Chapter IX Conductors, Capacitors
1.1 The balance of charges on conductors:
In conductors there are charged particles which can be freely
move under any small force. Therefore the balance of charges on
conductors can be observed under these circumstances:
The electric field equals zero everywhere inside the conductor
E = 0
The electric potential is constant inside the conductor
V = const
The electric field vector on the surface of conductors direct along the
normal of the surface at each point
E = En
The surface of conductors is equipotential
Inside conductors there is no charge. This conclusion can be proved
by applying the Gauss’s law for any arbitrary closed surface inside
conductor. All the charge...
GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
2/20/2008 1
Chapter IX
Conductors, Capacitors
§1. Charges and electric field on conductors
§2. Capacitance of conductors and capacitors
§3. Energy storage in capacitors and electric field energy
§4. Electric current, resistance and electromotive force
2/20/2008 2
§1. Charges and electric field on conductors:
1.1 The balance of charges on conductors:
In conductors there are charged particles which can be freely
move under any small force. Therefore the balance of charges on
conductors can be observed under these circumstances:
The electric field equals zero everywhere inside the conductor
E=0
The electric potential is constant inside the conductor
V = const
The electric field vector on the surface of conductors direct along the
normal of the surface at each point
E = En
The surface of conductors is equipotential
Inside conductors there is no charge. This conclusion can be proved
by applying the Gauss’s law for any arbitrary closed surface inside
conductor. All the charge is distributed on the surface of conductors.
2/20/2008 3
Since the distribution of charge on conductors
does not depend on the distribution of the
matter the distribution of charges is
the same for hollow and solid conductors.
The fact that the distribution of charges only
on the surface of conductors can be
understood as follows:
Suppose that we provide the conductor with
an amount of charges charges repulse
mutually and tend to leave as far as possible
each from other.
2/20/2008 4
1.2 The electric field at the surface of
a conductor:
Consider the red small cylindrical surface
σ
with the base dS. Applying the Gauss’s law
for this closed surface we have
The electric field in vacuum
near the surface
The electric field in dielectric (σis the surface charge density
environment near the surface at the considered point on the
surface)
+ Near convexes of the surface:
the equipotential surfaces are
dense E is large
the charge density is large
+ Near deepenings of the surface:
the equipotential surfaces are
2/20/2008 rare E is smaller 5
the charge density is small
§2. Capacitance of conductors and capacitors:
2.1 Charging by induction: Negatively
charged
Metal sphere
rod
A charge body (rod) can give
another body a charge of
opposite sign, without losing Isulating
any of its own charge. stand
a) b)
Pictures: wire
a) Metal sphere is initially uncharged
b) Charged rod brought nearby
c) Wire allows piled-up electrons to
flow to ground
d) Wire is disconnected from sphere
c) d)
e) Charged rod is removed: Electrons
on sphere rearrange themselves. e)
2/20/2008 6
2.2 Capacitors and capacitance:
A capacitor is a device whose purpose is to store electrical energy which
can then be released in a controlled manner during a short period of time.
A capacitor consists of 2 spatially separated conductors which can be
charged to +Q and -Q respectively.
Definition: The capacitance of the capacitor is the ratio of the charge on one
conductor of the capacitor to the potential difference between the
conductors:
Q
C [The unit of capacitance is the
V Farad: 1 F = 1C/V]
• The capacitance belongs only to the capacitor, independent of the charge
and voltage.
2/20/2008 7
Example 1: Parallel Plate Capacitor
• Calculate the capacitance. We assume
+ - charge densities on each plate
,
with potential difference V :
Q A
C ++++
V d -----
Need Q:
Need V: from definition:
Use Gauss’ Law to find E (in next slide)
2/20/2008 8
Recall the formula for the electric field of two infinite sheets:
+ -
• Field outside the sheets is zero
+ -
E=0 + - E=0
Gaussian surface encloses zero net
charge
+ -
+ -
• Field inside sheets is not zero: A + -
+ -
• Gaussian surface encloses non-zero
net charge + -
Q A
+ -
S AEinside
E d E + -
A -
0
+ -
(Note that here we consider a capacitor in vacuum)
+ -
E
2/20/2008 9
Q
E
A
0 0
b
Q A
Q
V a dl d
b V E Ed C 0
a
A0
V d
Remark:
• The capacitance of this capacitor depends only on its shape and size
• This formula is true for parallel-plate capacitor (shape), and C depends
on A, d (size) (for another shape one has other formula).
• When the space between the metal plates is filled with a dielectric
material, the capacitance increases by a factor k (see the previous chapter)
(Recall: k - dielectric constant; ε- permitivity of the dielectric)
In order to increase C d (limitatively), and one must increase A, ε
.
