Transcript lecture16

Chapter 15
Conductors (again)
What happens to a conductor when you
place it in an electric field and allow the
charges on it to attain equilibrium?
E
conductor
(Remember, charges are free to move
around on the surface of conductors.)
1) no electric field exists inside the
conductor.
What if it did?
Then an electrical force would be exerted on
the charges present in the conductor. In a
good conductor, charges are free to move
around, and will when a force is exerted on them.
If charges are moving around, we are not in
equilibrium.
2) Excess charges on an isolated
conductor are found entirely
on its surface.
The 1/r2 nature of the electrostatic repulsive
force is responsible for this one. The excess
charges are trying to get as far away from one
another as possible. It turns out, therefore, they
all end up on the surface of the conductor.
3) The electric field just outside of a
conductor must be perpendicular
to the surface of the conductor.
Again, what if this were not the case?
Then a component of the electric field would
exist along the conductor’s surface. This would
yield an electrical force along the surface. As a
good conductor, charges would move around in
the presence of the force...
4) On an irregularly shaped conductor,
charges build up near the points
(regions with smallest curvature).
--- -
E
Charges build up
here: not as much
room for them to
move apart!
-
-
-
k|q1||q2 |
|F| 
r2
k|q|
| E|  2
r
In both cases, you
are advised to THINK
about the direction!

kq 1q 2

F
r
2
r

kq
E = 2 r
r
What is r ?
To determine the direction of the r ,
simply join point A to point B. The
sign of their charges does not matter!
r
B
A
Just connect the dots! r points
from the charge creating the field
toward the charge of interest
regardless of their signs!
The notion of r is just that of a coordinate
system centered on the object of interest.
We’re probably most familiar with the
Cartesian coordinate system
z
q
r
(x,y,z)
or (r, q, f)
y
f
x
z
z
x
r
y
y
x
r
points radially outward from the
origin of the coordinate system.
When you use r (and only when you use
this formulation) the signs of the charges
become mathematically meaningful in the
formulae you apply to the problems!
If you use the book’s formulation, you
must take the absolute values of the signs
and THINK about the direction of the vector
quantities you are calculating!!!
Use whatever works for you!
Electrical Energy and Capacitance
Recall that in the presence of the gravitational
field, we defined potential energy and work
done. This work could be done by any
Newtonian force. We’ll now define similar
quantities for electrical forces. We also examine
the capacity of systems that hold charge.
A force is conservative if the
total work it does on a particle is
zero when the particle moves
exactly once around any closed
path.
The work done on a particle by a conservative
force is independent of the path a particle
takes to move from one point to another.
Work = Force through a distance
W = F|| d
For the electrostatic force...
W=qEd


