Transcript Diapositiva 1

Electrostatic Energy and Capacitance
Electrostatic Potential Energy
Capacitance.
Capacitors
Parallel-Plate Capacitor
The Storage of Electrical Energy
Electrostatic Field Energy
Capacitors. Series and parallel
Combination
Dielectrics
Energy stored
Molecular View of a Dielectric
U
U
Calculate the total work required to place
the point charges at each corner of the
square by separately calculating the
work required to move each charge to its
final position. Show that the previous
equation gives the total required work.
Apply to the case: q= 10 µC; a= 10 cm.
n
1
2
qV
i 1
i i
n
1
2
QV
i 1
i i
For a set of n single
point charges
For a set of n conductors
with the ith conductor at
potential Vi and charge Qi
Capacitance, Capacitors
Q
V
1C
1F 
1V
C
Capacitance is a measure of
the capacity to store charge
for a given potential difference
farad [F]
Capacitance of an isolated conductor
Find the radius of a spherical
conductor that has a capacitance of
220 µF
Capacitors: A device consisting of two
conductors carrying equal but opposite charges
Parallel-Plate Capacitors
Q 0 A
C 
V
d

Q
V  Ed  d 
d
0
0 A
A parallel-plate capacitor has
square plates of a length of 10
cm separated by 1 mm. (a)
Calculate the capacitance (b) If
this capacitor is charged to 12 V,
how much charge is transferred
from one plate to another
Find an expression for the capacitance of a cylindrical capacitor consisting of two
conductors, each of length L. Apply to the case: R1=.1 cm; R2= 1.2cm; L =50 cm.
To represent the capacitance against the quotient R2/R1
A coaxial cable is a long
cylindrical capacitor …
A variable air-gap capacitor (for tuning
circuits of old radios) . The
semicircular plates rotate through the
fixed plates, which changes the
amount of surface area between the
plates, and hence the capacitance
Capacitance switching in computer
keyboards. A metal plate attached to
each key acts as the top plate of a
capacitor. Depressing the key
decreases the separation between the
top and bottom plates and increases
capacitance, which triggers the
electronic circuitry of the compute to
acknowledge the keystroke
Some types of capacitors:
Air - Often used in radio tuning circuits
Mylar - Most commonly used for timer circuits like clocks, alarms and
counters
Glass - Good for high voltage applications
Ceramic - Used for high frequency purposes like antennas, X-ray and
magnetic resonance imaging (MRI) machines
Super capacitor - Powers electric and hybrid cars
In a way, a capacitor is a little like a battery. Although they work in completely
different ways, capacitors and batteries both store electrical energy
The storage of electrical energy
q
dU  V dq 
dq
C
Q q
1 Q2
1
1
U   dU  
dq 
 QV  CV 2
Work done by the
0 C
2 C
2
2
battery to charge the
1
1
capacitor
U  CV 2   o E 2 Ad
QV
2
2
U
1
u
 oE2
To consider the previous square parallel-plates capacitor
volum en 2
Energy density (energyper-unit volume) in an
electric field strength E
connected to a 12 V battery How much energy is stored in
the capacitor? .How much work is done by the battery to
charge the capacitor?. The battery is then disconnected
and the plate separation is then increased to 3 mm. How
much the stored energy now is?. To calculate the work
required to separate the plates.
Capacitors connected in parallel
Once the switch is closed , the
capacitors charge. When the
capacitors are fully charged (a) what
is the potential of each conductor in
the circuit? (b) What is the charge on
each capacitor plate? (c) What total
charge passed through a battery?
Equivalent Capacitance:
the capacitance of a single
capacitor that is operationally
equivalent to the combination
What is the equivalent capacitance
of the above described combination?
Capacitors connected in series
Once the switch is closed , the capacitors charge.
When the capacitors are fully charged (a) what is
the potential of each conductor in the circuit? (b)
What is the charge on each capacitor plate? (c)
What total charge passed through a battery?
What is the equivalent capacitance of the above
described combination?
Find the equivalent capacitance of the network of
three capacitors shown in the figure. (b) The are
initially uncharged . Find the charge on each
capacitor and the voltage drop across it if the
combination is connected to a 6-V battery.
The figure shows the so-called
“capacitance bridge”. The capacitors
are initially uncharged. Find the relation
between the four capacitances to obtain
zero potential difference between points
c and d when a voltage V is applied
between a and b
Storing electric energy by a
capacitors combination
Dielectrics
A no conducting material (air, paper, wood, glass,…) is called a dielectric
When the space between the two conductors of a capacitor is
occupied by a dielectric , the capacitances is increased by a factor κ,
that is characteristic of the dielectric. It was discovered by Faraday
A.- An isolated, charged capacitor
E
E0

V  Ed 
C
E0 d


V0

Q
Q

  C0
V
Vo
C 
o A
where
d

 o A
d

   o
Permitivity of the dielectric
 A
d
Dielectrics
A no conducting material (air, paper, wood, glass,…) is called a dielectric
When the space between the two conductors of a capacitor is
occupied by a dielectric , the capacitances is increased by a factor κ,
that is characteristic of the dielectric. It was discovered by Faraday
B.- Dielectric inserted while the battery
remains connected, the battery pumps
additional charge to maintain the
original potential difference.
Q   Qo
V  Vo
Qo
Q
C

  C0
V
V
C 
o A
where
d

 o A
d

 A
   o
Permitivity of the dielectric
d
Energy stored in the presence of a Dielectric
A no conducting material (air, paper, wood, glass,…) is called a dielectric
When the space between the two conductors of a capacitor is occupied by a dielectric , the
capacitances is increased by a factor κ, that is characteristic of the dielectric.
The stored energy when a voltage V is applied is increased κ times the stored
energy without dielectric for the same potential
Q   Qo
V  Vo
1
1
U  QV  CV 2
2
2
1
1  A 
1
2


U  CV 2  
E
d

E 2 ( Ad )

2
2 d 
2
U
1
 E 2
volum e 2
where     o
A 10 µF parallel-plate capacitor, with separation between plates about 1 mm, is
charged with a 12 V battery.
A.- Find the charge, the electric field and the total energy stored in the capacitor
B.- Once charged is disconnected from battery. Then, a dielectric slab of constant
κ= 2.5 is inserted between the plates, completely filling the gap. In this case, find
the charge, the electric field in the dielectric and the total energy stored in the
capacitors.
C.- The battery remains connected. Then the dielectric is inserted, in this case find
the charge, the electric field and the total energy stored
Molecular View of a Dielectric
Polarization of a dielectric in
presence of an electric field
When a dielectric is placed on a electric field, its molecules are polarized in such a way that there is a net
dipole moment parallel to the field. The net effect is the creation of a surface charge on the dielectric
faces near the plates. The surface charges is called a bound charge because it is bound to the
molecules of dielectric and cannot move.
By this way, the net
electric field in the
dielectric is weakened
The piezoelectric effect