Electrical Charge

1
Electrical Charge
Some Concepts Electrical charge (??) Electric
field (??) Electric current (??) Conductor
(??) Semiconductor (???) Insulator (???)
Superconductor (???) Two Fundamental Properties
of Electrical Charge Charge is quantized
Charge is
conserved Electrostatic Force permittivity
constant (????) Note that this satisfies
Newton's third law because it implies that
exactly the same magnitude of force acts on q2 .
Coulomb's law is a vector equation and includes
the fact that the force acts along the line
joining the charges. Like charges repel and
unlike charges attract. Coulomb's law describes a
force of infinite range which obeys the inverse
square law, and is of the same form as the
gravity force. Shell Theorem A shell of uniform
charge attracts or repels a charged particle that
is outside the shell as if all the shells charge
were concentrated at its center. A shell of
uniform charge exerts no electrostatic force on a
charged particle that is located inside the shell.
2
Electrical Charge
Spherical Conductors If excess charge is placed
on a spherical shell that is made of conducting
material, the excess charge spreads uniformly
over the external surface. Example The figure
shows two particles fixed in a place a particle
of charge q18q at the origin of an x axis and a
particle of charge q2-2q at xL. At what point
(other than infinitely far away) can a proton be
placed so that it is in equilibrium? Is that
equilibrium stable or unstable? Solution
Example The arrangement of six fixed charged
particles, where ?30?, is shown. All six
particles have the same magnitude of charge q
their electrical signs are as indicated. What is
the net electrostatic force acting on q1 due to
the other charges? Solution 0
3
Electrical Charge
Example Two identical, electrically isolated
conducting spheres A and B are separated by a
(center-to-center) distance a that is large
compared to the spheres. Sphere A has a positive
charge of Q sphere B is electrically neutral
and initially, there is no electrostatic force
between the spheres. (a) Suppose the spheres are
connected for a moment by a conducting wire. The
wire is thin enough so that any net charge on it
is negligible. What is the electrostatic force
between the spheres after the wire is removed?
(b) Next, suppose sphere A is grounded
momentarily, and then the ground connection is
removed. What now is the electrostatic force
between the spheres? Solution
4
Electrical Charge

• Homework
• The figure shows two charges, q1 and q2, held in
  a fixed distance d apart. (a)What is the
  magnitude of the electrostatic force that acts on
  q1? Assume that q1q220.0?C and d1.50m. (b)A
  third charge q320.0 ?C is brought in and placed
  as shown in the figure. What now is the
  magnitude of the electrostatic force on q1?
• In the basic CsCl (cesium chloride) crystal
  structure, Cs ions form the corners of a cube
  and a Cl? ion is at the cubes center. The edge
  length of the cube is 0.40 nm. The Cs ions are
  each deficient by one electron (and thus each has
  a charge of e), and the Cl? ion has one excess
  electron (and thus has a charge of e). (a) What
  is the magnitude of the net electrostatic force
  exerted on the Cl? ion by the eight Cs ions at
  the corners of the cube? (b)If one of the Cs
  ions is missing, the crystal is said to have a
  defect what is the magnitude of the net
  electrostatic force exerted on the Cl? ion by the
  seven remaining Cs ions?

