Physics 122B Electricity and Magnetism

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Physics 122B Electricity and Magnetism
Lecture 16 Review and Extension May 2, 2007

• Martin Savage

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Lecture 16 Announcements

• Lecture HW Assignment 5 is due by 10pm this
  evening.
• Lecture HW Assignment 6 is posted on Tycho and
  is due next Wednesday at 10pm.
• The second midterm exam is this Friday. It will
  cover everything up to the end of Wednesdays
  lecture, emphasizing the most recent material,
  but assumes understanding of all material
  inclusive.

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Current and Drift Velocity
If the electrons have an average driftspeed
vd, then on the average in a timeinterval Dt
they would travel a distance Dx in the wire,
where Dx vd Dt. If the wire has cross
sectional area A and there are n electrons per
unit volume in the wire, then the number of
electrons moving through the cross sectional area
in time Dt is Ne n A Dx n A vd Dt i Dt .
Therefore,
This table gives n for various metals.
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A Puzzle
We discharge a capacitor that has been
given a charge of Q 16 nC, using a copper wire
that is 2 mm in diameter and has a length of L
20 cm. Assume that the electron drift speed is
vd 10-4 m/s. How long does it take to
discharge the capacitor? (Note that L/vd
0.2m/10-4 m/s 2000 s 33.3 min.)

• Points to consider
• The wire is already full of electrons.
• The wire contains about 5x1022conduction
  electrons.
• Q 16 nC requires about 1011 electrons.
• A length L of wire that holds 16 nCof
  conduction electrons is 4x10-13 m.
• L/vd 4x10-9 s 4 ns. That is roughly the
  discharge time.

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Establishing theElectric Field in a Wire (2)

• The figure shows the region of the wire near the
  neutral midpoint. The surface charge rings
  become more positive to the left and more
  negative to the right.
• In Chapter 26, we found that a ring of charge
  makes an on-axis E field that
• Points away from a positive ring and toward a
  negative ring
• Is proportional to the net charge of the ring
• Decreases with distance from the ring.

The non-uniform surface charge distribution
creates an E field inside the wire. This pushes
the electron current through the wire
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A Model of Conduction (1)
Now turn on an E field. The straight-line
trajectories become parabolic, and because of the
curvature, the electrons begin to drift in the
direction opposite E, i.e., downhill.
axF/meE/m so vxvix axDt vix Dt eE/m
This acceleration increases an electrons kinetic
energy until the next collision, a friction
that heats the wire.energy is imparted to the
atoms of the lattice.
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The Current Density in a Wire
Example A current of 1.0 A passes through a 1.0
mm diameter aluminum wire. What is the drift
speed of the electrons in the wire?
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Kirchhoffs Junction Law
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Conductivity and Resistivity
The current density J nevd is directly
proportional to the electron drift speed vd. Our
microscopic conduction model gives vd etE/m,
where t is the mean time between collisions.
Therefore
The quantity ne2t/m depends only on the
properties of the conducting material, and is
independent of how much current density J is
flowing. This suggests a definition
J s E

• This result is fundamental and tells us
  three things
• Current is caused by an E-field exerting forces
  on charge carriers
• (2) Current density J and current IJA depends
  linearly on E
• (3) Current density J also depends linearly on
  s. Different materials have different s values
  because n and t vary with material type.

