From last time
This week Honor Lecture
More on Capacitors and Capacitance (Human capacitors)
Energy stored in Capacitors
Current
Resistance
Ohm’s Law
Detecting the Body’s electrical signal
Prof. R. Moss
Dept. of Physiology
Today:
DC circuits
R and C in parallel and series
Kirchhoff’s laws
EMF
MTE 2: Ch 2103 5:30-7pm on Mar 26
Quick Quiz
Two cylindrical conductors are made from the same
material. They are of equal length but one has twice
the diameter of the other.
A. R1 < R2
Contact me and Prof. Rzchowski
after this lecture for Alternate
Exams (also by email asap!)
B. R1 = R2
C. R1 > R2
R="
L
A
1
2
!
3
4
Ohm’s Law
Ohm’s Law:
! ! = resistivity
!
Dependence on Temperature
L
J = ' E or E = ! J
! = !0 +"(#$#0)
!
E=
"V
I
=#
L
A
Define: R = !L/A
! Resistance in ohms (%)
! Ohm’s
!
law becomes: &V = R I
!
!
!
!
ohmic device: relationship current and voltage is linear
! of a conductor varies approximately
linearly with T
!
!
" x (!-!0)
!0
T-T0
!o is the resistivity at To = 20° C
" = temperature coefficient in SI units of
T0
T
oC-1
The higher T the greater atomic vibrations that
increases collision probability
Similarly: R = Ro[1 + "(T - To)]
But other materials…
Resistivity
Resistivity:
!
Semiconductors:
!
!
" is negative ( ! decreases for
!=1/'
SI units of % . m
! The resistance depends
resistivity and
geometry:
increasing T
!
!
!
" = " 0 (1+ #$T )
Mercury
Superconductors
!
Below a certain temperature, TC = critical
temperature
! is zero
!
!
Once a current is set up in a
superconductor, it persists without any
applied voltage since R = 0
Resistors control
the current level in
circuits
!Resistors can be
composite or wirewound
!
Resistors in Series
I = I2 = I
! 1
Resistors in parallel
2 resistors in series:
R!L
Like summing lengths
R
!
&V = &V1 + &V2
!
Req = R1+R2
on
conductor
R
2R
=
R=#
L
A
!
&V = &V1 = &V2
!
I=I1+I2
A A
I1
I2
&V1
"
Neglect
resistance of
wires in circuits
respect to R
resistance
&V2
I = I1 + I2 =
R
R
"V1 "V2 # 1
1&
"V
+
= % + ("V =
R1
R2 $ R1 R2 '
Req
!
2A
Add areas
"V = "V1 + "V2 = R1I1 + R2 I2 = (R1 + R2 )I = Req I
R="
!
L
A
!
Capacitors in series
Current Conservation: 1st Kirchoff’s law
I2
Same charge: Q1 = Q2 = Q
Share potential difference: !V1+!V2=!V
Add d: C = "0 A
I1
Iin
I3
I1=I2+I3
d
"V = "V1 + "V2 =
I1
electrons migrate to +
!
+
I3
Iout
!
I2
Iout = Iin
Junction Rule:
!
!V1
!V
I1+I2=I3
!
#1
Q1 Q2
1& Q
+
= Q% + ( =
C1 C2
C
C
Ceq
$ 1
2'
) Iin = ) Iout
!V2
-
A statement of Conservation of Charge
11
Capacitors in parallel
•As a charge "Q moves from a to b, the
Same voltage: !V1=!V2=!V
Share Charge: Q = Q1+Q2
Add Areas:
! 0A
C=
Electrical power
electric potential energy of the
system increases by "Q!V and the
chemical energy in the battery
decreases by this same amount
d
•As the charge moves through the
resistor (c to d), the system loses this
electric potential energy during
collisions of the electrons with the
Power = rate of energy loss
atoms of the resistor
C1
!V
!V !V1=!V2
•This energy is transformed into internal
Q = Q1 + Q2 = C1"V1 + C2"V2 = (C1 + C2 )"V = Ceq "V
energy in the resistor (vibrational
motion of the atoms in the resistor)
•Power in units of Watts = J/s
"U "Q
=
"V
"t
"t
"V 2
P = I"V = RI 2 =
R
P=
14
!
!
Quick Quiz
2 Identical Light Bulbs in parallel and series
B)
A)
How does brightness of bulb B compare to that of A?
A)
A. B brighter than A
B)
B. B dimmer than A
"
"
C. Both the same
Which of these statements is true?
A) Parallel connection is more luminous
B) Series connection is more luminous
C) Both connections have the same luminosity
15
Emf and real battery
!
!
A battery: device supplying electrical energy to circuit
Ideal battery: const. voltage difference between terminals "V = # and
I =*/R
!
Real battery: &V = * - Ir = RI so the battery voltage at its terminals
Electrical measurements: Ammeter
!
!
A multimeter can measure currents (as an ammeter), potential
difference (as a voltmeter)
Electrical measuring devices must have minimal impact in the
circuit
!V
depends on the load resistance R, since !V higher for smaller I (higher R)
!
The electromotive force *: max possible voltage that the battery
can provide between its terminals (I=0)
!V
&V= terminal
voltage < *
In the absence of the ammeter: !VA
I = */R
R=R1+R2
In the presence of the ammeter:
I = */(R + rA) + */R
for rA +0 the ammeter does not
change the behavior of the circuit
I
IA
Electrical measurements: Voltmeter
Loop Rule:
&VV
Voltmeter
V
IV
R
IR
I
Kirchhoff’s Rules
!
!
&VR
I
*
In the absence of the voltmeter: IR = */R
In the presence of the voltmeter:
I = Iv+IR
"
I= +
&VV = &VR= *
R
V
"Vloop = # "Vk = 0
A statement of Conservation of Energy
k
Remember that a charge that moves around a closed loop back to the starting point
has potential energy difference "U=0 (conservative electric force)
-
+
I
+
" RV #$ "
% %%#
R
R
potential
! increases
potential decreases
-
+
potential decreases
-
If the resistance of the voltmeter is very large it will not a"ect the
behavior of the circuit
+
potential increases
!
Kirchoff’s laws application
What is the current?
25#
5#
I1
6V
12V
5#
40#
"1 # "2 # (r1 + R1 + R2 + r2 + R3 )I = 0
I=
I2
25#
I3
What happens if the
polarity of "2 is
reversed? Does I
increase or decrease?
I1 = I3 + I2 " I3 = I1 # I2
8V + 4V # 4V #1$I1 # 2$I1 # 2$(I1 # I2 ) = 0
#4V # 6$I2 # 2$(I2 # I1 ) = 0
"1 # "2
6
=
= 0.06A
r1 + R1 + R2 + r2 + R3 100
21
!
22
!
!
An example of DC circuit
Resistors control the
current level in circuits!
Which is the current in the circuit when S1 is open and S2 closed?
A)
I=
# 1 1&
"
= 18% + ( = 4.5A
$12 6 '
Req
B)
I=
"
18
=
= 3A
Req 2 + 4
!
C)
!
!
I=
"
18
=
= 8A
Req 2 + 1
4
2 loops
Assume 1 current verse
per loop
Which of this cases for
S1 closed and S2 closed?
23