5
Electric Fields
Electric Field Electric field is defined as the
electric force per unit charge. The direction of
the field is taken to be the direction of the
force it would exert on a positive test charge.
The electric field is radially outward from a
positive charge and radially in toward a negative
point charge. Conversely, the electrostatic
force acting on a particle with a charge q is
Electric Field of Point Charge
A positive number is taken to be an outward
field the field of a negative charge is toward
it. The electric field from any number of point
charges can be obtained from a vector sum of the
individual fields.
6
Electric Fields
Example The figure shows three particles with
charges q12Q, q2?2Q, and q3?4Q, each a
distance d from the origin. What net electric
field E is produced at the origin? Solution
Example The nucleus of a uranium atom has a
radius R of 6.8 fm. Assuming that the positive
charge of the nucleus is distributed uniformly,
determine the electric field at a point on the
surface of the nucleus due to that charge.
Solution The atomic number Z92
7
Electric Fields
Electric Field Lines (???) Electric field lines
provide a means for visualizing the direction and
magnitude of electric fields. The electric field
vector at any point is tangent to a field line
through that point. The density of field lines in
any region is proportional to the magnitude of
the electric field in that region. Field lines
originate on positive charges and terminate on
negative charges. A few examples are shown
below Electric Dipole (????) Field An
electric dipole consists of two particles with
charges of equal magnitude q but opposite sign,
separated by a small distance d. It can be
proved that the electric field generated by the
dipole along the dipole axis is where r is the
distance to the center of the dipole rgtgtd.
8
Electric Fields
Dipole Moment (???) pqd is called the electric
dipole moment. It is a vector with a direction
pointing from the negative to the positive charge
of the dipole. The electric field in a plane
perpendicular to the dipole axis that cuts
through the dipole by half is Dipole in an
Electric Field The electric field exerts a
torque on the dipole The dipole has a
potential energy where we choose the potential
energy to be zero when the angle ? is 90?.
9
Electric Fields
Electric Field Due to a Charged Ring
(Optional) As shown in the figure, ? is the
charge per unit length. The total charge on the
ring is q. Electric Field Due to a Charged
Disk (Optional) For a uniformly charged disk
with a radius R, we can cut the disk into small
pieces of rings. Suppose the charge per unit
area is ?, then for each piece of ring, it
carries a charge of Each ring contributes an
electric field The total field is
10
Electric Fields

• Homework
• A clock face has negative point charges q, ?2q,
  ?3q, , ?12q fixed at the positions of the
  corresponding numerals. The clock hands do not
  perturb the net field due to the point charge.
  At what time does the hour hand point in the same
  direction as the electric field vector at the
  center of the dial? (Hint Use symmetry).
• In the figure a uniform, upward-pointing electric
  field E of magnitude 2.00?103 N/C has been set up
  between two horizontal plates by charging the
  lower plate positively and the upper plate
  negatively. The plates have length L10.0 cm and
  separation d2.00 cm. An electron is then shot
  between the plates from the left edge of the
  lower plate. The initial velocity v0 of the
  electron makes an angle ?45.0? with the lower
  plate and has a magnitude of 6.00?106 m/s. (a)
  Will the electron strike one of the plate? (b)
  If so, which plate and how far horizontally from
  the left edge?

11
Electric Potential
Electrical Potential Energy The electrostatic
force is a conservative force. Thus we can