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Resistivity andConducting Materials
For many applications, it is more
convenient to use inverse of conductivity, which
is called the resistivity, denoted by the symbol
r
Thus, the current density is J Es E/r.
Here are the conducting properties of common
materials
Units of resistivity are W m
Unitsohms W Nm2/CA Nm2s/C2
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Resistors and Resistance
Conducting material that carries current
along its length can form a resistor,a circuit
element characterized by anelectrical resistance
R R rL/Awhere L is
the length of the conductor and A is its cross
sectional area. R has units of ohms ( W ).
Multiple resistors may be combined in series,
where resistances add, or in parallel, where
inverse resistances add.
For identical resistors can simply add the areas
For identical resistors can simply add the
lengths
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The Potential Energy ofLike and Unlike Charge
Pairs
This approach can be applied to pairs of
electrically charged particles, whether they have
the same or opposite charges. However, for
like-sign particles (a) the system energy is
positive and decreases with separation, while for
opposite-sign particles (b) the system is
typically bound, so that the net energy is
negative and increases (closer to zero) with
increasing separation.
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The Electric Force asa Conservative Force
The electrical force is a conservative
force, in that the amount of energy involved in
moving from point i to point f is independent of
the path taken. This can be demonstrated in
the field of a single point charge by observing
that tangential paths involve no change in energy
(because r is constant). Therefore, an arbitrary
path can be approximated by a succession of
radial and tangential segments, and the
tangential segments eliminated. What remains
is a straight line path from the initial to the
final position of the moving charge, indicating a
net work that will be the same for all possible
paths.
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Multiple Point Charges
We have established that both energy and
electrical forces obey the principle of
superposition, i.e., they can be added linearly
without cross terms. Therefore, for multiple
point charges
Here, iltj means that for summing over N
particles, the sum over i runs from 1 to N, and
the sum over j runs from i1 to N for each value
of i. This it a mathematical trick to avoid
counting pairs of point charges twice or having
ij terms, which would give a zero in the
denominator.
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The Electric Potential
In Chapter 25 we introduced the concept of
an electric field E, which can be though of as a
normalized force, i.e., E F/q, the field E
that would produce a force F on some test charge
q. We can similarly define the electric
potential V as a charge-normalized potential
energy, i.e., VUelec/q, the electric potential V
that would give a test charge q an electric
potential energy Uelec because it is in the field
of some other source charges.
We define the unit of electric potential as
the volt 1 volt 1 V 1 J/C 1 Nm/C. Other
units are kV103 V, mV10-3 V, and mV10-6
V. Example A D-cell battery has a potential of
1.5 V between its terminals.
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The Electric Potential Insidea Parallel Plate
Capacitor
Consider a parallel-plate capacitor with
(with U00)
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Graphical Representationsof Electric Potential
Distance from plate
This linear relation can be represented as a
graph, a set of equipotential surfaces, a contour
plot, or a 3-D elevation graph.
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Field Lines and Contour Lines
Field lines and equipotential contour lines
are the most widely used representations to
simultaneously show the E field and the electric
potential. The figure shows the field lines and
equipotential contours for a parallel plate
capacitor. Remember that for both the
field lines and contours , their spacing, etc, is
a matter of choice.
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Rules for Equipotentials

1. Equipotentials never intersectother
   equipotentials. (Why?)
2. The surface of any staticconductor is an
   equipotentialsurface. The conductor volumeis
   all at the same potential.
3. Field line cross equipotentialsurfaces at right
   angles. (Why?)
4. Dense equipotentials indicate astrong electric
   field. The potential V decreases in the
   direction in which the electric field E points,
   i.e., energetically downhill for a charge
5. For any system with a net charge, the
   equipotential surfaces become spheres at large
   distances.

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Visualizing the Potentialof a Point Charge
The potential of a point charge can be
represented as a graph, a set of equipotential
surfaces, a contour map, or a 3-D elevation
graph. Usually it is represented by a graph
or a contour map, possibly with field lines.

Spherical Shells
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The Electric Potentialof Many Charges
The principle of superposition allows us to
calculate the potentials created by many point
charges and then add the up. Since the potential
V is a scalar quantity, the superposition of
potentials is simpler than the superposition of
fields.
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Example The Potentialof Two Charges
p
What is the potential at point p?
Note that 1/4pe0 9.0 x 109 Nm2/C2
9.0 x 109 Vm/C,which, for problems like this,
are more convenient units.
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Potential of a Disk of Charge
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Example The Potential of a Dime

• A dime (diameter 17.5 mm) is given a charge
  of Q5.0 nC.
• What is the potential of the dime at its surface?
• What is the potential energy Ue of an electron
  1.0 cm above the dime (on axis)?

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Finding E from V
In other words, the E field components are
determined by how much the potential V changes in
the three coordinate directions.
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ExampleFinding E from the Slope of V
An electric potential V in a particular
region of space where E is parallel to the x axis
is shown in the figure to the right. Draw Ex
vs x.
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Example Finding the E-Field from Equipotential
Surfaces
The figure shows a contour map of a
potential. Estimate the strength and
direction of the electric field at points 1, 2,
and 3.

