section 25.4
25.5 Electric Potential for a Continuous Charge Distribution
Method 1: The charge distribution is
known.
Consider a small charge element dq
 Treat it as a point charge.
The potential at some point due to this
charge element is

dq
dV  ke
r

Section 25.5
V for a Continuous Charge Distribution, cont.
To find the total potential, you need to integrate to include the
contributions from all the elements.
dq
V  ke 
r
 This value for V uses the reference of V = 0 when P is infinitely far
away from the charge distributions.
V for a Continuous Charge Distribution, final
If the electric field is already known from other considerations, the
potential can be calculated using the original approach:
B  
V    E  ds
A

 If the charge distribution has sufficient symmetry, first find the field
from Gauss’ Law and then find the potential difference between any
two points,
 Choose V = 0 at some convenient point

Section 25.5
V for a Uniformly Charged Ring

P is located on the perpendicular

central axis of the uniformly
charged ring .
The symmetry of the situation
means that all the charges on the
ring are the same distance from
point P.
 The ring has a radius a and a
total charge Q.
The potential and the field are
given by dq keQ
V  ke  
r a2  x 2
ke x
Ex  Q
a 
3/2
2
x 2
Section 25.5
V for a Uniformly Charged Disk
The ring has a radius R and surface
charge density of σ.
P is along the perpendicular central
axis of the disk.
P is on the cental axis of the disk,
symmetry indicates that all points in a
given ring are the same distance from
P.
The potential and the field are given by
 
 
1
V  2πke σ  R 2  x 2 2
 x
 
 
x
E x  2πke σ 1  

  
1/2
R2  x2
 
Section 25.5
V for a Finite Line of Charge

A rod of line ℓ has a total charge of

Q and a linear charge density of λ.
 There is no symmetry to use,
but the geometry is simple.

keQ    a 2   2 
V ln  
  a 
 

Section 25.5
V Due to a Charged Conductor
Consider two points on the surface of
the charged conductor as shown.

E is always perpendicular
 to the
displacement ds .
 
Therefore, E  ds  0
Therefore, the potential difference
between A and B is also zero.

Section 25.6
V Due to a Charged Conductor, cont.
V is constant everywhere on the surface of a charged conductor in
equilibrium.
 ΔV = 0 between any two points on the surface
The surface of any charged conductor in electrostatic equilibrium is an
equipotential surface.
Every point on the surface of a charge conductor in equilibrium is at the
same electric potential.
Because the electric field is zero inside the conductor, we conclude that
the electric potential is constant everywhere inside the conductor and
equal to the value at the surface.

Section 25.6
Cavity in a Conductor
Assume an irregularly shaped cavity is
inside a conductor.
Assume no charges are inside the
cavity.
The electric field inside the conductor
must be zero.

Section 25.6
Cavity in a Conductor, cont
The electric field inside does not depend on the charge distribution on
the outside surface of the conductor.
For all paths between A and B,
B  
VB  VA    E  ds  0
A

A cavity surrounded by conducting walls is a field-free region as long as

no charges are inside the cavity.

Section 25.6
Chapter 26
Capacitance and Dielectrics
Circuits and Circuit Elements
Electric circuits are the basis for the vast majority of the devices used in
society.
Circuit elements can be connected with wires to form electric circuits.
Capacitors are one circuit element.
 Others will be introduced in other chapters

Introduction
Capacitors
Capacitors are devices that store electric charge.
Examples of where capacitors are used include:
 radio receivers
 filters in power supplies
 to eliminate sparking in automobile ignition systems
 energy-storing devices in electronic flashes

Introduction
26.1 Makeup of a Capacitor

A capacitor consists of two

conductors.
 These conductors are called
plates.
 When the conductor is
charged, the plates carry
charges of equal magnitude
and opposite directions.
A potential difference exists
between the plates due to the
charge.

Section 26.1
Definition of Capacitance
The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the
charge on either conductor to the potential difference between the conductors.