define a change of the electric potential energy
U of a charged particle in an electric field to
be,
where W is the work done by the electrostatic
force and is path independent. If the potential
energy is defined to be zero at infinity, the
electric potential energy U of a point charge is
U?W? , where W? is the work done by the electric
field on the point charge as the charge moves
from infinity to the particular point.
Concept In the figure, a proton moves from
point i to point f in a uniform electric field
directed as shown. (a) Does the electric field do
positive or negative work on the proton? (b) Does
the electric potential energy of the proton
increase or decrease? Electrical Potential The
electrical potential energy per unit charge is
defined as the electric potential difference
(???) The electrical potential at a point is
The unit is voltjoule per
coulomb.
12
Electric Potential
Equipotential Surfaces (???) Adjacent points
that have the same electric potential form an
equipotential surface, which can be either an
imaginary surface or a real, physical one.
(i) No work is done by the electric field when a
charge is moved on that surface. (ii) The
electric field E is always directed
perpendicularly to the equipotential surface.
Potential Calculated from the Field In a
uniform electric field E, the potential
difference is,
13
Electric Potential
Potential due to a Point Charge Example
(a) What is the electric potential V at a
distance r2.12?10-10 m from the nucleus of a
hydrogen atom (the nucleus consists of a single
proton)? (b) What is the electric potential
energy U in electron-volts of an electron at the
given distance from the nucleus? (c) If the
electron moves closer to the proton, does the
electric potential energy increase or decrease?
Solution (c) V
increases and U decreases.
Example What is the potential at point P,
located at the center of the square of point
charges shown in the figure? Assume that d1.3 m
and that the charges are q112 nC, q2?24 nC,
q331 nC, q417 nC. Solution
14
Electric Potential
Example (a) In figure a, 12 electrons are
equally spaced and fixed around a circle of
radius R. Relative to V0 at infinity, what are
the electric potential and electric field at the
center C of the circle due to these electrons?
(b) If the electrons are moved along the circle
until they are nonuniformly spaced over a 120?
are (figure b), what then is the potential at C?
Solution
Potential of Line Charge (Optional) It can be
found by superposing the point charge potentials
of infinitesmal charge elements. It is an
example of a continuous charge distribution.
15
Electric Potential
Potential for Ring of Charge (Optional) It can
be found by superposing the point charge
potentials of infinitesmal charge elements. The
ring potential can then be used as a charge
element to calculate the potential of a charged
disc.
Potential for Disc of Charge (Optional) It can
be found by superposing the point charge
potentials of infinitesmal charge elements. The
evaluation of the potential can be facilitated by
summing the potentials of charged rings.
Calculating E from V
16
Electric Potential
Electric Potential Energy of a System of Point
Charges The electric potential energy of a
system of fixed point charges is equal to the
work that must be done by an external agent to
assemble the system, bringing each charge in from
an infinite distance. For two point charges
For many point
charges Example Starting with the expression
for the potential at any point on the axis of a
charged disk, derive the expression for the
electric field at any point on the axis of the
disk. Solution Example The figure shows
three charges held in fixed positions by forces
that are not shown. What is the electric
potential energy of this system of charges?
Assume that d12 cm and that q1q, q2?4q, and
q32q, in which q150 nC. Solution
17
Electric Potential

