Transcript Document

Chapter 24 Capacitance and
Dielectrics
•Capacitance and dielectrics
•Capacitors in series and parallel
•Energy storage in capacitors
and electric field energy
•Dielectrics
•Molecular model / polarization
•R-C circuits
C 2009 J. F. Becker
(sec. 24.1)
(sec. 24.2)
(sec. 24.3)
(sec. 24.4)
(sec. 24.5)
(sec. 26.4)
A "charged" capacitor
can store charge.
When a capacitor is
being charged, negative
charge is removed from
one side of the
capacitor and placed
onto the other, leaving
one side with a negative
charge (-q) and the
other side with a
positive charge (+q).
Any two conductors insulated from one
another form a CAPACITOR.
Q = C V where C = eo A / d
for a parallel plate capacitor,
where eo is the permittivity of
the insulating material
(dielectric) between plates.
A charged parallel plate
capacitor.
Recall that we used Gauss's
Law to calculate the electric
field (E) between the plates
of a charged capacitor:
E = s / eo where there is a
vacuum between the plates.
Vab = E d, so E = Vab /d
The unit of capacitance is called the Farad (F).
1 / Ceq = 1 / C1 + 1 / C2
(a) Two capacitors in series,
(b) the equivalent capacitor.
Ceq = C1 + C2
(a) Two capacitors in parallel,
(b) the equivalent circuit.
Capacitors can store charge and ENERGY
dU = q dV, and the potential V increases as the
charge is placed on the plates (V = Q / C).
Since the V changes as the Q is increased, we
have to integrate over all the little charges
“dq” being added to a plate: dU = q dV gives
U =  V dq =  q/c dq = 1/C  q dq = Q2 / 2C.
And using Q = C V, we get
U = Q2 / 2C = C V2 / 2 = Q V / 2
So the energy stored in a capacitor can be
thought of as the potential energy stored in
the system of positive charges that are
separated from the negative charges, much
like a stretched spring has potential energy.
ELECTRIC FIELD ENERGY
Here's another way to think of the energy
stored in a charged capacitor: If we consider
the space between the plates to contain the
energy (equal to 1/2 C V2) we can calculate an
energy DENSITY (Joules per volume). The
volume between the plates is area x plate
separation, or A d. Then the energy density u
is
u = 1/2 C V2 / A d = eo E2 / 2
Recall C = eo A / d and V =E d.
C 2009 J. F. Becker
Energy density:
u =
eo E2 / 2
This is an important result because it tells us
that empty space contains energy if there is
an electric field (E) in the "empty" space.
If we can get an electric field to travel (or
propagate) we can send or transmit energy
and information through empty space!!!
C 2009 J. F. Becker
DIELECTRIC
CONSTANT:
K = C / Co
= ratio of the
capacitances
V = Vo / K
Effect of a dielectric between the plates of a
parallel plate capacitor. Note – the charge is
constant !
A dielectric is added between the plates of a
charged capacitor (battery not connected):
Q = Q, therefore Q = C V and Q = Co Vo
Co Vo = C V,
and if Vo decreases to V, Co must increase to
C to keep equation balanced, and
V = Vo Co/C
Definition of DIELECTRIC CONSTANT:
K = C / Co = ratio of the capacitances
V = Vo / K
C 2009 J. F. Becker
The charges induced on the surface of the
dielectric reduce the electric field.
“Polarization” of a
dielectric in an electric
field E gives rise to
thin layers of bound
charges on the
dielectric’s surfaces,
creating surface
charge densities
+si and –si.
“Polarization” of a dielectric in
an electric field E.
A neutral sphere
B in the electric
field of a charged
sphere A is
attracted to the
charged sphere
because of
polarization.
DISCHARGING:
An RC circuit
that can be used
to charge and
discharge a
capacitor
(through a
resistor).
CHARGING:
CHARGING A CAPACITOR:
current vs time
CHARGING A CAPACITOR:
charge vs time
DISCHARGING A CAPACITOR:
current vs time
DISCHARGING A CAPACITOR:
charge vs time
Review
See www.physics.edu/becker/physics51
C 2009 J. F. Becker