F = qE
Scalar quantity!
No directions!
Electrical work is “Quite Easily Done!”
NOTE: d is the distance parallel
to the Electric field only!!!
Project the electric field vector onto
the displacement vector.
r
E
q
d
W=qEd
NOT
q E r !!!
r
q
d
E
W=qEd
As long as the electric field is uniform, this is the answer!
Recall: Potential Energy and Work are Related...
DPE = -W = -q E d
Only valid in a
uniform electric
field!
Notice that work and potential energy are
scalar quantities, NOT vectors!
(i.e. they have no directional components)
Example: speed is a scalar
velocity is a vector
A
B
A useful quantity in examining problems with charges
is the electrical potential difference.
This quantity is NOT the potential energy. Rather,
the electrical potential difference is given by
DV = Vb-Va = DPE/q
DV is the change in potential energy per unit charge
as the charge moves from point A to point B.
So, what are the
units of V?
DV = DPE/q
So...
[DV]=[DPE]/[q]
[DV] = J/C = 1 Volt
1 Joule of energy is required to
move 1 Coulomb of charge through
a potential difference of +1 Volt.
DV = Vb - Va = DPE/q
DPE = -W = -q E d
REMEMBER, d is
measured in direction
of E only!!!
NOTE: This ONLY
works for a uniform
electric field!!
DV = - q E d/q = -E d
[DV] = [E] [d]
1 Volt = (N/C) m = J/C !
DV = - E d,
Wilbur!
It’s EASY!
Does DV = -Ed
in the field around
a point charge?
+
What happens to the potential energy of a
charge as it moves in the direction of the
electric force?
+q
d
The electric field DOES work
on + q, therefore, the potential
energy must…...
E
Remember, the change in potential energy
is minus the work done by the field force.
What happens to the potential of a charge as
it moves in the direction of the electric
force?
E
+q
d
Va
Vb
E
+q
d
Va
Vb
Vb - Va = - E d
It decreases in the direction of the electric field,
REGARDLESS OF THE SIGN OF THE CHARGE!
So, how’s this work for point charges?
First, assume that the potential in the field of
a point charge is ZERO at infinity (we did
something similar for gravity, right?)
Use Calculus….
VOILA!
kq
V
r
scalar!
kq
V
r
V>0
around positive charges
V<0
around negative charges
The superposition principle
applies to potentials!
Vtot = V1 + V2 + V3 + ...
How much work do we do in bringing a
point charge (q2) from infinity to a distance
r from point charge q1?
q1
r
q2
DPE  q 2 DV  q 2 ( Vr  V )
kq 1
Vr 
r
V  0
kq 1q 2
DPE 
r
kq 1 q 2
PE 
r
PE > 0
for LIKE charges
PE < 0
for opposite charges
You could probably have guessed that
PEtot = PE1 + PE2 + PE3 + ...
+q
A
Fe
d
B
E
What happens to the potential energy of a
positive charge (+q) as it moves in the
direction of the electric field?
DPE = - q E d = - (+q) E d = - q E d
-q
A
Fe
d
B
E
What happens to the potential energy of a
negative charge (-q) as it moves in the
direction of the electric field?
DPE = - q E d = - (-q) E d = + q E d
+q
A
Fe
B
d
E
What happens to the potential of a positive
charge (+q) as it moves in the direction of the
electric field?
DV = - E d
-q
A
Fe
B
d
E
What happens to the potential of a negative
charge (-q) as it moves in the direction of the
electric field?
DV = - E d
The electrical potential ALWAYS
decreases in the direction of the
electric field! It does not depend
upon the sign of the charge.
The electrical potential energy
depends upon the sign of the charge.
It decreases in the direction of the
electrical force.
kq
V
r
What is the potential
at point p?
p
1m
q1
5m
Vtot = V1 + V2
2m
q2
(9  109 Nm2 / C2 )( 2  106 C)
V1 
(1m)
q1 = +2 mC
q2 = -5 mC
(9  109 Nm2 / C2 )(-5  106 C)
V2 
( 5m)
kq
V
r
p
1m
q1
What is the potential
at point p?
5m
2m
q1 = +2 mC
q2 = -5 mC
q2
Vtot = V1 + V2
V1 = 18,000 V
V2 = -20,125 V
Vtot = -2,125 V
W = q DV
p
1m
q1
5m
2m
q1 = +2 mC
q2 = -5 mC
q2
How much work is
required to bring a
5 mC charge from
infinity to point p?
W= (5 X 10-6 C) (-2,125 V)
= -1.1 X 10-2 J
An equipotential surface is a set of points in an
electric field which are all at the same electrical
potential. One example is the locus of points
equidistant from an isolated point charge.
Electric field lines
Equipotential surfaces
Notice that the equipotential surfaces are
perpendicular to the electric field lines
everywhere! Remember that work is only
done when a charge moves parallel to the
electric field lines. So no work is done by
the electric field as a charge moves along
an equipotential surface.
This is just like GRAVITY, right?
The potential difference between any
two points on an equipotential surface
(Vb - Va ) must be...
When in electrostatic equilibrium (i.e.,
no charges are moving around), all
points on and inside of a conductor
are at the same electrical potential!
Remember that the electric field is everywhere
perpendicular to the surface of a conductor,
and is zero inside of a conductor. So, from the
surface of a conductor throughout its interior