Q
C
V
The SI unit of capacitance is the farad (F).
The farad is a large unit, typically you will see microfarads (mF) and picofarads (pF).
Capacitance will always be a positive quantity
The capacitance of a given capacitor is constant.
The capacitance is a measure of the capacitor’s ability to store charge .
 The capacitance of a capacitor is the amount of charge the capacitor can store
per unit of potential difference.

Section 26.1
Parallel Plate Capacitor
Each plate is connected to a terminal of
the battery.
 The battery is a source of potential
difference.
If the capacitor is initially uncharged,
the battery establishes an electric field
in the connecting wires.

Section 26.1
Parallel Plate Capacitor, cont
This field applies a force on electrons in the wire just outside of the
plates.
The force causes the electrons to move onto the negative plate.
This continues until equilibrium is achieved.
 The plate, the wire and the terminal are all at the same potential.
At this point, there is no field present in the wire and the movement of the
electrons ceases.
The plate is now negatively charged.
A similar process occurs at the other plate, electrons moving away from
the plate and leaving it positively charged.
In its final configuration, the potential difference across the capacitor
plates is the same as that between the terminals of the battery.
Section 26.1
26.2 Capacitance – Isolated Sphere
Assume a spherical charged conductor with radius a.
The sphere will have the same capacitance as it would if there were a conducting
sphere of infinite radius, concentric with the original sphere.
Assume V = 0 for the infinitely large shell
Q Q R
C    4πεo a
V keQ / a ke
Note, this is independent of the charge on the sphere and its potential.

Section 26.2
Capacitance – Parallel Plates
The charge density on the plates is σ = Q/A.
 A is the area of each plate, the area of each plate is equal
 Q is the charge on each plate, equal with opposite signs
The electric field is uniform between the plates and zero elsewhere.
The capacitance is proportional to the area of its plates and inversely
proportional to the distance between the plates.
Q Q Q ε A
C    o
V Ed Qd / εo A d

Section 26.2
Capacitance of a Cylindrical Capacitor

V = -2keln (b/a)
 = Q/l
The capacitance is
Q 
C 
V 2ke ln  b / a 

Section 26.2
Capacitance of a Spherical Capacitor

The potential difference will be

 1 1
V  keQ   
b a

The capacitance will be

Q ab
C 
V k e  b  a 

Section 26.2
Capacitors in Parallel
When capacitors are first connected in
the circuit, electrons are transferred
from the left plates through the battery
to the right plate, leaving the left plate
positively charged and the right plate
negatively charged.

Section 26.3
Capacitors in Parallel, 2
The flow of charges ceases when the voltage across the capacitors
equals that of the battery.
The potential difference across the capacitors is the same.
 And each is equal to the voltage of the battery
 V1 = V2 = V
 V is the battery terminal voltage
The capacitors reach their maximum charge when the flow of charge
ceases.
The total charge is equal to the sum of the charges on the capacitors.
 Qtot = Q1 + Q2

Section 26.3
Capacitors in Parallel, 3

The capacitors can be replaced

with one capacitor with a
capacitance of Ceq.
 The equivalent capacitor must
have exactly the same external
effect on the circuit as the
original capacitors.

Section 26.3
Capacitors in Parallel, final
Ceq = C1 + C2 + C3 + …
The equivalent capacitance of a parallel combination of capacitors is
greater than any of the individual capacitors.
 Essentially, the areas are combined

Section 26.3
Capacitors in Series
When a battery is connected to the
circuit, electrons are transferred from
the left plate of C1 to the right plate of
C2 through the battery.

As this negative charge accumulates

on the right plate of C2, an equivalent
amount of negative charge is
removed from the left plate of C2,
leaving it with an excess positive
charge.
All of the right plates gain charges of
–Q and all the left plates have
charges of +Q.

Section 26.3
Capacitors in Series, cont.

An equivalent capacitor can be

found that performs the same
function as the series combination.
The charges are all the same.
Q1 = Q2 = Q

Section 26.3
Capacitors in Series, final
The potential differences add up to the battery voltage.
ΔVtot = V1 + V2 + …
The equivalent capacitance is
1 1 1 1
   
Ceq C1 C2 C3

The equivalent capacitance of a series combination is always less than

any individual capacitor in the combination.