2/20/2008 10
Example 2: Cylindrical Capacitor
• Calculate the capacitance:
• Assume +Q, -Q on surface of cylinders with r
potential difference V. a
b
• Gaussian surface is cylinder of radius r (a L
< r < b) and length L
Q Q
• Apply Gauss' Law: S 2
E d rLE E 2 Lr
0
0
If we assume that inner cylinder has +Q, then the potential V is positive if we
take the zero of potential to be defined at r = b:
a Q 2 0 L
b C
a b
Q Q
V dl
E Edr dr ln V b
ln
b b a 2
0 rL 20 L
a a
2/20/2008 11
Summary on Capacitance
• A capacitor is an object with two spatially separated conducting
surfaces.
• The definition of the capacitance of such an object is:
Q
C
V
• The capacitance depends on the geometry :
-Q
-Q a
A r
+Q
++++ a
d +Q
----- b b
L
Parallel Plates Cylindrical Spherical
A L ab
C C C
2/20/2008 d ln( b / a ) b a 12
2.3 Capacitors in Parallel:
a a
Q1 Q2 Q
VC C V C
1 2
-Q1 -Q2 -Q
b b
• Find “equivalent” capacitance C in the sense that no
measurement at a, b could distinguish the above two situations.
• The voltage across the two is the same….
Q1 Q 2 C
Parallel Combination: V Q 2 Q1 2
C1 C 2 C1
Q Q Q Q (C )
C
Equivalent Capacitor: C 1 2 1 1 2
V V C1V
2/20/2008 C12
C C 13
2.4 Capacitors in Series:
+Q -Q
C2 +Q -Q
a b a b
C1 +Q -Q C
• Find “equivalent” capacitance C in the sense that no measurement at
a, b could distinguish the above two situations.
• The charge on C1 must be the same as the charge on C 2 since
applying a potential difference across ab cannot produce a net
charge on the inner plates of C1 and C2
assume there is no net charge on node between C1 and C2
Q
RHS: V ab
1 1 1
C
Q Q C C1 C 2
LHS: Vab 1 2
V V
C1 C2
2/20/2008 14
Examples: Combinations of Capacitors
a
C3 a b
C1 C2 C
b
• How do we start??
• Recognize C3 is in series with the parallel combination on C1 and
C 2. i.e.,
C 3 ( C1 2 )
C
1 1 1 C
C C3 C1 2
C C1 2 3
C C
2/20/2008 15
§3.Energy of a Capacitor and electric field energy:
3.1 Energy of a charged capacitor:
• How much energy is stored in a charged capacitor?
– Calculate the work provided (usually by a battery) to charge a
capacitor to +/- Q:
Calculate incremental work dW needed to add charge dq to capacitor at
voltage V (there is a trick here!): - +
q
dW (q) dq
V dq
C
• The total work W to charge to Q is then given by:
Q
1 1 Q2
W
qdq Look at this!
C0 2 C Two ways to write W
1
• In terms of the voltage V: W CV 2
2/20/2008 2 16
3.2 Energy storage in capacitors:
Where is the Energy Stored?
• Energy is stored in the electric field itself. Think of the energy needed
to charge the capacitor as being the energy needed to create the field.
• To calculate the energy density in the field, first consider the
constant field generated by a parallel plate capacitor, where
-Q
-------- -- ----
1 Q2 1 Q2
U
2 C 2 ( A / d )
++++++++ +++++++
+Q 0 This is the energy
density, u, of the
• The electric field is given by: electric field….
Q
E 1
A U E2 Ad
0
0 0 2
• The energy density u in the field is given by:
U U 1 2 J
u 0E Units:
volume Ad 2 m3
2/20/2008 17
Note 1: The expression for the energy density of the electrostatic
field 1
u E 20
2
is general and is not restricted to the special case of the
constant field in a parallel plate capacitor.
Note 2: For the electric field in a dielectric
(In a small volume surrounding the considered point we can consider
the electric field as constant, and apply the formulas of the electric field
inside paralell-plates capacitor).
2/20/2008 18
§4. Electric current, resistance and electromotive force
4.1 The definition of current and current density:
Charges, e.g. free electrons, exists in conductors with a density, ne (ne
approx 1029 m-3 )
“Somehow” put that charge in motion:
effective picture -- all charge moves with a velocity, ve
real picture -- a lot of “random motion” of charges with a small average
equal to ve
We need a quantity which can characterize flows of moving charges.
Definition of current: The rate at which charge flows
unit of I :
2/20/2008 19
l=vt
Definition of current density:
In metal wires, the electrons are the carriers
Cross section
of charge. We have the following equations:
area A
Volume
= A.l
(N – number of electrons which pass through the cross-section A during t ).
where
n is free electron density
The formula gives relation between
current density, electron density and
2/20/2008 velocity of electrons. 20
