Transcript phys1442-lec11

PHYS 1442 – Section 004
Lecture #10-Rev
Monday February 17 2014
Dr. Andrew Brandt
•
•
Weds. Feb. 5, 2014
CH 18
Alternating Current
CH 16-19
Short review
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Alternating Current
• Does the direction of the flow of current change when a battery
is connected to a circuit?
– No. Why?
• Because its source of potential difference is constant.
– This kind of current is called the Direct Current (DC
• How would DC look as a function of time?
– A horizontal line
• Electric generators at electric power plant produce alternating
current (AC)
– AC reverses direction many times a second
– AC is sinusoidal as a function of time
• Most currents supplied to homes and
business are AC.
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Alternating Current
• The voltage produced by an AC electric generator is
sinusoidal
– This is why the current is sinusoidal
• Voltage produced can be written as
V = V0 sin 2p ft = V0 sin wt
• What are the maximum and minimum voltages?
 V0 and –V0
 The potential oscillates between +V0 and –V0, the peak voltages or
amplitude
 What is f ?
• The frequency, the number of complete oscillations made per second. What is
the unit of f ? What is the normal size of f in the US?
– f = 60 Hz in the US and Canada.
– Many European countries have f = 50Hz.
 w=2pf
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Alternating Current
• Since V=IR, if a voltage V exists across a resistance R, the
What is this?
current I is
V V0
I= =
sin 2p ft = I 0 sin wt
R R
• What are the maximum and minimum currents?
– I0 and –I0
– The current oscillates between +I0 and –I0, the peak currents or
amplitude. The current is positive when electron flows in one
direction and negative when they flow in the opposite direction.
– What is the average current?
• Zero. So there is no power and no heat produced in a heater?
– Wrong! The electrons actually flow back and forth, so power is delivered.
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Power Delivered by Alternating Current
• AC power delivered to a resistance is:
P = I 2 R = I 02 R sin 2 wt
– Since the current is squared, the power is always positive
1
P = I 02 R
2
• The average power delivered is
• Since the power is also P=V2/R, we can obtain
P=