• Homework
• In a given lightning flash, the potential
  difference between a cloud and the ground is
  1.0?109 V and the quantity of charge transferred
  is 30 C. (a) What is the change in energy of
  that transferred charge? (b) If all the energy
  released by the transfer could be used to
  accelerate a 1000 kg automobile from rest, what
  would be the automobiles final speed? (c) If
  the energy could be used to melt ice, how much
  ice would it melt at 0?C? The heat of fusion of
  ice is 3.33?105 J/kg.
• For the charge configuration of the figure, show
  that V(r) for points such as P on the axis,
  assuming rgtgtd, is given by
• (Note the charges should be treated as
  point charges and their size being neglected.)

18
Capacitance
Capacitor A capacitor is a device that stores
electric potential energy by storing separated
positive and negative charges. It consists of
two conductors separated by either vacuum or an
insulating material. The simplest case is a
parallel plate capacitor.
Capacitance is defined in terms of charge
storage
(Farad, F) where, Q magnitude of charge
stored on each plate. V voltage
applied to the plates.
Cylindrical Capacitor (Optional) The charge
resides on the outer surface of the inner
conductor and the inner wall of the outer
conductor. Assume the length of the cylinder
Lgtgtb.
19
Capacitance
Spherical Capacitor (Optional) An Isolated
Sphere (Optional) By taking the limits a?R and
b??, Concept For capacitors charged by the
same battery, does the charge stored by the
capacitor increase, decrease, or remain the same
in each of the following situations? (a) The
plate separation of a parallel capacitor is
increased. (b) The radius of the inner cylinder
of a cylindrical capacitor is increased. (c) The
radius of the outer spherical shell of a
spherical capacitor is increased.
Example The plates of a parallel-plate
capacitor are separated by a distance d1.0 mm.
What must be the plate area if the capacitance is
to be 1.0 F? Solution Example A storage
capacitor on a random memory (RAM) chip has a
capacitance of 55fF. If the capacitor is charged
to 5.3 V, how many excess electrons are on its
negative plate? Solution
20
Capacitance
Capacitors in Parallel and Series Equivalent
capacitance Example (a) Find the equivalent
capacitance of the combination as shown. Assume
C112.0 ?F, C25.30 ?F, C34.50 ?F. (b) A
potential difference V12.5 V is applied to the
input terminals. What is the charge on C1?
Solution (a) (b) Example A
3.55 ?F capacitor C1 is charged to a potential
difference V06.30 V, using a 6.30 V battery. The
battery is then removed and the capacitor is
connected as in the figure to an uncharged 8.95
?F capacitor C2. When switch S is closed, charge
flows from C1 to C2 until the capacitors have the
same potential difference V. What is the common
potential difference? Solution
21
Capacitance
Potential Energy and Energy Density The electric
potential energy of a capacitor is the energy
stored in the electric field between the two
plates (electrodes). It is the work required to
charge the capacitor. The energy density is the
potential energy per unit volume. The above
results hold generally for types of
capacitors. Example An isolated conducting
sphere whose radius R is 6.85 cm has a charge
q1.25 nC. (a) How much potential energy is
stored in the electric field of this charged
conductor? (b) What is the energy density at the
surface of the sphere? (c) What is the radius R0
of an imaginary spherical surface such that half
of the stored potential energy lies within it?
Solution
22
Capacitance
Dielectrics Dielectric material contains polar
molecules (or being polarized under an electric
field), they will generally be in random
orientations when no electric field is applied.
An applied electric field will polarize the
material by orienting the dipole moments of polar
molecules. This decreases the effective electric
field between the plates and will increase the
capacitance of the parallel plate structure. The
dielectric must be a good electric insulator so
as to minimize any DC leakage current through a
capacitor.
? is called the dielectric constant of a
material. In a region completely filled by a
material of dielectric constant ?, all
electrostatic equation containing the
permittivity constant ?0 are to be replaced by
??0. The Guass law need to be generalized to,
23
Capacitance
Example A parallel-plate capacitor whose
capacitance C is 13.5 pF is charged to a
potential difference V12.5 V between its plates.
The charging battery is now disconnected and a
porcelain slab is slipped between the plates.
What is the potential energy of the device, both
before and after the slab is introduced?
Solution
Room temperature dielectric constants for some
materials
Material ?
Air (1atm) 1.00054
Polystyrene 2.6
Paper 3.5
Transformer oil 4.5
Pyrex 4.7
Ruby mica 5.4
Porcelain 6.5
Silicon 12
Germanium 16
Ethanol 25
Water (20C) 80.4
Water (25C) 78.5
Titania ceramic 130
Strontium titanate 310
24
Capacitance
Example The figure shows a parallel-plate
capacitor of plate area A and plate separation d.
A potential difference V0 is applied between the
plates. The battery is then disconnected, and a
dielectric slab of thickness b and dielectric
constant ? is placed between the plates as shown.
Assume, A115cm2, d1.24cm, V085.5V, b0.780cm,
?2.61. (a) What is the capacitance C0 before the
dielectric slab is inserted? (b) What free charge
appears on the plates? (c) What is the electric
field E0 in the gaps between the plate and the
dielectric slab? (d) What is the electric field
E1 in the dielectric slab? (e) What is the
potential difference V between the plates after
the slab has been introduced? (f) What is the
capacitance with the slab in place? Solution
25
Capacitance

• Homework (continued)
• The figure shows a variable air gap capacitor.
  Alternate plates are connected together one
  group is fixed in position and the other group is
  capable of rotation. Consider a pile of n plates
  of alternate polarity, each having an area A and
  separated from adjacent plates by a distance d.
  Show that this capacitor has a maximum
  capacitance of C(n-1)?0A/d.
• In the figure, battery B supplies 12 V. (a) Find
  the charge on each capacitor first when only
  switch S1 is closed and (b) later when switch S2
  is also closed. Take C11.0 ?F, C22.0 ?F,
  C33.0 ?F, and C44.0 ?F.
• A parallel-plate capacitor of plate area A is
  filled with two dielectrics as shown in the
  figure. Show that the capacitance is
• A parallel-plate capacitor of plate area A is
  filled with two dielectrics of the same thickness
  as shown in the figure. Show that the
  capacitance is

