MTE 2: Ch 2103 5:30-7pm on Mar 26 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 1
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)
RC Circuits and Magnetism This Lecture: More on Kirchhoff s laws RC Circuits Magnets and B-field Lorentz force and motion of charge in a B-field 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
Current Conservation: 1 st Kirchoff s law I 2 I in I 1 I 1 =I 2 +I 3 I 3 I 1 I 3 I out I out = I in I 2 I 1 +I 2 =I 3 Junction Rule: Σ I in = Σ I out A statement of Conservation of Charge 4
Kirchhoff s Rules: energy conservation Loop Rule: ΔV loop = ΔV k = 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 ΔU=0 (conservative electric force) I potential increases potential decreases - + + - I potential decreases + - - potential increases +
What is the current and the power? 5Ω 25Ω 25Ω 12V 6V 5Ω ε 1 ε 2 (r 1 + R 1 + R 2 + r 2 + R 3 )I = 0 I= 40Ω ε 1 ε 2 = 6 r 1 + R 1 + R 2 + r 2 + R 3 100 = 0.06A Power dissipated in resistors is P = RI 2 Power produced by battery is P = ε1 I - ε2 I 6
Kirchoff s laws application I1 I2 2 loops Assume 1 current verse per loop I3 I 1 = I 3 + I 2 I 3 = I 1 I 2 8V + 4V 4V 1ΩI 1 2ΩI 1 2Ω(I 1 I 2 ) = 0 4V 6ΩI 2 2Ω(I 2 I 1 ) = 0 7
RC Circuits In the Lab this week 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)
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 Ohm s law Current definition [ RC] = ΔV I Q ΔV = QΔt Q 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 100kΩ and C 1µF RC 0.1 s 9
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) = 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 = dq dt ε ε = ΔV R + ΔV C = RI + q C I = I(t ) = 0 q = Cε The current becomes exponentially zero when the max charge is reached because the potential difference across the capacitor matches that supplied by the battery
Current and charge vs time ε = RI + q Differentiate C 0 = R di dt + I C di I = dt RC 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 Cε After 3τ, C is 95% charged Energy stored in C is U = Q2 provided by the battery 2C = C 2 ε 2 2C = 1 2 Cε2 I = dq dt I from I0 = ε/r exponentially goes to zero while the charge builds up on C 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 I0 = I(t=0) = ε/r. When C is fully charged I = 0 and it behaves as an open circuit I = I(t 0) = 0 12
Discharging a Capacitor in an RC Circuit Exclude battery close S2 ε q/c = RI I = -dq/dt q = Qe -t/rc I(t) = dq dt = Q RC e t / RC q decreases exponentially In 1τ = RC, q decreases from initial value Q to 36.8% Q In 1τ = RC, current decreases from I 0 = Q/CR = ε/r to 36.8% I 0
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 14
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 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!
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/cm 2 R results from the behavior of ion channels: R = ρl/a R A = ρl = 10-10 6 Ω cm 2. 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 Minocqua, WI, Aug 2005 Antarctica, July 1993 18
Magnets 13 th century BC: Chinese already used a compass with a magnetic needle 800 BC: Greeks discovered magnetite (Fe 3 O 4 ) Like poles repel each other N-N or S-S Unlike poles attract each other N-S
Let s Break A Magnet! Magnetic poles are always found in pairs! A monopole has never been observed (but )!
Magnetic Fields in ordinary life 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
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.
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 = 10 4 G (Earth surface 0.5 G) Refrigerator magnet 5 x 10-3 T 23
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 - +