MTE 2: Ch 2103 5:30-7pm on Mar 26

Contact me and Prof. Rzchowskiafter this lecture for Alternate Exams (also by email asap!)

2:30-4pm6:00-7:30pm

on Mar 26Office hrs change this week

Wed morning

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Contents of MTE2

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)

This Lecture:

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

RC Circuits and Magnetism

•More on Kirchhoff’s laws•RC Circuits•Magnets and B-field•Lorentz force and motion of charge in a B-field

Current Conservation: 1st Kirchoff’s law

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Iin

Iout

Iout = Iin

I1

I2

I3I1=I2+I3

I2

I3

I1

I1+I2=I3

Junction Rule: Σ Iin = Σ Iout A statement of Conservation of Charge

Kirchhoff’s Rules: energy conservation

Loop Rule: A statement of Conservation of Energy Remember that a charge that moves around a closed loop back to the starting point

has potential energy difference ΔU=0 (conservative electric force)

€

ΔVloop = ΔVk = 0k∑

I

+-

potential increases potential decreases

-+

I

-+

potential decreases

potential increases+-

What is the current and the power?

6

€

ε1 −ε2 − (r1 + R1 + R2 + r2 + R3)I = 0

12V 6V

5Ω

5Ω

25Ω25Ω

40Ω

€

I= ε1 −ε2r1 + R1 + R2 + r2 + R3

=6100

= 0.06A

Power dissipated in resistors is P = RI2Power produced by battery is P = ε1 I - ε2 I

Kirchoff’s laws application

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2 loopsAssume 1 current verse per loop

I3

€

I1 = I3 + I2 ⇒ I3 = I1 − I28V + 4V − 4V −1ΩI1 − 2ΩI1 − 2Ω(I1 − I2) = 0−4V − 6ΩI2 − 2Ω(I2 − I1) = 0

I1 I2

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.

ε

Charge: S2 open and S1 closed (C connected to battery)Discharge: S1 open, S2 closed (battery disconnected from C)

In the Lab this week

How fast does a C charge or discharge?

The time it takes to charge or discharge a capacitor in an RC circuit depends on the time constant

9

€

RC[ ] =ΔVI×QΔV

=QΔtQ

It is a time! It is easily measurable by you in the lab if it is of the order of fractions of secondsEg R ~100kΩ and C ~1µF ➔ RC ~ 0.1 s

Ohm’s law Current definition

RC Circuits: charge

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

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) = ε/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:

€

I∞ = I(t→∞) = 0

€

q∞ = Cε

€

I =dqdt

€

ε = ΔVR + ΔVC = RI +qC

ε

Current and charge vs time

€

ε = RI +qC⇒ 0 = R dI

dt+IC⇒

dII

= −dtRC

Differentiate

I from I0 = ε/R exponentially goes to zero while the charge builds up on C

€

I =dqdt

I(t)=I0 e-t/RC

q(t) = Cε(1 – e-t/RC)VC=q/C VR=RI

After 1τ, charge increases from 0 to Cε(1-e-1) = 63.2% of its max value CεAfter 3τ, C is 95% chargedEnergy stored in C is provided by the battery

€

U =Q2

2C=C2ε2

2C=12Cε2

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31

A simple rule of thumb

When the capacitor is initially connected to the battery it is uncharged and it behaves as a short circuit

When C is fully charged I = 0 and it behaves as an open circuit

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I0 = I(t=0) = ε/R.

I∞ = I(t→∞0) = 0

Discharging a Capacitor in an RC Circuit

q = Qe-t/RC

q decreases exponentially In 1τ = RC, q decreases from initial

value Q to 36.8% Q In 1τ = RC, current decreases from

I0 = Q/CR = ε/R to 36.8% I0

q/C = RI I = -dq/dt

€

I(t) = −dqdt

= −QRC

e−t /RC

ε

Exclude batteryclose S2

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

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Question:

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?

A B C All 3 are equally bright.

15

Cell Membranes

Lipid bilayers of cell membranes like capacitorsTypical 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!

Biological Membrane Electrical Model

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 = ρL/A ⇒ R A = ρL = 10-106 Ω 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

Magnetism

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Minocqua, WI, Aug 2005Antarctica, July 1993

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!

Magnetic Fields in ordinary lifeWilliam 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.

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.

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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

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

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