Ben Gurion University of the Negev

1
Ben Gurion University of the Negev
www.bgu.ac.il/atomchip
Physics 2B for Materials and Structural
Engineering
Lecturer Daniel Rohrlich Teaching Assistants
Oren Rosenblatt, Shai Inbar
Week 4. Potential, capacitance and capacitors E
from V equipotential surfaces E in and on a
conductor capacitors and capacitance
capacitors in series and in parallel Source
Halliday, Resnick and Krane, 5th Edition, Chaps.
28, 30.
2
With 100,000 V on her body, why is this girl
smiling???
3
E from V Let U(r2) be the potential energy of a
charge q at the point r2. We found that it is
minus the work done by the electric force Fq in
bringing the charge to r2 If we divide both
sides by q, we get (on the left side) potential
instead of potential energy, and (on the right
side) the electric field instead of the electric
force
4
E from V Let V(r2) be the electric potential at
the point r2. We have just obtained it from the
electric field E Now lets see how to
obtain E from V. For r2 close to r1 we can write
r2 r1 ?r and expand V(r2) in a Taylor series
5
E from V We can write this expansion more
compactly using the del (gradient)
operator so with all the derivatives
evaluated at the point r1. We also write
6
E from V So now we can write and therefore
7
E from V The equation is a scalar equation,
but since we can vary each component of ?r
independently, it actually yields three equations
8
E from V The equation is a scalar equation,
but since we can vary each component of ?r
independently, it actually yields three
equations, i.e.
9
E from V which reduce to a single vector
equation
10
E from V Example 1 (Halliday, Resnick and
Krane, 5th Edition, Chap. 28, Exercise 34)
Rutherford discovered, 99 years ago, that an atom
has a positive nucleus with a radius about 105
times smaller than the radius R of the atom. He
modeled the electric potential inside the atom (r
lt R) as follows where Z is the atomic
number (number of protons). What is the
corresponding electric field?
11
E from V Answer since
12
E from V Example 2 Electric field of a
dipole We found that the electric potential
V(x,y,z) of a dipole made of charges q at
(0,0,d/2) and q at (0,0,d/2) is
z
d/2
r
(x,y,z)
r
d/2
13
E from V
z
d/2
r
(x,y,z)
r
d/2
14
E from V Now we will consider the case d ltlt r
, r and use these rules for 0 lt a ltlt 1 (derived
from Taylor or binomial expansions) to
approximate E
15
E from V For d ltlt r , r we have
z
r
d/2
(x,y,z)
r
d/2
16
E from V For d ltlt r , r we have
z
r
d/2
(x,y,z)
r
d/2
17
E from V So
z
r
d/2
(x,y,z)
r
d/2
18
E from V We can check that these results
coincide with the results we obtained for the
special cases x 0 y and z 0
z
r
d/2
(x,y,z)
r
d/2
19
E from V This may look complicated but it
is easier than calculating Ex, Ey and Ez
directly!
z
r
d/2
(x,y,z)
r
d/2
20
Equipotential surfaces We have already seen
equipotential surfaces in pictures of lines of
force and in connection with potential
energy
r2
r2
F(r)
r
r1
r1
21
Equipotential surfaces All points on an
equipotential surface are at the same electric
potential. Electric field lines and
equipotential surfaces meet at right angles.
Why?
r2
r2
F(r)
r
r1
r1
22
Equipotential surfaces The surface of a
conductor at electrostatic equilibrium when all
the charges in the conductor are at rest is an
equipotential surface, even if the conductor is
charged
Small pieces of thread (in oil) align with the
electric field due to two conductors, one pointed
and one flat, carrying opposite charges. From
Halliday, Resnick and Krane
23
Equipotential surfaces We can also visualize the
topography of the electric potential from the
side (left) and from above (right) with
equipotentials as horizontal curves.
V
y
x
24
Equipotential surfaces Quick quiz In three
space dimensions, rank the potential differences
V(A) V(B), V(B) V(C), V(C) V(D) and V(D)
V(E).
