RC Circuits. RC Circuits: Charging a Capacitor

Transcription

1 Circuits Circuits: Charging a Capacitor C initially uncharged Connect switch at t = 0 Calculate current and charge as function of time Apply Kirchhoff s Voltage Law: ε IR Q C = 0 I t = 0, Q = Q 0 = 0: t =, I C = 0: ε I 0 R 0 = 0 I 0 = ε R ε 0 R Q C = 0 Q = Cε While charging ε dq R Q(t) C = 0 dq = ε R Q(t) 1

2 Circuits: Charging a Capacitor Convert to differential equation dq = ε R Q(t) ubstitution of variables X = ε / R Q / Q t dq = ε / R Q / Change integration limits ε R Q 0 t dx = X ε R 0 dx = 1 dq e t = 1 Q εc 0 ε ε R ln X Q ε = ln R Q = t ε R R I Circuits: Charging a Capacitor e t = 1 Q Q = εc(1 e t/ ) εc Capacitive time constant (t) Larger t => longer charging time τ = Q = εc(1 e t/τ ) I = dq = ε R e t/τ V C = Q C = ε(1 e t/τ ) Units of t : ΩF = V A I C V = C C/s = s 2

3 Q = Cε(1 e t/τ ) at t = 0 t = t = τ I = ε R e t/τ Charging a Capacitor Q 0 = 0 Q = Q max = Cε Q τ = 0.63Cε at t = 0 t = t = τ I 0 = I max = ε R I = 0 I τ = 0.37I max 0.37 I max Circuits: Discharging a Capacitor C initially charged with Q 0 =Ce Connect switch at t = 0 Calculate current and charge as function of time Apply Kirchhoff s Voltage Law: Q C IR = 0 I = dq t = 0, Q = Q 0 = Ce : Q 0 C I R = 0 0 I 0 = Q 0 t =, I C = 0: Q C 0 R = 0 Q = 0 Q decreasing, dq/ is negative While discharging Q(t) C + dq R = 0 dq I = Q(t) 3

4 Circuits: Discharging a Capacitor Convert to differential equation dq = Q(t) Q dq = 1 Q Cε t 0 I Integrate t = lnq Q Cε = ln Q Cε = ln Q Q 0 Q = Q 0 e t/ = Cεe t/τ I = dq = ε R e t/ Q = Cεe t/τ at t = 0 t = t = τ I = ε R e t/τ Discharging a Capacitor Q 0 = Q max = Cε Q = 0 Q τ = 0.37Q max 0.37 Q 0 at t = 0 t = t = τ I 0 = I max = ε R I = 0 I τ = 0.37I max 0.37 I 0 4

5 Demo Charging and Discharging a Capacitor I 6C-01 Behavior of Capacitors Charging Initially Capacitor behaves like a wire After a long time Capacitor behaves like an open switch Discharging Initially Capacitor behaves like a battery After a long time Capacitor behaves like an open switch 5

6 Question 1 Bulb 2 R V Bulb 1 R C What will happen after the switch is closed? A. Both bulbs come on and stay on. B. Both bulbs come on but then bulb 2 fades out. C. Both bulbs come on but then bulb 1 fades out. D. Both bulbs come on and then both fade out. Question 2 Bulb 2 R V Bulb 1 R C uppose the switch has been closed a long time. ow what will happen after open the switch? A. Both bulbs come on and stay on. B. Both bulbs come on but then bulb 2 fades out. C. Both bulbs come on but then bulb 1 fades out. D. Both bulbs come on and then both fade out. 6

7 Magnetic Fields Large Magnetic fields are used in MRI 2003 obel prize for medicine Extremely large magnetic fields are found in some stars Earth has a magnetic field Magnetic Field Vector field (B) produced by moving electric charge (current) Magnetic dipole field due to spin Intrinsic property of elementary particles e.g., electrons, protons, neutrons ome materials act like permanent magnets Magnetism known for a long time Greeks recorded observations more than 2500 years ago The word magnetism comes from the Greek word for a type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece. English magnetite 7

8 Bar magnet Like poles repel Unlike poles attract Magnetic field lines Bar Magnets Defined in same way as electric field lines Direction and density Attraction Repulsion From orth to outh Demo: Magnetic Field Lines Electric field lines of an electric dipole Magnetic field lines of a bar magnet 8

9 Question Which drawing shows the correct field lines for a bar magnet? 1 A. 1 B. 2 C Magnetic Monopoles One explanation for magnetic fields There exists magnetic charge, just like electric charge Magnetic monopole (+ or - magnetic charge) A postulated entity which carries this magnetic charge BUT How can you isolate this magnetic charge? Try cutting a bar magnet in half o success to date in finding magnetic monopoles in nature cientists are looking for them 9

10 ource of Magnetic Fields? What is the source of magnetic fields, if not magnetic charge? Moving electric charge Current in wire surrounding a cylinder (solenoid) produces very similar field to that of bar magnet ource of field generated by bar magnet eed to understand currents at atomic level within bulk matter Orbits of electrons about nuclei Intrinsic spin of electrons (more important effect) By convention Earth s Magnetic Field end of a bar magnet is what points to the Earth s orth Geomagnetic Pole Opposite poles attract, analagous to opposite electrical charges orth Geomagnetic Pole is in fact a magnetic OUTH pole, by convention Confusing, but just a convention Just remember that for bar magnets defined as pointing to geomagnetic orth 10

11 Earth s Magnetic Field Generated by convection currents in outer core Crust Mantle Temperature and compositional differences Movement of molten iron Generates electric currents, which generate magnetic field k m Moving charged metal passes through field Generates electric fields Liquid Outer Core olid Inner Core And so on... = Geodynamo elf-sustaining 6/06/02/ironing-out-themystery-of-earths-magneticfield/ Earth s Magnetic Field ince 1904: Pole moved 750 km Average of 9.4 km/yr From 1973 to late 1983: Pole moved 120 km Average of 11.6 km/yr Accelerating! Yellow squares Measured Colored dots Modeled

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