Section 26.3
Energy Stored in a Capacitor
Assume the capacitor is being charged
and, at some point, has a charge q on
it.
The work needed to transfer a charge
from one plate to the other is
q
dW  Vdq  dq
C
The work required is the area of the tan
rectangle.
The total work required is
Q q Q2
W  dq 
0 C 2C

Section 26.4
Energy, cont
The work done in charging the capacitor appears as electric potential
energy U:
Q2 1 1
U  QV  C (V )2
2C 2 2

This applies to a capacitor of any geometry.

The energy stored increases as the charge increases and as the
potential difference increases.
In practice, there is a maximum voltage before discharge occurs
between the plates.

Section 26.4
Energy, final
The energy can be considered to be stored in the electric field .
For a parallel-plate capacitor, the energy can be expressed in terms of
the field as U = ½ (εoAd)E2.
It can also be expressed in terms of the energy density (energy per unit
volume)

uE = ½ oE2.

Section 26.4
26.5 Capacitors with Dielectrics
A dielectric is a nonconducting material that, when placed between the
plates of a capacitor, increases the capacitance.
 Dielectrics include rubber, glass, and waxed paper
With a dielectric, the capacitance becomes C = κCo.
 The capacitance increases by the factor κ when the dielectric
completely fills the region between the plates.
 κ is the dielectric constant of the material.
If the capacitor remains connected to a battery, the voltage across the
capacitor necessarily remains the same.
If the capacitor is disconnected from the battery, the capacitor is an
isolated system and the charge remains the same.

Section 26.5
Dielectrics, cont
For a parallel-plate capacitor, C = κ (εoA) / d
In theory, d could be made very small to create a very large capacitance.
In practice, there is a limit to d.
 d is limited by the electric discharge that could occur though the
dielectric medium separating the plates.
For a given d, the maximum voltage that can be applied to a capacitor
without causing a discharge depends on the dielectric strength of the
material.

Section 26.5
Dielectrics, final
Dielectrics provide the following advantages:
 Increase in capacitance
 Increase the maximum operating voltage
 Possible mechanical support between the plates
 This allows the plates to be close together without touching.
 This decreases d and increases C.

Section 26.5
Some Dielectric Constants and Dielectric Strengths

Section 26.5
26.6 Electric Dipole
An electric dipole consists of two
charges of equal magnitude and
opposite signs.
The charges are separated by 2a.

The electric dipole moment (p) is
directed along the line joining the
charges from –q to +q.

Section 26.6
Electric Dipole, 2
The electric dipole moment has a magnitude of p ≡ 2aq.

Assume the dipole is placed in a uniform external field, E

 E is external to the dipole; it is not the field produced by the
dipole
Assume the dipole makes an angle θ with the field

Section 26.6
Electric Dipole, 3
Each charge has a force of F = Eq
acting on it.
The net force on the dipole is zero.
The forces produce a net torque on the
dipole.
The dipole is a rigid object under a net
torque.

Section 26.6
Electric Dipole, final
The magnitude of the torque is:
= 2Fa sin θpE sin θ
The torque can also be expressed as the cross product of the moment
and the field:
  
   pE
The system of the dipole and the external electric field can be modeled
as an isolated system for energy.
The potential energy can be expressed as a function of the orientation of
the dipole with the field:
Uf – Ui = pE(cos θi – cos θfU = - pE cos θ
This expression
  can be written as a dot product.
U  p E

Section 26.6
Polar vs. Nonpolar Molecules
Molecules are said to be polarized when a separation exists between
the average position of the negative charges and the average position of
the positive charges.
Polar molecules are those in which this condition is always present.
Molecules without a permanent polarization are called nonpolar
molecules.

Section 26.6
Polar Molecules and Dipoles
The average positions of the positive and negative charges act as point
charges.
Therefore, polar molecules can be modeled as electric dipoles.

Section 26.6
Induced Polarization

A linear symmetric molecule has no

permanent polarization (a).
Polarization can be induced by
placing the molecule in an electric
field (b).
Induced polarization is the effect
that predominates in most materials
used as dielectrics in capacitors.

