MTE 2: Ch 2103 5:30-7pm on Mar 26
Contents of MTE2
!
Contact me and Prof. Rzchowski
after this lecture for Alternate
Exams (also by email asap!)
2:30-4pm
6:00-7:30pm
on Mar 26
Office hrs change this week
Wed morning
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Work of the electric force and potential energy
Electric Potential and Field
Capacitors and capacitance
Current and resistance, Ohm’s law
DC Circuits and Kirchoff’s laws
RC circuits
Lorentz force and motion of charge in a magnetic field
Biot and Savart
No Ampere’s law, No Magnetic Properties of matter (32.6,
32.10)
Study chapters 27-32 (no 32.6, no 32.10)
1
RC Circuits and Magnetism
Current Conservation: 1st Kirchoff’s law
This Lecture:
I2
•More on Kirchhoff’s laws
•RC Circuits
•Magnets and B-field
•Lorentz force and motion of charge in a B-field
I1
Iin
I3
I1=I2+I3
I1
I3
From previous lecture:
•Connections of resistors and capacitors
•Batteries and Electromotive Force
•DC circuits and measurements of currents
and potential difference
•1st Kirchoff’s laws
Iout
I2
Iout = Iin
I1+I2=I3
!
Junction Rule:
!
! Iin = ! Iout
A statement of Conservation of Charge
4
Kirchhoff’s Rules: energy conservation
Loop Rule:
!
!
"V
What is the current and the power?
= # "V = 0
loop
k
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
-
potential increases
!
+
25!
5!
potential decreases
12V
+
6V
-
+
40!
I
25!
5!
potential decreases
"1 # "2 # (r1 + R1 + R2 + r2 + R3 )I = 0
-
+
I=
potential increases
"1 # "2
6
=
= 0.06A
r1 + R1 + R2 + r2 + R3 100
Power
dissipated in
resistors is
P = RI2
Power
produced by
battery is
P = "1 I - "2 I
6
!
!
Kirchoff’s laws application
I1
2 loops
Assume 1 current verse
per loop
I2
In the Lab this week
RC Circuits
Until now, circuits with resistors and batteries where the
current is constant in wires (Direct Current Circuits).
Connections of R and C: current varies with time during
charge and discharge of C.
Charge: S2 open and S1
closed (C connected to
battery)
Discharge: S1 open, S2
closed (battery
disconnected from C)
I3
"
I1 = I3 + I2 " I3 = I1 # I2
8V + 4V # 4V #1$I1 # 2$I1 # 2$(I1 # I2 ) = 0
#4V # 6$I2 # 2$(I2 # I1 ) = 0
7
!
How fast does a C charge or
discharge?
!
RC Circuits: charge
At t=0 C is uncharged and S1 is closed (S2 open). Current flows in C
and it starts to charge it. C behaves like a short circuit (VC=q/C=0
because q=0) and I0 = I(t=0) = "/R.
At any t the potential difference at the battery terminals equals the
potential difference on R and C and the charge increases on C:
The time it takes to charge or discharge a
capacitor in an RC circuit depends on the time
constant
Ohm’s law
Current definition
$ "V
Q ' $ Q"t '
#
=&
)
% I
"V )( % Q (
[RC] = &
I=
It is a time! It is easily measurable by you in the
lab if it is of the order of fractions of seconds
!
Eg R 100k% and C 1µF ! RC 0.1 s
" = #VR + #VC = RI +
I" = I(t # ") = 0
!
9
!
Current and charge vs time
Differentiate
" = RI +
!
q
dI I
dI
dt
#0=R + #
=$
C
dt C
I
RC
I(t)=I0 e-t/RC
q(t) = C"(1 – e-t/RC)
VC=q/C VR=RI
2
2 2
Energy stored in C is U = Q = C " = 1 C" 2
2C
2C
2
provided by the battery
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31
"
q" = C#
!
!
dq
dt
A simple rule of thumb
!
When the capacitor is initially connected to the
battery it is uncharged and it behaves as a short
circuit
I0 = I(t=0) = "/R.
