February 6 th , 2013 Kirchhoff’s rules, RC circuits, R vs T

1. Chapter 19 February 6th, 2013 Kirchhoff’s rules, RC circuits, R vs T

2. summary of last class Ohm’s Law • V = IR Resistance R(Ω) = V(V)/I(A) • Kirchhoff’s Loop Rule: The total change in electric potential around any closed circuit loop must be zero (conservation of energy) • Kirchhoff’s Junction Rule: The amount of current entering a junction must be equal to the current leaving it (conservation of charge) Kirchhoff’s Loop and Junction Rules

3. resistors summary Ohm’s Law: V =IR if there are many resistors in series, power is mostly dissipated by the resistor with largest resistance (e.g. light bulb filament) Resistance: R = r L/A,1Ω=1V/1A Power Dissipated: P = IV = I2R = V2/R Series Parallel R1 R1 R2 R2 R2 Voltage Differentfor each resistor. Vtotal = V1 + V2 Samefor each resistor. Vtotal = V1 = V2 Current Same for each resistor Itotal= I1= I2 Different for each resistor Itotal= I1+ I2 Resistance Increases Req = R1 + R2 Decreases 1/Req = 1/R1 + 1/R2

4. capacitors summary Capacitance: q = CV,C = 0A/d, 1F = 1C/1V if there are many capacitors in series, energy is mostly stored by the capacitor with smallest capacitance Voltage across capacitor DV = Ed = Q/C Energy stored in a capacitor: PEcap =½QDV = ½C(DV)2 = ½Q2/C C1 Parallel Series C1 C2 C2 Samefor each capacitor. Vtotal = V1 = V2 Voltage Differentfor each capacitor. Vtotal = V1 + V2 Currentor Charge Same for each capacitor. Itotal= I1= I2qtotal = q1 = q2 Different for each capacitor.Itotal= I1+ I2 qtotal = q1 + q2 Decreases 1/Ceq= 1/C1+ 1/C2 Increases Ceq= C1+ C2 Capacitance

5. demo Series Ohm’s Law demo Voltage Differentfor each resistor. Vtotal = V1 + V2 R1 R2 Current Same for each resistor Itotal= I1= I2 Resistance Increases Req = R1 + R2

6. demo Parallel Ohm’s Law demo R1 R2 R2 Samefor each resistor. Vtotal = V1 = V2 Different for each resistor Itotal= I1+ I2 Decreases 1/Req = 1/R1 + 1/R2

7. Light bulb in series with coil resistor in liquid nitrogen demo Series Voltage Differentfor each resistor. Vtotal = V1 + V2 R1 R2 Current Same for each resistor Itotal= I1= I2 Resistance Increases Req = R1 + R2

8. Question 1 2 3 Which configuration has the smallest resistance? 1 2 3 R2R R/2 Which configuration has thelargestresistance? 123

9. Question • An electric shock due to a 10,000 Vpotential difference is always fatal: • True • False e. g. the static shocks from hand to door knob, in 1 cm of air: Dielectric breakdown of air occurs when the Emax = 3 x 106V/m Voltage difference for 1 cm gap to knob = 3 x 104V = 30,000 V

10. Question 25 A 10 A 5 A 10 A Coffee Pot Microwave Johnny “Danger” Powells uses one power strip to plug in his microwave, coffee pot, space heater, toaster, and guitar amplifier all into one kitchen outlet, using a power strip. This is dangerous because… (hint: power strips are wired in parallel.) Toaster 1. The resistance of the kitchen circuit is too high. • parallel connectionthecurrents add up • P = IV • Ptotal= ItotalV=(I1+I2…+I5)V gets too large, and it burns the wires 2. The voltage across the kitchen circuit is too high. 3. The current in the kitchen circuit is too high.

11. demo 3 light bulbs with resistance R are connected to the power source in parallel (top) and in series (bottom) The effective resistance of the 3 in parallel is lower, thus the currents are greater, and the light bulbs are brighter. (The current is different in each resistor in parallel.)