Section 26.6
Dielectrics – An Atomic View
The molecules that make up the
dielectric are modeled as dipoles.
The molecules are randomly oriented in
the absence of an electric field.

Section 26.7
Dielectrics – An Atomic View, 2
An external electric field is applied.
This produces a torque on the
molecules.

The molecules partially align with the

electric field.
 The degree of alignment depends
on temperature and the magnitude
of the field.
 In general, the alignment increases with
decreasing temperature and with
increasing electric field.

Section 26.7
Dielectrics – An Atomic View, 4
If the molecules of the dielectric are nonpolar molecules, the electric field
produces some charge separation.
This produces an induced dipole moment.
The effect is then the same as if the molecules were polar.

Section 26.7
Dielectrics – An Atomic View, final
An external field can polarize the
dielectric whether the molecules are
polar or nonpolar.
The charged edges of the dielectric act
as a second pair of plates producing an
induced electric field in the direction
opposite the original electric field.

Section 26.7
Induced Charge and Field

The electric field due to the plates

is directed to the right and it
polarizes the dielectric.
The net effect on the dielectric is an
induced surface charge that results
in an induced electric field.
If the dielectric were replaced with
a conductor, the net field between
the plates would be zero.

Section 26.7
Chapter 27
Current and Resistance
Electric Current
Most practical applications of electricity deal with electric currents.
 The electric charges move through some region of space.
The resistor is a new element added to circuits.
Energy can be transferred to a device in an electric circuit.
The energy transfer mechanism is electrical transmission, TET.

Introduction
Electric Current
Electric current is the rate of flow of charge through some region of
space.
The SI unit of current is the ampere (A).
 1A=1C/s
The symbol for electric current is I.

Section 27.1
Average Electric Current

Assume charges are moving

perpendicular to a surface of area
A.
If ΔQ is the amount of charge that
passes through A in time Δt, then
the average current is
Q
I avg 
t

Section 27.1
Instantaneous Electric Current
If the rate at which the charge flows varies with time, the instantaneous
current, I, is defined as the differential limit of average current as Δt→0.
dQ
I
dt

Section 27.1
Current and Drift Speed
Charged particles move through a
cylindrical conductor of cross-sectional
area A.
n is the number of mobile charge
carriers per unit volume.
nAΔx is the total number of charge
carriers in a segment.

Section 27.1
Current and Drift Speed, cont
The total charge is the number of carriers times the charge per carrier, q.
 ΔQ = (nAΔx)q
Assume the carriers move with a velocity parallel to the axis of the
cylinder such that they experience a displacement in the x-direction.
If vd is the speed at which the carriers move, then
 vd = Δx / Δt and x = vd t
Rewritten: ΔQ = (nAvd Δt)q
Finally, current, Iave = ΔQ/Δt = nqvdA
vd is an average speed called the drift speed.

Section 27.1
Charge Carrier Motion in a Conductor
When a potential difference is applied
across the conductor, an electric field is
set up in the conductor which exerts an
electric force on the electrons.
The motion of the electrons is no longer
random.
The zigzag black lines represents the
motion of a charge carrier in a
conductor in the presence of an electric
field.
 The net drift speed is small.
The sharp changes in direction are due
to collisions.
The net motion of electrons is opposite
the direction of the electric field.

Section 27.1
Motion of Charge Carriers, cont.
In the presence of an electric field, in spite of all the collisions, the
charge carriers slowly move along the conductor with a drift velocity,

vd
The electric field exerts forces on the conduction electrons in the wire.
These forces cause the electrons to move in the wire and create a
current.

Section 27.1
Drift Velocity, Example
Assume a copper wire, with one free electron per atom contributed to the
current.
The drift velocity for a 12-gauge copper wire carrying a current of 10.0 A
is
2.23 x 10-4 m/s
 This is a typical order of magnitude for drift velocities.

Section 27.1
Current Density
J is the current density of a conductor.
It is defined as the current per unit area.
 J ≡ I / A = nqvd
 This expression is valid only if the current density is uniform and A is
perpendicular to the direction of the current.
J has SI units of A/m2
The current density is in the direction of the positive charge carriers.

