III B. Conductors, Capacitors, Dielectrics
B.
Conductors, Capacitors, Dielectrics
1.
Electrostatics with Conductors
a. Students should understand the nature of electric fields in and
around conductors so they
can:
(1) Explain the mechanics responsible for the absence of electric
field inside a conductor, and why all
excess charge must reside on the surface of the conductor.
(2) Explain why a conductor must be an
equipotential, and apply this principle in analyzing
what happens when conductors are
joined by wires.
(3) Determine the direction of the
force on a charged particle brought near an uncharged or
grounded conductor
b. Students should be able to describe and sketch a graph of the
electric field and potential
inside and outside a charged conducting sphere so they can:
(1) Describe qualitatively the process
of charging by induction.
(2) Determine the direction of the
force on a charged particle brought near an uncharged or
grounded conductor
2.
Capacitors and Dielectrics
a. Students should know the definition of capacitance so they can relate
stored charge and
voltage for a capacitor.
b. Students should understand energy storage in capacitors so they can:
(1) Relate voltage, charge, and stored
energy for a capacitor.
(2) Recognize situations in which
energy stored in a capacitor is converted to other forms.
c. Students should understand the physics of the parallel-plate
capacitor so they can:
(1) Describe the electric field inside
the capacitor, and relate the strength of this field to the potential
difference between the plates and the plate separation.
(4) Determine how changes in dimension will affect the value of
capacitance.
Sharing of
Charge
·
All
systems, mechanical and electrical, come to equilibrium when the energy
of the system is at a minimum. When a conductor is in equilibrium, the electric
field everywhere inside the conductor is zero. This is fairly obvious for a
neutral conductor where the electric field emanating from all of the positive
charges end on the negative charges. But, what happens when an excess charge is
placed on a conductor. Since the charges within a conductor are free to move
about, the excess charge moves to the outside surface of the conductor, reestablishing
equilibrium and the internal field reduces to zero
·
Charges move until all parts of a
conductor are at the same potential.
·
A
charged sphere shares charge equally with a neutral sphere of equal
size. Charges flow until all parts of a conducting body, two touching spheres
in this case, are at the same potential.
Which way will the charge flow when each of the
following conducting spheres are connected?
- Q - Q
- Q - Q - Q
Which way will the charge flow when each of the
above conducting spheres are connected to the Earth?
·
The
potential on the larger sphere is lower than the potential on the smaller
sphere because the charges are farther apart thus the repulsive force between them
is reduced. If the two spheres are touched together, charges will move to the
sphere with the lower potential;
that is from the smaller to the larger sphere. The result is
a greater charge
on the larger sphere when the two different-sized spheres are at the same potential.
grounding- touching an object to Earth to eliminate excess charge.
·
Earth
is a very large sphere so a charged
body that touches it will result in practically any amount of charge to flow
from the body
to the Earth,
allowing the body to become neutral.
Electric
Fields Near Conductors
·
The
electric field around a conducting body depends on the structure and shape
of the body.
·
The
charges on a conductor are spread apart as far apart as possible in order to
make the energy
of the system as low as possible.
The result is that all charges are on the surface
of a solid conductor. If the conductor is hollow, excess charges will move to
the outside
surface. In this way, a closed metal container shields the inside from electric
fields. (Faraday
cage)
·
The
electric field around the outside of a conductor depends on the shape
of the body as well as its potential. The charges are closer together at
the sharp
points of a conductor, therefore, the field lines are closer together; the field is stronger.
·
Application: To reduce corona and
sparking, conductors that are highly charged or operate at high potentials are
made smooth
in shape. Lightning rods are pointed so that the electric field will be
strong near the end of the rod thus attracting lightning to it rather than the
building it is protecting.
Storing
Electric Energy- The Capacitor P 59
1746- Musschenbroek- invented a device that
could store electric charge.
Leyden jar (see our Leyden jar)
·
Two
isolated conductors having a potential difference of DV
between them with equal and
opposite charges + Q and - Q ,
constitute what is called a capacitor
because as long as they
are isolated they have the capacity for storing charge.
·
The
difference
in potential
is directly
proportional to the charge of the objects. This should be rather
obvious since doubling the charge also doubles the electric field in the region
about the objects and hence, the work-per-unit charge (that is, the potential difference, DV) required to move
a test charge between two points in the field also doubles.
Q µ DV or Q = CDV Where
C is a constant of proportionality
·
For
a given size and shape of a capacitor, the ratio of stored charge to potential
difference, Q/V, is a constant called the
capacitance (C).
·
The
capacitance is independent of the charge on it. It can be
measured by placing charge +q on one
plate and -q on the other, and
measuring the potential difference, DV,
that results. The AP equation is...
C = Q/V unit: =
1C/1V = farad (F) (Eq
***)
·
A
capacitor in electronics lingo is- a device designed to have a specific
capacitance.
·
All
capacitors used in circuitry are made up of two conductors of equal and
opposite charges, separated by an insulator called
a dielectric.
Capacitors have become very important devices in electronic circuits,
electrical machinery, etc. since they are devices which
store charge and electrical energy.
Applications: Tuners in radio
receivers, “condensers” in ignition systems of
cars, camera flash, strobe light
Demo
Practice
Problem 10 The Charge on the Plates of a Capacitor
A 3.0-mF capacitor is
connected to a 12-V battery. What is the magnitude of the charge on each
plate of the capacitor?
C = Q /V
Q = C V
= (3 x 10^{-6 }C)(12
V)
= 36
mC
The
Parallel-Plate Capacitor
·
Recall
that a constant
electric field can be made by placing two flat plates with equal and opposite
charges parallel
to each other. Represent this electric field on the diagram above.
·
For
the parallel plate capacitor shown in the figure above, the voltage V is given by Ed where E is the electric
field
strength
between the plates and d is the
plate separation.
·
The
electric field E depends on the amount of charge on the plates which in turn depends on the area
of the plates (more area can hold more charge) with the proportionality
constant = 1/ e_{o} . Thus E = Q/(e_{o} A). The capacitance of a
parallel-plate capacitor, therefore, is directly proportional to the area
of one of the plates and inversely proportion to their separation.
C = Q/V V = Ed C = Q/Ed
C = e_{o} A/d (Eq) ***
A =
the area of one
of the plates.
e_{o} = constant = permittivity of free space = 8.85 x 10^{-12} C^{2}/N m^{2} = 1/4 p k
Practice
Problem 11 Calculating Capacitance for a Parallel Plate Capacitor
C = e_{o} A/d
= (8.85 x 10^{-12} C^{2}/N m^{2})(.0002 m^{2}) / .001 m
= 1.77 x 10^{-10 }F^{}
Energy Stored
in a Charged Capacitor
·
The
energy
stored in a charged capacitor is equal to the work required to charge it.
The work required to transfer a small amount of charge DQ from the negative to the
positive plate at potential V is DW = VDQ. Since V increases linearly as more charge is added, the total
work must be the average V, which is (1/2) (0 + V_{f}), times Q. So, the total work done and
thus the electric potential energy (U_{c}) stored in the
capacitor is
U_{c}
= 1/2 QV = 1/2 CV^{2 }(Eq) ***
Practice
Problem 12 Energy Stored in a Charged Capacitor
U_{c}
= ˝ CV^{2}
= ˝ (1.77 x 10^{-10 }F)((120
V)^{ 2 } = .036 J
·
Thus
the usefulness of a capacitor lies in its ability to store electrical energy
which can, upon discharging the capacitor, then be converted to other useful
forms such as sound, light, heat, and mechanical energy.