B
A
9 V
E
8 V
D
C
7 V
6 V
25
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. Explanation A conductor contains
free charges (electrons) that move in response to
an electric field at electrostatic equilibrium,
then, the electric field inside a conductor must
vanish. Example An infinite conducting sheet
in a constant electric field develops surface
charges that cancel the electric field inside
the conductor.
- - - - - - -

26
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. 2. Any net charge on an isolated
conductor lies on its surface.
27
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. 2. Any net charge on an isolated
conductor lies on its surface. Explanation A
charge inside the conductor would imply, via
Gausss law, that the electric field could not be
zero everywhere on a small surface enclosing it,
contradicting 1.
28
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. 2. Any net charge on an isolated
conductor lies on its surface. 3. The electric
field on the surface of a conductor must be
normal to the surface and equal to s/e0, where s
is the surface charge density.
29
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. 2. Any net charge on an isolated
conductor lies on its surface. 3. The electric
field on the surface of a conductor must be
normal to the surface and equal to s/e0, where s
is the surface charge density. Explanation A
component of E parallel to the surface would move
free charges around. (Hence the surface of a
conductor at electrostatic equilibrium is an
equipotential surface, even if the conductor is
charged.)
30
E in and on a conductor Lets compare the
electric fields due to two identical surface
charge densities s, one on a conductor with all
the excess charge on one side (e.g. the outside
of a charged sphere) and the other on a thin
insulating sheet. The figure shows a short
Gaussian can straddling the thin charged sheet.
If the can is short, we need to consider only
the electric flux through the top and
bottom. Gausss law gives 2EA ?E sA/e0, so E
s/2e0. But if there is flux only through the
top of the can, Gausss law gives EA ?E
sA/e0 and E s/e0.
Gaussian can
31
E in and on a conductor Four rules about
conductors at electrostatic equilibrium 1. The
electric field is zero everywhere inside a
conductor. 2. Any net charge on an isolated
conductor lies on its surface. 3. The electric
field on the surface of a conductor must be
normal to the surface and equal to s/e0, where s
is the surface charge density. 4. On an
irregularly shaped conductor, the surface charge
density is largest where the curvature of the
surface is largest.
32
E in and on a conductor Explanation If we
integrate Edr along any electric field line,
starting from the conductor and ending at
infinity, we must get the same result, because
the conductor and infinity are both
equipotentials. But E drops more quickly from a
sharp point or edge than from a smooth surface
as we have seen, E drops as 1/r2 from a point
charge, as 1/r near a charged line, and scarcely
drops near a charged surface. The integral far
from the conductor is similar for all electric
field lines near the conductor, if Edr drops
more quickly from a sharp point or edge, it must
be that E starts out larger there. Then, since E
is proportional to the surface charge s, it must
be that s, too, is larger at a sharp point or
edge of a conductor.
33
Capacitors and capacitance A capacitor is any
pair of isolated conductors. We call the
capacitor charged when one conductor has total
charge q and the other has total charge q.
(But the capacitor is then actually neutral.)
Whatever the two conductors look like, the
symbol for a capacitor is two parallel lines.
34
Capacitors and capacitance Here is a circuit
diagram with a battery to charge a capacitor, and
a switch to open and close the circuit.
35
Capacitors and capacitance The charge q on the
capacitor (i.e. on one of the two conductors) is
directly proportional to the potential difference
?V across the battery terminals q C ?V.
36
Capacitors and capacitance The charge q on the
capacitor (i.e. on one of the two conductors) is
directly proportional to the potential difference
?V across the battery terminals q C ?V.
(Note that q and ?V both scale the same way as
the electric field.)
37
Capacitors and capacitance The charge q on the
capacitor (i.e. on one of the two conductors) is
directly proportional to the potential difference
?V across the battery terminals q C ?V.
(Note that q and ?V both scale the same way as
the electric field.) The constant C is called
the capacitance of the capacitor. The unit of
capacitance is the farad F, which equals Coulombs
per volt F C/V.