?1
?2
?2
?1
26
Current and Resistance
Current An electric current in a conductor is
(ampere, A) A current direction is the
one in which positive charge carriers would move.
The negative charges move in the opposite
direction. Current Density Drift of Charge
Carriers When an electric field E is established
in a conductor, the charge carriers (assumed
positive) acquire a drift speed vd in the
direction of E, where n is the number of charge
carriers per unit volume. Example (a) The
current density in a cylindrical wire of radius
R2.0 mm is uniform across a cross section of the
wire and is given by J2.0?105 A/m2. What is the
current through the outer portion of the wire
between radial distance R/2 and R? (b) Suppose,
instead, that the current density through a cross
section varies with radial distance r as Jar2,
in which a3.0?1011 A/m4 and r is in meters. What
now is the current through the same outer portion
of the wire? Solution
27
Current and Resistance
Example One end of an aluminum wire whose
diameter is 2.5 mm is welded to one end of copper
wire whose diameter is 1.8 mm. The composite wire
carries a steady current i of 17 mA. (a) What is
the current density in each wire? (b) What is the
drift speed of the conduction electrons in the
copper wire? Assume that, on the average, each
copper atom contributes one conduction electron.
Solution Example Consider a strip of
silicon that has a rectangular cross section with
width w3.2 mm and height h250 ?m, and through
which there is a uniform current i of 5.2 mA. The
silicon has a number of charge carriers
(electrons) per unit volume n1.5?1023 m-3. (a)
What is the current density in the strip? (b)
What is the drift speed? Solution
Concept The figure shows conduction electrons
moving leftward through a wire. Are the following
leftward or rightward (a) the current i, (b) the
current density J, (c) the electric field E in
the wire?
28
Current and Resistance
Resistivity of some materials at room temperature
Resistance (ohm,
?) Resistivity unit of ?
??m Conductivity (?-1?m-1) The
resistivity R of a conducting wire of length L
and uniform cross-section A is The resistivity
for most materials changes with temperature ? is
the mean temperature coefficient of
resistivity. Ohms Law The current through a
device is always directly proportional to the
potential difference applied to the device.
Material ? (?m) ? (K-1)
Silver 1.62?10-8 4.1?10-3
Copper 1.69?10-8 4.3?10-3
Aluminium 2.75?10-8 4.4?10-3
Tungsten 5.25?10-8 4.5?10-3
Iron 9.68?10-8 6.5?10-3
Platinum 10.6?10-8 3.9?10-3
Manganin 48.2?10-8 -70?10-3
Pure Silicon 2.5?103
n-type Silicon 8.7?10-4
p-type Silicon 2.8?10-3
Glass 1010-1014
Fused quartz 1016
A conducting material or device obeys Ohms law
when the resistivity is independent of the
magnitude and direction of the applied electric
field.
29
Current and Resistance
Example A rectangular block of iron has
dimension 1.2 cm?1.2 cm?15 cm. (a) What is the
resistance of the block measured between the two
square ends? (b) What is the resistance between
two opposite rectangular faces?
Solution Power Example A wire has a
resistance R of 72 ?. At what rate is energy
dissipated in each of the following situations?
(1) A potential difference of 120 V is applied
across the full length of the wire. (2) The wire
is is cut in half, and a potential difference of
120 V is applied across the length of each half.
Solution Example A wire of length L2.35 m
and diameter d1.63 mm carries a current i of
1.24 A. The wire dissipates electrical energy at
the rate P48.5 mW. Of what is the wire made?
Solution Its aluminum.
30
Current and Resistance
Semiconductor (optional) Semiconductors are
materials with intermediate conductivity between
conductors and insulators. From quantum
mechanics, the electrons of an atom may occupy
quantized energy levels. When a large amount of
atoms form a solid, the discrete energy levels
may be merged to form energy bands. The electrons
are only allowed to sit within the energy bands
but they cannot have an energy value within the
gaps separating the gaps. When excited (by
thermal activation, for example), electrons can
jump from the valence band to the conduction band
and leave holes in the valence band. The
conductivity of the material is
enhanced. Superconductors (optional) Superconduct
ivity is the lost of any conductivity at low
temperatures. It was first discovered in mercury
by K.Onnes in 1911. Since 1986, high temperature
(90K) superconductor in ceramics have been
discovered and developed.
31
Current and Resistance

• Homework
• A charged belt, 50 cm wide, travels at 30 m/s
  between a source of charge and a sphere. The
  belt carries charge into the sphere at a rate
  corresponding to 100 ?A. Compute the surface
  charge density on the belt.
• In the figure, a resistance coil, wired to an
  external battery, is placed inside a thermally
  insulated cylinder fitted with a frictionless
  piston and containing an ideal gas. A current
  i240 mA exists in the coil, which has a
  resistance R550 ?. At what speed v must the
  piston, of mass m12 kg, move upward to keep the
  temperature of the gas unchanged?