V02

1  V0
P= 
2  R
2
R sin wt
2
Average power



• The average of the square of current and voltage are
important in calculating power: 2 1 2
1 2
2
I =
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I0
V = V0
2
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Power Delivered by Alternating Current
• The square root of each of these are called root-mean-square, or rms:
I rms = I 2 =
I0
2
= 0.707 I 0
Vrms = V 2 =
V0
2
= 0.707V0
• rms values are sometimes called effective values
– These are useful quantities since they can substitute current and voltage directly in
power equations, as if they were DC values
1 2
2
P = I 0 R = I rms
R
2
2
2
1 V0 Vrms
P=
=
2 R
R
P = I rmsVrms
– In other words, an AC of peak voltage V0 or peak current I0 produces as much
power as DC voltage of Vrms or DC current Irms.
– So normally, rms values in AC are specified or measured.
• US uses 115 to 120 V rms voltage. What is the peak voltage?
V0 = 2V =
rms
2  120V = 170V
• Europe uses 240V
2  240V = 340V
V0 = 2Vrms =
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Superconductivity
• At temperatures near absolute 0K, the resistivity of certain
materials approaches 0.
– This state is called the “superconducting” state.
– Observed in 1911 by H. K. Onnes when he cooled mercury to 4.2K
(-269oC).
• Resistance of mercury suddenly dropped to 0.
– In general superconducting materials become superconducting
below a transition temperature.
– The highest temperature superconductor so far is 160K
• First observation above the boiling temperature of liquid nitrogen is in 1987 at
90K observed from a compound of yttrium, barium, copper and oxygen.
• Since a much smaller amount of material can carry just as
much current more efficiently, superconductivity can make
electric cars more practical, computers faster, and capacitors
store higher energy (not to mention LHC magnets)
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Electric Hazards: Leakage Currents
• How does one feel an electric shock?
– Electric current stimulates nerves and muscles, and we feel a shock
– The severity of the shock depends on the amount of current, how
long it acts and through what part of the body it passes
– Electric current heats tissues and can cause burns
• Currents above 70mA on a torso for a second or more is fatal,
causing heart to function irregularly, “ventricular fibrillation”
• Dry skin has a resistance of 104 to 106 W.
• When wet, it could be 103W.
• A person in good contact with the ground who touches 120V
DC line with wet hands can receive a fatal current
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V 120V
= 120mA
I= =
R 1000W 11
Magnetism
• So are magnet poles analogous to electric charge?
– No. Why not?
– While the electric charges (positive and negative) can be isolated,
magnet poles cannot.
– So what happens when a magnet is cut?
• You get two magnets!
• The more they get cut, the more magnets are made
– Single-pole magnets are called “monopoles,” but to date none have
been observed…
• Ferromagnetic materials: Materials that show strong
magnetic effects
– Iron, cobalt, nickel, gadolinium and certain alloys
• Other materials show very weak magnetic effects
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Magnetic Field
• Just like an electric field surrounds electric charge, a magnetic field
surrounds a magnet
• What does this mean?
– Magnetic force is also a field force
– The force one magnet exerts on another can be viewed as the interaction
between the magnet and the magnetic field produced by the other magnet
– What kind of quantity is the magnetic field? Vector or Scalar? Vector
• So one can draw magnetic field lines, too.
– The direction of the magnetic field is tangent to a line at
any point
– The direction of the field is the direction the north pole
of a compass would point to
– The number of lines per unit area is proportional to the
strength of the magnetic field
– Magnetic field lines continue inside the magnet
– Since magnets always have both poles, magnetic field
lines form closed loops, unlike electric field lines
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Earth’s Magnetic Field
• Which way does a compass point?
• So the magnetic pole of the geographic North pole is …
– Yep South!
– Since the magnetic north pole points to the geographic north, the
geographic north must have magnetic south pole
• The pole in the north is still called geomagnetic north pole just because it
is in the north
– Similarly, south pole has magnetic north pole
• To add confusion: the Earth’s magnetic poles
do not coincide with the geographic poles 
magnetic declination
– Geomagnetic north pole is in northern Canada,
some 1300km off the true north pole
• Earth’s magnetic field line is not tangent to
the earth’s surface at all points
– The angle the Earth’s field makes to the
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October 16, 2012
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horizontal
line is called
the
angleDr.of
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Electric Charge and Conservation
• Two types of electric charge
– Like charges repel while unlike charges attract
• The net amount of electric charge
produced in any process is ZERO!!
• When a positively charged metal object is
brought close to an uncharged metal object
– If the objects touch each other, the free charges
