Nobel Prize in Physics, 2011The Nobel Prize in Physics 2011 has been awarded to Saul Perlmutter, Brian P Schmidt and Adam G Riess for discovering the accelerating expansion of the universe

Nobel Prize in Physics, 2010"for groundbreaking experiments regarding the two-dimensional material graphene"Since it is practically transparent and a good conductor, graphene is suitable for producing transparent touch screens, light panels, and maybe even solar cells. Graphene transistors are predicted to be substantially faster than todays silicon transistors and result in more efficient computers.

Andre Geim Konstantin NovoselovGeim and Novoselov extracted the graphene from a piece of graphit such as is found in ordinary pencils. Using regular adhesive tape they managed to obtain a flake of carbon with a thickness of just one atom.

Lectures 10-11 (Ch. 26)R-C circuitsI. Steady state regime (R and C in series, R and C in parallel)II. Transient regimes1.Charging the capacitorinitial conditions temporal dynamics2.Discharging the capacitorinitial conditions temporal dynamicsIII. Applications

2. C and R are in parallel

Switch has been closed for a long time. R1R2 I=i(t)=0 but I10 !I1= /(R1+R2)Vab= R1 /(R1+R2)Vbc = R2 /(R1+R2)q(t ) = Q=CVbc =CR2 /(R1+R2) (capacitor is fully charged)

I1I1I=0abcSteady state regime (t)

=0!=0!NB: In steady state regime current does not go through the capacitor: I=i(t)=0!

1.C and R are in seriesSwitch has been closed for a long time. I=i(t)=0 Vab=IR=0, Vbc =, q(t ) = Q=C (capacitor is fully charged)

Example. (Problem 26.74) Switch is open. In the steady state regime find: a)voltage at the points a and b; b) charge on each capacitor. V=0Current does no go through the capacitor in the steady state regime. Hence there is no current in this circuit.

a) Vb=0x3=0, Va=18V-0x3=18Vb) Q1=3Fx18V=54 C Q2=6Fx18V=108 C 2. Switch is closed. In the steady state regime find: a)voltage at the points a and b; b) charge on each capacitor.

Current does not go through the capacitors but it does go through 6 and 3 resistors.

I=18V/9=2A , Va=Vb=2Ax3=6VQ1=3Fx6V=18 CQ2=6Fx12V=72 C

Example. Switch is open. In the steady state regime find: a) voltage at the points a and b; b) charge on each capacitor. V=0Current does not go through the capacitors but it does go through 14 and 10 resistors.

I=24V/24=1A

Va=1Ax10=10V Vb=24V/2=12V b) Q1=Q2=Q=24VxCeq=24Vx1F=24C 2. Switch is closed. In the steady state regime find: a) voltage at the points a and b; b) charge on each capacitor.Current is the same as in the previous part:I=24V/24=1AVa=1Ax10=10V (also the same)Vb=Va=10V Q2=10Vx2F=20C; Q1=14Vx2F=28C

V=24VS14102 F2 FabQ1Q2

Example. Switch is closed at t=0. Find the final charge on the capacitor at t.12VS110 951.5Fabi1+i2i1i2o

Charging the capacitor

at t=0

q( t=0)= 0

Vbc(t=0)=q( t=0)/C=0

Vab (t= 0)= i(t=0)= /R

I.C and R in seriesCurrent appears at t=0.1. Initial conditions

Switch is closed at t=0. Capacitor is initially uncharged:

Example. All capacitors are initially uncharged. Switch is closed at t=0. a)Find Vab(t=0); b) Find Vab(t).12V2F54F3F1abcS

Example. Problem 26.72 . The capacitor is initially uncharged. The switch is closed at t=0. a) Immediately after the switch is closed, what is a current through each resistor? b) what are the final currents and final charge ?42V84F63Si1i2i1+i2

2. Transient regime

Example. Verify an energy balance in the process of charging the capacitor.One half of the energy provided by the battery is stored in capacitor, another half is dissipated in the resistor.

Example. Which fraction of the maximum possible energy was stored in capacitor by the moment when current droped 3 times?

Example. Switch is closed at t=0. a)Find Vab(t*) if q(t*)=Q/3, where Q is the charge of the capacitor when t; b) Find characteristic time constant of this circuit.12V2F54F3F1abcS

Charging the capacitor.

1. Initial conditions.Capacitor is initially uncharged. Switch is closed at t=0. Current i1 appears at t=0. R1R2q( t=0)= 0

Vbc(t=0)=q( t=0)/C=0 i2 (t=0)=0

Vab(t=0)= i1+i2 (t=0)=i1= /R1

i1+i2

i2bc2. C and R in parallelai1

Example. Capacitor is initially uncharged. At t=0 the switch is closed. Find a)the initial and the final current; b)the final charge.10VS20400.5Fcb

R1R2i1i2bcai1-i22. Transient regime.Charge increases, current, i1-i2 decreases till steady state regime considered above is established at t .

Discharging the capacitor. at t=0 q( t=0)= Q

Vbc(t=0)=Q/C=0

Vab (t= 0)=-Q/C

i(t=0)=-Q/CR

I.C and R in series

1.Initial conditions.

Capacitor is initially charged and disconnected from the battery. Switch is closed at t=0. Current appears at t=0.

2.Transient regime

Example. Verify an energy balance in the process of discharging the capacitor.

Example. The switch was closed for a long time so that the capacitor was fully charged (Q=72 C, see solution of this part of the problem above).Then the switch is opened at t=0. a) Immediately after the switch is opened, what is a current through each resistor? b) what are the currents through each resistor when charge decreased twice?42V84F63S

Discharging the capacitor.

Initial conditions.Capacitor is initially charged and disconnected from the battery. Switch is closed at t=0. Current appears at t=0.

R1R2q( t=0)= Q Vbc(t=0)=Q/Ci2 (t=0)=Q/CR2 Vba(t=0)= Q/C i1 (t=0)=Q /CR1

i1i2bc2. C and R in parallelai1+i22. Transient regime.Charge decreases , currents also decrease to zero.

Example. Switch was closed for a long time. Then it was opened at t=0. At which moment of time charge decreases up to 5% of its value at t=0.12VS110 951.5F

Applications of R-C circuits2. Frequency filtration

~(t)HF output,LF filtr (1/RC) do not go through)1. Sawtooth voltage (pacemakers, turn signals, windshield wipes, etc.)Gas filled tubeVtT

Example. You need to built a pacemaker with 72beats/min. You have capacitor with C=2.5F and it is known that the discharge starts at v=0.45 . Which resistance you should select?Gas filled tubeVtT

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