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UW-Madison PHYSICS 208 - RC Circuits and Magnetism

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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 weekWed morning1Contents 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 lawsRC Circuits and Magnetism•More on Kirchhoff’s laws•RC Circuits•Magnets and B-field•Lorentz force and motion of charge in a B-fieldCurrent Conservation: 1st Kirchoff’s law4IinIoutIout = IinI1I2I3I1=I2+I3I2I3I1I1+I2=I3!Junction Rule: ! Iin = ! Iout!A statement of Conservation of ChargeKirchhoff’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 !U=0 (conservative electric force)! "Vloop= "Vk= 0k#I+-potential increases potential decreases-+I-+potential decreasespotential increases+-What is the current and the power?6! "1#"2# (r1+ R1+ R2+ r2+ R3)I = 012V6V5!5!25!25!40!! I="1#"2r1+ R1+ R2+ r2+ R3=6100= 0.06APower dissipated in resistors is P = RI2Power produced by battery is P = "1 I - "2 IKirchoff’s laws application72 loopsAssume 1 current verse per loopI3! I1= I3+ I2" I3= I1# I28V + 4V # 4V #1$I1# 2$I1# 2$(I1# I2) = 0#4V # 6$I2# 2$(I2# I1) = 0I1I2RC CircuitsUntil now, circuits with resistors and batteries where the current is constant in wires (Direct Current Circuits).Connections of R and C: current varies with time during charge and discharge of C. "Charge: S2 open and S1 closed (C connected to battery)Discharge: S1 open, S2 closed (battery disconnected from C)In the Lab this weekHow fast does a C charge or discharge?!The time it takes to charge or discharge a capacitor in an RC circuit depends on the time constant9! RC[ ]="VI#Q"V$ % & ' ( ) =Q"tQ$ % & ' ( ) It is a time! It is easily measurable by you in the lab if it is of the order of fractions of secondsEg R ~100k% and C ~1µF ! RC ~ 0.1 sOhm’s law Current definitionRC Circuits: chargeThe current becomes exponentially zero when the max charge is reached because the potential difference across the capacitor matches that supplied by the batteryAt 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: ! I"= I(t # ") = 0! q"= C#! I =dqdt! "= #VR+ #VC= RI +qC"Current and charge vs time! "= RI +qC# 0 = RdIdt+IC#dII= $dtRCDifferentiateI from I0 = "/R exponentially goes to zero while the charge builds up on C! I =dqdtI(t)=I0 e-t/RCq(t) = C"(1 – e-t/RC)VC=q/C VR=RIAfter 1#, charge increases from 0 to C"(1-e-1) = 63.2% of its max value C"After 3#, C is 95% chargedEnergy stored in C is provided by the battery! U =Q22C=C2"22C=12C"2http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31A simple rule of thumb!When the capacitor is initially connected to the battery it is uncharged and it behaves as a short circuit!When C is fully charged I = 0 and it behaves as an open circuit12I0 = I(t=0) = "/R.I# = I(t$#0) = 0Discharging a Capacitor in an RC Circuit!q = Qe-t/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% I0q/C = RI I = -dq/dt! I(t) = "dqdt= "QRCe"t / RC"Exclude batteryclose S2Question:!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 B14Question:!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.15Cell MembranesLipid 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 = $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 axonMagnetism18Minocqua, WI, Aug 2005Antarctica, July 1993Magnets!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 BorealisMagnetic 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.23Magnetic 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


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UW-Madison PHYSICS 208 - RC Circuits and Magnetism

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