Section 27.2
Conductivity
A current density and an electric field are established in a conductor
whenever a potential difference is maintained across the conductor.
For some materials, the current density is directly proportional to the
field.
The constant of proportionality, σ, is called the conductivity of the
conductor.

Section 27.2
Ohm’s Law
Ohm’s law states that for many materials, the ratio of the current density
to the electric field is a constant σ that is independent of the electric field
producing the current.
 Most metals obey Ohm’s law
 Mathematically, J = σ E
 Materials that obey Ohm’s law are said to be ohmic
 Not all materials follow Ohm’s law
 Materials that do not obey Ohm’s law are said to be nonohmic.
Ohm’s law is not a fundamental law of nature.
Ohm’s law is an empirical relationship valid only for certain materials.

Section 27.2
Resistance
In a conductor, the voltage applied across the ends of the conductor is
proportional to the current through the conductor.
The constant of proportionality is called the resistance of the conductor.
V
R
I

SI units of resistance are ohms (Ω).

 1Ω=1V/A
Resistance in a circuit arises due to collisions between the electrons
carrying the current with the fixed atoms inside the conductor.

Section 27.2
Resistor Color Code Example

Red (=2) and blue (=6) give the first two digits: 26
Green (=5) gives the power of ten in the multiplier: 105
The value of the resistor then is 26 x 105 Ω (or 2.6 MΩ)
The tolerance is 10% (silver = 10%) or 2.6 x 105 Ω

Section 27.2
Resistivity
The inverse of the conductivity is the resistivity:
 ρ=1/σ
Resistivity has SI units of ohm-meters (Ω . m)
Resistance is also related to resistivity:

Rρ
A

Section 27.2
Resistivity Values

Section 27.2
Resistance and Resistivity, Summary
Every ohmic material has a characteristic resistivity that depends on the
properties of the material and on temperature.
 Resistivity is a property of substances.
The resistance of a material depends on its geometry and its resistivity.
 Resistance is a property of an object.
An ideal conductor would have zero resistivity.
An ideal insulator would have infinite resistivity.

Section 27.2
Ohmic Material, Graph
An ohmic device
The resistance is constant over a wide
range of voltages.
The relationship between current and
voltage is linear.
The slope is related to the resistance.

Section 27.2
Nonohmic Material, Graph
Nonohmic materials are those whose
resistance changes with voltage or
current.
The current-voltage relationship is
nonlinear.
A junction diode is a common example
of a nonohmic device.

Section 27.2
Resistance of a Cable, Example

Assume the silicon between the

conductors to be concentric
elements of thickness dr.
The resistance of the hollow
cylinder of silicon is
ρ
dR  dr
2πrL

Section 27.2
Resistance of a Cable, Example, cont.
The total resistance across the entire thickness is
b ρ b
R   dR  ln  
a 2πL  a 

This is the radial resistance of the cable.

The calculated value is fairly high, which is desirable since you want the
current to flow along the cable and not radially out of it.

Section 27.2
Electrical Conduction – A Model
Treat a conductor as a regular array of atoms plus a collection of free
electrons.
 The free electrons are often called conduction electrons.
 These electrons become free when the atoms are bound in the solid.
In the absence of an electric field, the motion of the conduction electrons
is random.
 Their speed is on the order of 106 m/s.

Section 27.3
Conduction Model, 2
When an electric field is applied, the conduction electrons are given a
drift velocity.
Assumptions:
 The electron’s motion after a collision is independent of its motion
before the collision.
 The excess energy acquired by the electrons in the electric field is
transferred to the atoms of the conductor when the electrons and
atoms collide.
 This causes the temperature of the conductor to increase.

Section 27.3
Conduction Model – Calculating the Drift Velocity
The force experienced by an electron is
 
F  qE

From Newton’s Second Law, the acceleration is

 

a
F

qE
m me
Applying a motion equation

     qE
v f = vi + at or v f = vi + t
me

 Since the initial velocities are random, their average value is zero.

Section 27.3
Conduction Model, 4
Let  be the average time interval between successive collisions.
The average value
 of the final velocity is the drift velocity.
  qE
vf ,avg  vd  
me

This is also related to the current density: J = nqvd = (nq2E / me)

 n is the number of charge carriers per unit volume.

