This Lecture:

•More on DC circuits•RC circuits

•Membrane electrical currents

From previous lecture:

•Resistors and capacitors in series and parallel and examples of DC circuits• EMF•Kirchhoff’s rules

More on DC Circuits and RC Circuits

Next Honor Lecture

ICECUBE:EXPLORING THE UNIVERSE FROM THE SOUTH POLE

Kirchhoff’s Rules

! Junction Rule: $ Iin = $ Iout! A statement of Conservation of Charge

! Loop Rule:

! A statement of Conservation of Energy

I1 = I2 + I3

!

"Vloop = "Vk = 0k

#

Kirchoff’s laws application

4

I1

I2

2 loopsAssume 1 current verse per loop

I3

(you can consider 1 current per loop)

!

I1 = I2 + I3 " I3 = I1 # I2

8V + 4V # 4V #1$I1 # 2$I1 # 2$(I1 # I2) = 0

4V # 2$(I2 # I1) # 6$I2 = 0

Ground wire

5

A ground wire provides a conducting path to the earth which is independent of the normal current-carrying path in an electrical appliance. It normally carries no current and it is connected to the appliance.

It avoids shock hazards in the case

the hot wire gets in contact with the case.

If the case is not properly grounded the body

in contact with the case can be shocked. path with lower resistance

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

The internal resistance of the ammeter must be very smallI = IA != "V+"VA => I = !/(R + rA) # !/R

for rA #0

IA

I!VA

!V

R=R1+R2

Electrical measurements: Voltmeter

The internal resistance of the voltmeter must be very largeI = Iv+IR

"VV = "VR= !

!

I ="

RV

+"

R

RV#$

% # % % "

R

R

!

VVoltmeter

"VV

IV

IR

I

"VR

RC Circuits: charge

The current becomes zero when the max charge is reached because the potential difference across the capacitor matches that supplied by the battery

The current varies with time during the charge and discharge of C. When C is completely uncharged it behaves as a short circuit, when it is fully charged as an open circuit.At t=0 C is uncharged and S1 is closed (S2 open) and I0 = I(t=0) = !/R.An any t:

!

dI

I= "

dt

RC0

t

#I 0

I

# $ lnI

I0

= "t /RC$ I(t) = I0e" t /RC

!

I" = I(t#") = 0

!!

" = RI +q

C# 0 = R

dI

dt+I

C#

dI

I= $

dt

RC

!

q = C" # RCI0e# t /RC

= C"(1# e#t /RC )

!

q" = C#

! q = C!(1 – e-t/RC)

! The time constant, %=RC

! In 1% the charge increases from

zero to C!(1-e-1) = 63.2% of its

maximum C!

! In 1% the current decreases from

I0=!/R to !/R e-1 = 36.8% I0

! The energy stored in the charged capacitor is 1/2 Q! = 1/2 C!2

Charging a Capacitor in an RC Circuit Discharging a Capacitor in an RC Circuit

! When a charged capacitor is placed in the circuit, it can be discharged! q = Qe-t/RC

! The charge decreases exponentially

! In 1% = RC, the charge decreases from Q to 36.8% Q

! In 1% = RC, the current decreases from I0 = Q/CR = !/R to 36.8% I0

q/C = RI I = -dq/dt

!

I(t) = "dq

dt= "

Q

RCe"t /RC

Question:

! This circuit contains 3 identical light bulbs and a capacitor. Which light bulb(s) is (are) dimmest?The capacitor is fully charged

! A

! B

! C

! A and B

11

Question:

! The circuit contains 3 identical light bulbs and a capacitor. At the instant the switch S is closed (C uncharged), which light bulbs is brightest?

! A

! B

! C

! All 3 are equally bright.

12

C=1µF

Time Constant. Question:

! Which is the time constant of this circuit?

! 0.5 ms

! 0.3s

! 0.9 µs

13

0.5" 0.5"

1.00"

!