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Kirchhoffs Loop Law
Since the electric field isconservative,
any path betweenpoints 1 and 2 finds the same
potential difference. Any path can be
approximated by segments parallel and
perpendicular to equipotential surfaces, and the
perpendicular segments must cross the same
equipotentials. Since a closed loop starts
and ends at the same point, the potential around
the loop must be zero. This is Kirchhoffs Loop
Law, which we will use later.
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A Conductor inElectrostatic Equilibrium
A conductor is in electrostatic equilibrium
if all charges are at rest and no currents are
flowing. In that case, Einside0. Therefore,
all of it is at a single potential
Vinsideconstant.
Rules for conductor.
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Forming a Capacitor
Any two conductors can form a capacitor,
regardless of their shape.
The capacitance depends only on the
geometry of the conductors, not on their present
charge or potential difference.
(In fact, one of the conductors can be moved
to infinity, so the capacitance of a single
conductor is a meaningful concept.)
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Combining Capacitors
Parallel Same DV, but different Qs.
Series Same Q, but different DVs.
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Energy Stored in a Capacitor
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Energy in the Electric Field
Volume of E-field
Example d1.0 mm, DVC500 V
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Dielectric Materials
There is a class of polarizable dielectric
materials that have an important application in
the construction of capacitors. In an electric
field their dipoles line up, reducing the E field
and potential difference and therefore increasing
the capacitance
E off
E on
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Electric Fields and Dielectrics
In an external field EO, neutral molecules
can polarize. The induced electric field E
produced by the dipoles will be in the opposite
direction from the external field EO. Therefore,
in the interior of the slab the resulting field
is E EO-E. The polarization of the
material has the net effect of producing a sheet
of positive charge on the right surface and a
sheet of negative charge on the left surface,
with E being the field made by these sheets of
charge.
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Capacitors and Dielectrics
If a capacitor is connected to a battery, so
that it has a charge q, and then a dielectric
material of dielectric constant ke is placed in
the gap, the potential is unchanged but the
charge becomes keq.
If a capacitor is given a charge q, and then
a dielectric material of dielectric constant ke
is placed in the gap, the charge q is unchanged,
but the potential drops to V/ke.
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Resistors and Ohms Law
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Ohmic and Non-ohmic Materials
Despite its name, Ohms Law is not a law of
Nature (in the sense of Newtons Laws). It is a
rule about the approximately linear
potential-current behavior of some materials
under some circumstances.

• Important non-ohmic devices
• Batteries, where DVE is determined by chemical
  reactions independent of I
• Semiconductors, where I vs. DV can be very
  nonlinear
• Light bulbs, where heating changes R
• Capacitors, where the relation between I and DV
  differs from that of a resistor.

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The Ideal Wire Model

• In considering electric circuits, we will make
  the following assumptions
• Wires have very small resistance, so that we can
  take Rwire0 and DVwire0 in circuits. Any wire
  connections are ideal.
• Resistors are poor conductors with constant
  resistance values from 10 to 108 W.
• Insulators are ideal non-conductors, with R8 and
  I0 through the insulator.

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Circuit Elements Diagrams
These are some of the symbols we will use to
represent objects in circuit diagrams.
Other symbols inductance, transformer, diode,
transistor, etc.
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Applying KirchhoffsLoop Law to Many Loops
i1 i3

1. Define a minimum set of current loops. Label all
   elements.
2. Write a loop equation for each loop. (Battery or
   0 SDV).
3. Solve equations for currents
4. Calculate other variables of interest.

R1
i3
R2
i1
Vbat
R4
R3
R5
Loop equations
i2
i1 i2
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Applying Kirchhoffs JunctionLaw to Many
Junctions

1. Define a minimum set of junction potentials. You
   can select one ground point. Label all elements.
2. Write a junction equation for each unknown
   junction .
3. Solve these equations for the unknown junction
   potentials.
4. Calculate the other variables of interest.

J1
R1
R5
Vbat
V1
R3
R2
R4
Junction equations
J2
V2
V0
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Energy and Power (1)
Example A 90 W load resistance is connected
across a 120 V battery. How much power is
delivered by the battery?
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Real Batteries (2)
DVbat
Question How can you measure rint? Answer One
(rather brutal) way is to vary an external load
resistance R until the potential drop across R is
½E. Then Rrint because each drops ½E.
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Voltmeters vs. Ammeters
An ideal voltmeter has infinite internal
resistance. It must be connected between circuit
elements to measure the potential difference
between two points in the circuit.
An ideal ammeter has zero internal
resistance. It must be inserted by breaking a
circuit connection to measure the current flowing
through that connection in the circuit.
X
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ExampleAnalyzing a Complex Circuit (2)
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Lecture 16 Announcements

• Lecture HW Assignment 5 is due by 10pm this
  evening.
• Lecture HW Assignment 6 is posted on Tycho and
  is due next Wednesday at 10pm.
• The second midterm exam is this Friday. It will
  cover everything up to the end of Wednesdays
  lecture, emphasizing the most recent material,
  but assumes understanding of all material
  inclusive.