flow until an equilibrium state is reached
(charges flow in a conductor.)
– If the objects are close, the free electrons in the
neutral object still move within the metal toward
charged
object leaving the opposite end of
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the object positively charged.(induced charge)
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Coulomb’s Law – The Formula
Q11
Q22
Q
F
2
r
Formula
A vector quantity. Newtons
Q1Q2
F =k
2
r
• Direction of electric (Coulomb) force (Newtons) is always
along the line joining the two objects.
• Unit of charge is called Coulomb, C, in SI.
• Elementary charge, the smallest charge, is that of an
electron: -e where
19
e = 1.602  10
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C
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Vector Problems
• Calculate magnitude of vectors (Ex.
force using Coulomb’s Law)
• Split vectors into x and y components
and add these separately, using
diagram to help determine sign
• Calculate magnitude of resultant
|F|=(Fx2+Fy2)
• Use = tan-1(Fy/Fx) to get angle
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Angle:
After calculating magnitudes, take x+y components and then get total force
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Example 21-8
Calculate the total
electric field (a) at
point A and (b) at point
B in the figure due to
both charges, Q1 and
Q2.
Solution: The geometry is shown in the figure. For each point, the process is:
calculate the magnitude of the electric field due to each charge; calculate the x and
y components of each field; add the components; recombine to give the total field.
a.Tuesday,
E = 4.5October
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x axis. Dr. Andrew Brandt
2012 76° above
PHYS
19
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b. E = 3.6 x 10 N/C, along the x axis.
Example
•
Electron accelerated by electric field. An electron (mass m = 9.1x10-31 kg)
is accelerated from rest in a uniform field E (E = 2.0x104 N/C) between two
parallel charged plates (d=1.5 cm), andpasses through a tiny hole in the
positive plate.
(a) With what speed does it leave the hole?
F = qE = ma
v2 = v02  2ax
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Electric Potential Energy
• Concept of energy is very useful solving
mechanical problems
• Conservation of energy makes solving complex
problems easier.
• Defined for conservative forces (independent of
path)
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The Electric Field
• The electric field at any point in space is defined as the
force exerted on a tiny positive test charge
r divided by
r
magnitude of the test charge E = F = 1 Q
q
4p 0 r 2
• The electric field inside a conductor is ZERO in a
static situation
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Electric Potential Energy
• What is the definition of change in electric potential energy Ub –Ua?
– The potential gained by the charge as it moves from point a to point b.
– The negative work done on the charge by the electric force to move it from a
to b.
• Parallel plates w/ equal but opposite charges
– The field between the plates is uniform since the gap is
small and the plates are infinitely long…
• What happens when we place a small charge, +q,
on a point at the positive plate and let go?
– The electric force will accelerate the charge toward
negative plate and it gains kinetic energy
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Electric Potential
• The electric field (E) is defined as electric force
per unit charge: F/q (vector quantity)
• Electric potential (V) is defined as electrical
potential energy per unit charge U/q (scalar)
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Comparisons of Potential Energies
• Let’s compare gravitational and electric potential energies
m
•
2m
What are the potential energies of the rocks?•
– mgh and 2mgh
•
– +QVba and +2QVba
Which rock has a bigger potential energy? •
– The rock with a larger mass
•
Why?
– It’s got a bigger mass.
What are the potential energies of the charges?
Which object has a bigger potential energy?
– The object with a larger charge.
•
Why?
– It’s got a bigger charge.
Tuesday, October
2012same but the
PHYSheavier
1444-003 rock
Dr. Andrew
Brandt charge can do a greater25work.
The “potential”
is 2the
or larger
Properties of the Electric Potential
• What are the differences between the electric potential and
the electric field?
1 Q
– Electric potential (U/q)
V=
4p0 r
• Simply add the potential from each of the charges to obtain the total potential
from multiple charges, since potential is a scalar quantity
r
1 Q
– Electric field (F/q)
E=
4p0 r2
• Need vector sums to obtain the total field from multiple charges
• Potential for a positive charge is large near a positive charge
and decreases to 0 at large distances.
• Potential for the negative charge is small (large magnitude but
negative) near the charge and increases with distance to 0
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Electrostatic Potential Energy; Three Charges
• Work is needed to bring all three charges together
– Work needed to bring Q1 to a certain place without the presence
of any charge is 0.
1 QQ
1 2
– Work needed to bring Q2 to a distance to Q1 is U12 =
4p0 r12
– Work need to bring Q3 to a distance to Q1 and Q2 is
U 3 = U13  U 23
1 Q
1 QQ
1Q
3
2 3