38
Capacitors and capacitance Example 1
Parallel-plate capacitor If we can neglect
fringing effects (that is, if we can take the
area A of the conducting plates to be much larger
than the distance d between the plates) then E
s/e0 where s q/A and ?V Ed qd/e0A. By
definition, q C ?V, hence C e0A/d is the
capacitance of an ideal parallel-plate capacitor.
39
Capacitors and capacitance Example 2
Cylindrical capacitor Again, if we can neglect
fringing effects (that is, if we can take the
length L of the capacitor to be much larger than
the inside radius b of the outer tube) then
a
b
40
Capacitors and capacitance Example 3 Spherical
capacitor The diagram is unchanged, only E(r)
is different
a
b
41
Capacitors in series and in parallel Combinations
of capacitors have well-defined capacitances.
Here are two capacitors in series
42
Capacitors in series and in parallel Combinations
of capacitors have well-defined capacitances.
Here are two capacitors in series
Since the potential across both capacitors is ?V,
we must have q1 C1?V1 and q2 C2?V2 where ?V1
?V2 ?V. But the charge on one capacitor
comes from the other, hence q1 q2 q.
43
Capacitors in series and in parallel Combinations
of capacitors have well-defined capacitances.
Here are two capacitors in series
Since the potential across both capacitors is ?V,
we must have q C1?V1 and q C2?V2 where ?V1
?V2 ?V. If the effective capacitance is
Ceff, then we have hence
44
Capacitors in series and in parallel For n
capacitors in series, the generalized rule is
C1
C2
Cn
45
Capacitors in series and in parallel Combinations
of capacitors have well-defined capacitances.
Here are two capacitors in parallel
46
Capacitors in series and in parallel Combinations
of capacitors have well-defined capacitances.
Here are two capacitors in parallel
Since the potential across each capacitor is
still ?V, the charge on the capacitors is q1
C1?V and q2 C2?V. If the effective capacitance
is Ceff, then we have C1?V C2?V q1 q2
Ceff?V, thus capacitances in parallel add C1
C2 Ceff .
47
Capacitors in series and in parallel For n
capacitors in parallel, the generalized rule is
Ceff C1 C2 Cn .
C1
C2
Cn
48
Halliday, Resnick and Krane, 5th Edition, Chap.
30, MC 9 The capacitors have identical
capacitance C. What is the equivalent
capacitance Ceff of each of these combinations?
A
B
D
C
49
Halliday, Resnick and Krane, 5th Edition, Chap.
30, MC 9 The capacitors have identical
capacitance C. What is the equivalent
capacitance Ceff of each of these combinations?
Ceff 2C/3
Ceff 3C
A
B
Ceff C/3
Ceff 3C/2
D
C
50
Halliday, Resnick and Krane, 5th Edition, Chap.
30, Prob. 9 Find the charge on each capacitor
(a) with the switch open and (b) with the switch
closed.
C1 1 µF
C2 2 µF
?V 12 V
C3 3 µF
C4 4 µF
51
Answer (a) We have C1?V1 q1 and C3?V3 q1
since those two capacitors are equally charged.
Now ?V1 ?V3 12 V and so q1/C1 q1/C3 12 V
and we solve to get q1 q3 9 µC. In just the
same way we obtain q2 q4 16 µC.
C1 1 µF
C2 2 µF
?V 12 V
C3 3 µF
C4 4 µF
52
Answer (b) We have C1?V12 q1 and C2?V12
q2 and in the same way C3?V34 q3 and C4?V34
q4. Two more equations ?V12 ?V34 12 V and
q1 q2 q3 q4. We solve these six equations
to obtain ?V12 8.4 V and ?V34 3.6 V, and
charges q1 8.4 µC, q2 16.8 µC, q3 10.8 µC
and q4 14.4 µC.
C1 1 µF
C2 2 µF
?V 12 V
C3 3 µF
C4 4 µF