!
After 3#, C is 95% charged
q
C
The current becomes exponentially zero when the max charge is reached because the
potential difference across the capacitor matches that supplied by the battery
I from I0 = "/R exponentially goes to
zero while the charge builds up on C
After 1#, charge increases from 0 to
C"(1-e-1) = 63.2% of its max value C"
!
I=
dq
dt
!
When C is fully charged I = 0 and it behaves as
an open circuit
I# = I(t$#0) = 0
12
Question:
Discharging a Capacitor in an RC Circuit
Exclude battery
close S" 2
!
!
dq
Q "t / RC
="
e
dt
RC
!
q decreases exponentially
In 1# = RC, q decreases from initial
value Q to 36.8% Q
In 1# = RC, current decreases from
I0 = Q/CR = "/R to 36.8% I0
!
!
!
!
A
B
C
All 3 are equally bright.
!
!
!
15
Biological Membrane Electrical Model
!
!
!
!
A
B
C
A and B
Cell Membranes
The circuit contains 3 identical light bulbs and a
capacitor. At the instant the switch S is closed (C
uncharged), which light bulb/s is/are brightest?
!
This circuit contains 3 identical light bulbs and a
capacitor. Which light bulb(s) is (are) dimmest?
The capacitor is fully charged
14
Question:
!
!
!
q = Qe-t/RC
I(t) = "
!
q/C = RI I = -dq/dt
Lipid bilayers of cell membranes like capacitors
Typical capacitance: 35 pF (in reality factor of 10
larger because it is not an empty capacitor)
Resting potential (voltage of inactive cell)= excess of negative charge inside the cell.
The cell becomes depolarized when it undergoes an action potential = rapid change of
polarity from - to + due to an influx of Na+ and back from + to - due to K+
outflux. Current of 70 ions per ms
Prof Moss lecture!
Magnetism
The cell membrane can be modeled as an RC circuit with time constants in
the range from 10 µs to 1 s = (RA)(C/A)
C results from the separation of charges across the bilayer of lipids: C/A = 1
µF/cm2
R results from the behavior of ion channels: R = $L/A % R A = $L = 10-106
& cm2. In reality ion channels have a variable resistance.
the battery accounts for the cell’s resting potential
•Nobel prize
1963: A.
Hodgkin & A.
Huxley on the
giant squid
axon
Minocqua, WI, Aug 2005
Antarctica, July 1993
18
Magnets
Let’s Break A Magnet!
13th century BC: Chinese already used a compass
with a magnetic needle
800 BC: Greeks discovered magnetite (Fe3O4)
!
!
!
Like poles repel
each other
!
!
Magnetic poles are
always found in
pairs!
A monopole has never been
observed (but…)!
N-N or S-S
Unlike poles
attract each other
!
!
N-S
Magnetic Fields in ordinary life
Magnetotactic bacteria
Magnetotactic
bacteria (MTB)
(Blakemore, 1975)
orient and migrate
along the
geomagnetic field
towards favorable
habitats, a behavior
known as
magnetotaxis. MTB
are aquatic
microorganisms
inhabiting freshwater
and marine
environments.
William Gilbert (1600) :
Earth is a gigantic magnet!
Magnetic disc (floppy or hard disk): a memory device covered with a
magnetic coating on which digital information is stored in the form of
microscopically small, magnetized needles.
Aurora Borealis
Magnetic interaction and field
!
!
there is a ‘field’ associated with the magnetic
interaction.
B = magnetic field vector
!
!
!
!
Has both magnitude and direction
Magnitude = magnetic field strength
Magnetic Fields
!
!
!
!
A vector quantity (B)
compass needle traces B field lines and points towards N
B-field lines start in N and go to S
they do not start or stop (no magnetic monopoles
Electric dipole
Iron filings show pattern of B-field lines
SI unit of magnetic field: tesla (T)
CGS unit: gauss (G): 1 T = 104 G (Earth surface 0.5 G)
Refrigerator magnet 5 x 10-3 T
23
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+