=
4p0 r13
4p0 r23
• So the total electrostatic potential of the three charge
system is


Q
Q
Q
Q
1
1
3Q
2
3
1
2Q
Vr
=
0
a
t=



U
U
U= U
 
1
2
1
3
2
3=
4
r
0
1
2 r
1
3 r
2
3


p

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Capacitors
• A simple capacitor consists of a pair of parallel plates
of area A separated by a distance d.
– A cylindrical capacitors are essentially parallel plates
wrapped around as a cylinder.
Circuit
Diagram
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Capacitors
• If a battery is connected to a capacitor, the capacitor gets
charged quickly, one plate positive and the other negative
with an equal amount..
• For a given capacitor, the amount of charge stored in the
capacitor is proportional to the potential difference Vba
between the plates. C is a proportionality constant, called
capacitance of the device.
Q = CVba
C is a property of a capacitor so does not depend on Q or V.
• See Ex. 24.1 for example
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Dielectrics
• Capacitors generally have an insulating sheet of
material, called a dielectric, between the plates to
– Increase the breakdown voltage above that in air
– Allows the plates get closer together without touching
• Increases capacitance ( recall C=0A/d)
– Also increases the capacitance by the dielectric constant
C = KC0
• Where C0 is the intrinsic capacitance when the gap is vacuum,
and K or  is the dielectric constant
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Electric Current
• Electric Current: Any flow of charge
– Current can flow whenever there is potential difference between the
ends of a conductor
– Electric current in a wire can be defined as the net amount of charge
that passes through a wire’s full cross section at any point per unit
time
– Average current is defined as: I = Q t
– Current is a scalar
1A=1C/s
C/s
– Current is flow of charge, charge is conserved,
so current in equals current out at a given point on circuit
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Ohm’s Law: Resistance
• The exact amount of current flow in a wire depends on
– The voltage
– The resistance of the wire to the flow of electrons
• The higher the resistance the less the current for the given
potential difference V
V = IR
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1.0W = 1.0V / A
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Resistivity
• It is experimentally found that the resistance R of a metal wire
is directly proportional to its length l and inversely proportional
to its cross-sectional area A
l
A
R=r
A
l
– The proportionality constant r is called the resistivity and depends
on the material used. The higher the resistivity the higher the
resistance
– The reciprocal of the resistivity is called the conductivity, s,
s=
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1
r
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Electric Power
• Power -the rate at which work is done or the energy is
transferred
• P=IV can apply to any devices while the formulae
involving resistance only apply to Ohmic resistors.
I2R used for heat loss
2
V
P = I2R =
R
Temperature
dependence
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rT = r0 1   T  T0 
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1444 Test I Eq. Sheet
Electron charge:
e = 1.602  10
19
C
Electron mass: me = 9.1x10-31 kg
Proton mass: mp =1.67x10-27 kg
Colomb’s Law:
Q1Q2 k = 8.988  109 N  m2 C 2
F =k
 0 = 1 4p k = 8.85 1012 C 2 N  m2
r2
Dipole:
r
r
r
r r
 = p E U = p  E

El =
For a point charge: | E |=
1 Q
V=
4p 0 r
dV
dl
1
Q
4p 0 r 2
V=Ed (uniform field)
Eqs.
ofOctober
motion:
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2 2012
vxf = vxi  axt
xf
K.E.=mv2/2
Ugrav=mgh g=9.8 m/s2
Uelec=qV
Q=CV C=capacitance
 A
parallel plate: C =  0
d
dielectric:   1
Cap. stored energy:U = Q2C
Ohm’s Law: V=IR
Power: P=IV
Current: I=q/t
AC: V = V0 sin 2p ft
l s=1
R
=
r
Resistivity:
r
A
2
r r
Flux:  E = E  dAr r Q
encl
E

dA
=
Gauss Law: Ñ

0
r
r
Electric Field: E = F
q
|F|=(Fx2+Fy2)
= tan-1(Fy/Fx)
rT = r0 1   T  T0 
Ceq=C1+C2 (parallel)
1 2
1/Ceq=1/C1+1/C2 (series)
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=PHYS
xi 1444-003
vxit  Dr. Andrew
axt Brandt 2
2
2
0
v = v  2ax
A quiz
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Example
Will a 30A fuse blow?
Determine the total current drawn
by all the devices in the circuit in
the figure.
The total current is the sum of current
drawn by the individual devices.
P = IV
Bulb
Solve for I
I =PV
I B = 100W 120 V = 0.8 A
Stereo I S = 135W 120 V = 2.9 A
Heater I H = 1800W 120 V = 15.0 A
Dryer I D = 1200W 120 V = 10.0 A
Total current
I T = I B  I H  I S  I D = 0.8 A  15.0 A  2.9 A  10.0 A = 28.7 A
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What
is the
total
power?
PB  PH Dr.
 PAndrew
= 100W  1800W  350W  1200W37= 3450W
PPHYS
S  PD Brandt
T = 1444-003