Req =9 "1

9 +1=9

10= 0.9#

Cell MembranesLipid bilayers of cell membranes can be modeled as a conductor with plates made of polar lipid heads separated by a dielectric layer of hydrocarbon tails.

Difference in electrostatic potential inside and outside a cell due to ion distribution. Typical capacitance: 35 pF

In reality between the ‘plates’ there is a dielectric that increases C by a factor of 10.

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 + and viceversa. Current of 70 ions per ms Prof Moss lecture!

!

"oA

d=8.85 #10

$12F /m( )4% 50 #10$6m( )

2

8 #10$9m

= 3.5 #10$11F = 35pFproteins behave as ion channels

Biological Membrane Electrical Model

! RC circuit: the capacitance results from the separation of charges across the bilayer of lipids, the resistance mimics the behavior of ionic channels and the battery accounts for the cell’s resting potential

! C/A = 1 µF/cm2

! RC time constants range from 10 µs to 1 s = (RA)(C/A)

! Ion channels allow only the passage of specific ions (Na+, K+,Cl-...) and open and close in response to the membrane potential. They are responsible for the formation of the action potential

! Hence, specific resistance of membranes:

R = 'L/A & R A = 'L = 10-106 ( cm2.

Action potential

16

•When the membrane becomes depolarized the cell undergoes an action potential

•The action potential can travel long in the axon or nerve fiber

•Nobel prize 1963: A. Hodgkin & A. Huxley on the giant squid axon

action potential in the giant squid

! Between squid tentacles there is a siphon through which water can be expelled by the fast contractions of the body muscles of the animal.

! This contraction is initiated by action potentials in the giant squid axon.

Membrane depolarization

! let’s imagine a membrane permeable only to K+ ions

! K+ diffuse from compartment 1 to 2, but other ions cannot because the membrane is not permeable to them.

! 2 becomes electrically positive with respect to compartment 1.

! The trans-membrane potential difference #V will tend to push K+ ions from 2 to 1.

!Equilibrium establishes when #V is such to move K+ ions to the left at the same rate as they tend to diffuse to the right ! #Vequilibrium = Nernst potential

Resting potential of most cells is equal to K+ Nernst potential because K conductance dominates. During a typical action potential Na channels open and the membrane potential becomes close to the Nernst potential of Na

Membranes Electrical Model with ion channels

"V

! If the transmembrane potential E = ENernst &

inward and outward flow of ion A are equal

! Variable resistors are used for voltage-gated ion channels, whose resistance changes with voltage.

! If E " ENernst there is a net

flow of A one way

Magnetism

19

Magnets

! 13th century BC: Chinese already used a compass with a magnetic needle

! 800 BC: Greeks discovered magnetite (Fe3O4)

! Like poles repel each other

! N-N or S-S! Unlike poles

attract each other

! N-S

Let’s Break A Magnet!

A monopole has never been

observed (but…)!

! Magnetic poles are always found in pairs!

22

A complicated question…

! there is a ‘field’ associated with the magnetic interaction.

! B = magnetic field vector! Has both magnitude and direction

! Magnitude = magnetic field strength

! What is the magnetic field?! Another complicated question…

Magnetic Fields in ordinary life

William Gilbert (1600) :

Earth is a gigantic magnet!

Aurora Borealis

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.

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)

Iron filings show pattern of B-field lines

+-

Electric dipole

25

Magnetic dipole field lines

! Magnetic field is non unifom

! Changes direction

! Interaction of two dipoles quite complicated

! But far from the dipole, can think about ~ uniform magnetic field.

Earth’s A Magnet!

!N geographic pole almost at magnetic S pole

!S geographic pole almost at magnetic N pole

Compass needle aligns with local Earth field

Electric vs Magnetic Field Lines

! Similarities! Density gives strength

! Arrow gives direction! Leave +, North

! Enter -, South

! Differences! Start/Stop on electric charge

! No Magnetic Charge &

lines are continuous!

Magnetotactic bacteriaMagnetotactic

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.

J. Meisel lecture