Section 27.3
Conduction Model, final
Using Ohm’s Law, expressions for the conductivity and resistivity of a
conductor can be found:
nq 2 1 m
    2e
me  nq 

Note, according to this classical model, the conductivity and the

resistivity do not depend on the strength of the field.
 This feature is characteristic of a conductor obeying Ohm’s Law.

Section 27.3
Resistance and Temperature

Over a limited temperature range, the resistivity of a conductor

varies approximately linearly with the temperature.
ρ  ρ o [1  α ( T  T o )]
 ρo is the resistivity at some reference temperature To
 To is usually taken to be 20° C
 α is the temperature coefficient of resistivity
 SI units of α are oC-1
The temperature coefficient of resistivity can be expressed as
1 

 o T

Section 27.4
Temperature Variation of Resistance
Since the resistance of a conductor with uniform cross sectional area is
proportional to the resistivity, you can find the effect of temperature on
resistance.
R = Ro[1 + α(T - To)]
Use of this property enables precise temperature measurements through
careful monitoring of the resistance of a probe made from a particular
material.

Section 27.4
Resistivity and Temperature, Graphical View

For some metals, the resistivity is

nearly proportional to the
temperature.
A nonlinear region always exists at
very low temperatures.
The resistivity usually reaches
some finite value as the
temperature approaches absolute
zero.

Section 27.4
Residual Resistivity
The residual resistivity near absolute zero is caused primarily by the
collisions of electrons with impurities and imperfections in the metal.
High temperature resistivity is predominantly characterized by
collisions between the electrons and the metal atoms.
 This is the linear range on the graph.

Section 27.4
Semiconductors
Semiconductors are materials that exhibit a decrease in resistivity with
an increase in temperature.
α is negative
There is an increase in the density of charge carriers at higher
temperatures.

Section 27.4
Superconductors
A class of materials and compounds
whose resistances fall to virtually zero
below a certain temperature, TC.
 TC is called the critical
temperature.
The graph is the same as a normal
metal above TC, but suddenly drops to
zero at TC.

Section 27.5
Superconductors, cont
The value of TC is sensitive to:
 chemical composition
 pressure
 molecular structure
Once a current is set up in a superconductor, it persists without any
applied voltage.
 Since R = 0

Section 27.5
Superconductor Application

An important application of
superconductors is a
superconducting magnet.
The magnitude of the magnetic
field is about 10 times greater than
a normal electromagnet.
These magnets are being
considered as a means of storing
energy.
Are currently used in MRI units

Section 27.5
Electrical Power
Assume a circuit as shown
The entire circuit is the system.
As a charge moves from a to b, the
electric potential energy of the system
increases by QV.
 The chemical energy in the battery
must decrease by this same
amount.
This electric potential energy is
transformed into internal energy in the
resistor.
 Corresponds to increased
vibrational motion of the atoms in
the resistor

Section 27.6
Electric Power, 2
The resistor is normally in contact with the air, so its increased
temperature will result in a transfer of energy by heat into the air.
The resistor also emits thermal radiation.
After some time interval, the resistor reaches a constant temperature.
 The input of energy from the battery is balanced by the output of
energy by heat and radiation.
The rate at which the system’s potential energy decreases as the charge
passes through the resistor is equal to the rate at which the system gains
internal energy in the resistor.
The power is the rate at which the energy is delivered to the resistor.

Section 27.6
Electric Power, final
The power is given by the equation P = I ΔV.
Applying Ohm’s Law, alternative expressions can be found:
 V 
2

P  I  V  I2 R 
R

Units: I is in A, R is in Ω, ΔV is in V, and P is in W

Section 27.6
Some Final Notes About Current
A single electron is moving at the drift velocity in the circuit.
 It may take hours for an electron to move completely around a
circuit.
The current is the same everywhere in the circuit.
 Current is not “used up” anywhere in the circuit
The charges flow in the same rotational sense at all points in the circuit.
Definition of Current
Definition of Current
Definition of Current Density
Definition of Resistance
Definition of Power