12. Problem 19.111 For muscle rehabilitation patients can be fitted with a small electrical device designed to strengthen muscle tissue. A small electrical current is passed through the muscle tissue, to stimulate contractions. Involuntary muscle stimulation requires a current of 10 mA. Estimate the minimum potential difference required to stimulate the contraction of a bicep. Assume a bicep has a diameter of 10 cm and a resistivity of 150 Ωm.

13. Problem 19.111 cont.

14. Real Batteries • An ideal battery always maintains a constant voltage across its terminals • A real battery is equivalent to an ideal battery in series with a resistor, Rbattery • Rbattery is the internal resistance of the battery

15. Real Battery, cont. • The current through the internal and the external resistor is • The potential difference across the real battery’s terminals is

16. Kirchhoff’s Rules for RC circuit • Kirchhoff’s Rules can be applied to all kinds of circuits • The change in the potential around the circuit is +ε – I R – q/C = 0 • Solving a circuit means solving for I • Iand q are not constant, but time–dependent

17. RC circuit • When the switch is closed, there is a current carrying a positive charge to the top plate of the capacitor • When the capacitor plates are charged, there is a nonzero voltage across the capacitor

18. Charging the Capacitor in an RC circuit • The current in the circuit is described by • The voltage across the capacitor is • The charge is given by • τ= RCand is called the time constant of the RC circuit

19. Discharging the Capacitor in an RC circuit • Current in the circuit: • Voltage across the capacitor: Vcap= εe-t/τ • Charge on the capacitor plates: q = Cε e-t/τ • Time constant: τ = RC, the same as for charging

20. Filters • It is often desirable to filter out time-dependent fluctuations in a voltage signal • Circuits that can do so are called filters • They can be constructed with RC combinations • A filter is useful in many applications • Noise in a radio signal • The amount of filtering depends on the values of R and C

21. Ammeters • An ammeteris a device that measures current • An ammeter must be connected in series with the desired circuit branch • An ideal ammeter will measure current without changing its value • This the ammeter itself must have a very low resistance

22. Voltmeters • A voltmetermeasures the voltage across a circuit element • It must be connected in parallel with the element • An ideal voltmeter should measure the voltage without changing its value • The voltmeter should have a very high resistance

23. Up to here

24. Electric Currents in Nerves • Many nerves are long and thin, much like wires • The conducting solution inside the nerve fiber acts as a resistor • The nerve wall acts as a capacitor • The nerve fiber behaves as an RC circuit • Needs a small time constant • Small R through large radius • Small Cthrough a layer of myelin increasing the distance between the capacitor plates

25. Electric Currents and Health • Your body is a moderately good conductor of electricity • The body’s resistance when dry is about 1500 Ω • When wet, the body’s resistance is about 500 Ω • Current is carried by different parts of the body • Skin • Internal organs

26. Household Currents • The voltage in your home is an AC voltage oscillating at a frequency • Frequency is 60 Hz in the US • Most modern outlets and plugs have three connections • Two are flat • One is round and connects to ground • Polarized plugs have flat connectors of different sizes • The larger connects to the lower potential

27. Fuses and Circuit Breakers • In a fuse, current passes through a thin metal strip • This strip acts as a resistor with a small resistance • If a failure causes the current to become large, the power causes the strip to melt and the current stops • A circuit breaker also stops the current when it exceeds a predetermined limit • The circuit breaker can be reset for continued use

28. Temperature Dependence of Resistance • As temperature increases, the ions in a metal vibrate with larger amplitudes • This causes more frequent collisions and an increase in resistance (T then R) • For many metals near room temperature, ρ = ρo [1 + α(T – To) • α is called the temperature coefficient of the resistivity • The resistivity and resistance vary linearly with temperature

29. Superconductivity • At very low temperatures, the linearity of resistance breaks down • The resistivity of metals approachsa nonzero value at very low temperatures • In some materials [e.g. Mercury, and the cuprates bismuth strontium calcium copper oxide (BSCCO), and yttrium barium copper oxide (YBCO)], the resistivity drops abruptly and is zero below a critical temperature (~100 K) • Materials for which the resistivity goes to zero are called superconductors