the electric field is 0 when the conductor is in
electrostatic equilibrium.
DV = - E d = 0
+Q
d
-Q
A common type of capacitor consists of a
pair of parallel conducting plates, one
charged positively, one charged negatively.
+Q
d
E
-Q
An electric field exists between the
plates of a capacitor. Therefore….
Capacitance is the ability to store charge.
A
d
C = eoA/d
On what could capacitance possibly depend?
Think geometrically….
C = eoA/d
eo = permittivity of free space
= 8.85 X 10-12 C2/Nm2
This quantity is related to another constant
with which we are already familiar in
electrostatics:
k = 1 / 4p eo
C = eoA/d
[C] = [eo] [A] / [d]
[C] =
C2
N m2
m2
m
=
C2
J
=
1 Farad = C / V
C
V
What happens when you connect a capacitor
to a battery?
_
+
+
_
V
C
Circuit
Diagram
+
_
+Q
V
C
-Q
The battery converts internal chemical
energy to electrical energy, pulling
electrons off the top plate of the capacitor
and pushing them onto the lower plate of
the capacitor until the capacity of the
capacitor is reached. At that point, each
plate of the capacitor holds a charge Q.
The battery maintains a potential
difference across its terminals (and hence,
the capacitor) of V.
C=Q/V
Notice this has the same
units as the quantity we
derived earlier!
1 Farad = C / V
C=Q/V
This definition of capacitance is particularly
useful since it does not require us to have
any knowledge about the geometry of the
capacitor. Using this definition, capacitance
can be determined solely from the behavior
of the electrical circuit.
+
_
V
C1
C2
So, what happens after the battery is
connected to this circuit?
Charge of Q1 = C1 V accumulates on capacitor C1.
Charge of Q2 = C2 V accumulates on capacitor C2.
+
V
_
C1
C2
We can construct an equivalent circuit with
a single capacitor...
+
_
V
Ceq
+(Q1+Q2)
- (Q1+Q2)
Ceq = (Q1 + Q2) / V = C1 + C2
Capacitors in parallel ADD.
+
_
V
+Q
-Q
C1
+Q
C2
-Q
So, what happens after the battery is
connected to this circuit?
V1 + V2 = V
Q1 = Q2 due to conservation of charge!
+
+Q -Q
C1, V1
V
_
+Q
C2, V2
-Q
We can construct an equivalent circuit with
a single capacitor...
+
_
+Q
V
Ceq
-Q
Ceq = Q / V
V1 = Q / C1
V2 = Q / C2
V = V1 + V2
+
_
+Q
V
Ceq
-Q
Ceq = Q / V
Q
Ceq =
Q / C1 + Q / C2
1
Ceq
C2
=
1
C1
+
1
Capacitors in series ADD INVERSELY.
DV = 1 Volt
+
e-
_
An electron is
accelerated
across a potential
difference of 1 Volt.
It gains kinetic
energy as it moves
across the potential,
losing potential
energy. This
particular amount
of energy is known
as...
The amount of energy 1 electron gains
when accelerated through a potential
difference of 1 Volt.
1 eV = (1.6 X 10-19 C) (1 V) = 1.6 X 10-19 J
Can you name some
examples of devices
that use the energy
stored in capacitors?
Capacitors store charge.
It takes work to put charges on capacitors.
That work becomes the potential energy of
the capacitor. So capacitors store energy.
We take advantage of this all the time!
How much work
does it take to
charge a capacitor?
W=qV
Start with uncharged plates. V = 0
+Dq
So it requires almost
no work to bring up the
first bits of charge, Dq.
-Dq
Now that our capacitor
has a charge Dq, what
is the potential difference
between the plates?
+Dq
-Dq
V=Q/C
V
As we bring up more and more
charge, V increases with Q at
the rate 1/C, so we can plot V
as a function of Q:
Slope = 1 / C
Dq
Q
The total amount of work to bring charge Q
onto an initially uncharged capacitor,
therefore, is simply the area under the curve:
V
V=Q/C
Q
2
Q=CV
1
Q
1
2
W  U  QV 
 CV
2
2C 2
U = INTERNAL ENERGY of the capacitor.
This is where the energy comes from to
power many of our cordless, rechargeable
devices…When it’s gone, we have to plug
the devices into the wall socket to recharge
the capacitors!
Just so you know...
Insulating materials between capacitor plates are
known as dielectrics. In the circuits we have dealt
with, that material is air. It could be other insulators
(glass, rubber, etc.).
Dielectric materials are characterized by a dielectric
constant k such that when placed between the plates
of a capacitor, the capacitance becomes
C = k Co
Co is the vacuum (air) capacitance.
Dielectrics, therefore, increase the charge a
capacitor can hold at a given voltage, since...
Q = C V = k Co V
k Co
We’ve already used this concept, even though
we haven’t formally introduced it.
What do you think of when you hear
the word “current?”
Electrical Current is simply the
flow of electrical charges.
charge carriers
moving charges.
can be + or -
A
The current is the number of charges flowing
through a surface A per unit time.
DQ
I
Dt
DQ
I
Dt
By convention, we say that the
direction of the current is the
direction in which the positive
charge carriers move.
Note: for most materials we examine, it’s
really the negative charge carriers that move.
Nevertheless, we say that electrons move in
a direction opposite to the electrical current.
Leftover from Ben Franklin!