Physics 102: Lecture 04 Capacitors

Recall from last lecture….. Electric Fields, Electric Potential Electric potential energy Potential difference PED = qV V = Ed

Comparison: Electric Potential Energy vs. Electric Potential q A B DVAB : the difference in electric potential between points B and A DUAB : the change in electric potential energy of a charge q when moved from A to B DUAB = q DVAB

Electric Potential: Summary E field lines point from higher to lower potential For positive charges, going from higher to lower potential is “downhill” For a battery, the (+) terminal is at a higher potential than the (–) terminal Positive charges tend to go “downhill”, from + to - Negative charges go in the opposite direction, from - to + DUAB = q DVAB

Important Special Case Uniform Electric Field + - Two large parallel conducting plates of area A +Q on one plate -Q on other plate Then E is uniform between the two plates: E=4kQ/A zero everywhere else This result is independent of plate separation This is call a parallel plate capacitor

Parallel Plate Capacitor: Potential Difference Charge Q on plates Charge 2Q on plates V =VA – VB = +E0 d V =VA – VB = +2E0 d E=2 E0 E=E0 + - + - + - A B A B Use fire extinguisher analogy? d d Potential difference is proportional to charge: Double Q  Double V E0=4kQ/A

Capacitance: The ability to store separated charge CQ/V Any pair conductors separated by a small distance. (e.g. two metal plates) Capacitor stores separated charge Positive Q on one conductor, negative Q on other Net charge is zero Q=CV E d + - Stores Energy U =(½) Q V Stores energy….electrical potential energy. If the plates were free to move, they would collapse on each other. Just like releasing a brick from height h above the floor. Or, if charges were free to move, the capacitor would “discharge”. Demo. Can I make U more plausible? Units: 1 Coulomb/Volt = 1 Farad (F)

Why Separate Charge? Analogy to a waterfall Camera Flash Defibrillator—see ex. 17.12 in text AC → DC Tuners Radio Cell phones Analogy to a waterfall

Capacitance Practice Example How much charge is on a 0.9 F capacitor which has a potential difference of 200 Volts? How much energy is stored in this capacitor?

Capacitance of Parallel Plate Capacitor V = Ed E=4kQ/A (Between two large plates) So: V = 4kQd/A Recall: CQ/V So: C = A/(4kd) Define: e0=1/(4pk)=8.85x10-12 C2/Nm2 V + E - A A d C =e0A/d Parallel plate capacitor

Parallel Plate Capacitor C = e0A/d Calculate the capacitance of a parallel plate capacitor made from two large square metal sheets 1.3 m on a side, separated by 0.1 m. Example A A Here note that this is small number for such a big device. How do they make “real” capacitors? d

Dielectric Placing a dielectric between the plates increases the capacitance. C = k C0 Dielectric constant (k > 1) Same Q, smaller E, therefore smaller V, larger Q Capacitance without dielectric Capacitance with dielectric

ACT: Parallel Plates +q -q pull 1) Increases 2) Constant 3) Decreases A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. What happens to the charge q on each plate of the capacitor? 1) Increases 2) Constant 3) Decreases Remember charge is real/physical. There is no place for the charges to go.

Preflight 4.1 +q -q pull + - d A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved? 1)The capacitance increases True False 2)The electric field increases True False 3)The voltage between the plates increases True False

ACT/Preflight 4.1 +q -q pull The energy stored in the capacitor A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved? The energy stored in the capacitor A) increases B) constant C) decreases

ACT/Preflight 4.2 1) 2C 2) C 3) C/2 Two identical parallel plate capacitors are shown in end-view in A) of the figure. Each has a capacitance of C. A ) B ) If the two are joined as in (B) of the figure, forming a single capacitor, what is the final capacitance? 1) 2C 2) C 3) C/2

Voltage in Circuits Example Elements are connected by wires. Any connected region of wire has the same potential. The potential difference across an element is the element’s “voltage.” Example Vwire 1= 0 V Vwire 2= 5 V Vwire 3= 12 V Vwire 4= 15 V C1 C2 C3 VC1= 5-0 V= 5 V VC2= 12-5 V= 7 V VC3= 15-12 V= 3 V

Capacitors in Parallel Both ends connected together by wire Same voltage: V1 = V2 Add Areas: Ceq = C1+C2 remember C=0A/d Share Charge: Qeq = Q1+Q2 = Veq 15 V 15 V 15 V Ceq C1 C2 10 V 10 V 10 V

Parallel Practice Example A 4 mF capacitor and 6 mF capacitor are connected in parallel and charged to 5 volts. Calculate Ceq, and the charge on each capacitor. Example Ceq = C4+C6 = 4 mF+6 mF = 10 mF Q4 = C4 V4 = (4 mF)(5 V) = 20 mC Q6 = C6 V6 = (6 mF)(5 V) = 30 mC Qeq = Ceq Veq = (10 mF)(5 V) = 50 mC = Q4+Q6 V = 5 V 5 V 5 V 5 V C4 C6 Ceq 0 V 0 V 0 V

Capacitors in Series Connected end-to-end with NO other exits Same Charge: Q1 = Q2 = Qeq Add d: Share Voltage:V1+V2=Veq + + +Q -Q + +Q + + Ceq C1 + - + - + + +Q -Q + + + C2 -Q - - + - + + + +

Series Practice Example Q = CV 5 V +Q -Q 5 V 0 V +Q -Q C4 Ceq +Q -Q C6 A 4 mF capacitor and 6 mF capacitor are connected in series and charged to 5 volts. Calculate Ceq, and the charge on the 4 mF capacitor. Example Q = CV 5 V +Q -Q 5 V 0 V +Q -Q + C4 + - - Ceq +Q -Q + C6 - 0 V

Comparison: Series vs. Parallel Can follow a wire from one element to the other with no branches in between. Parallel Can find a loop of wire containing both elements but no others (may have branches). C1 C2 C2 C1

Electromotive Force + - Battery Maintains constant potential difference V Does NOT produce or supply charges, just “pushes” them.

Preflight 4.4 A circuit consists of three initially uncharged capacitors C1, C2, and C3, which are then connected to a battery of emf E. The capacitors obtain charges q1, q2,q3, and have voltages across their plates V1, V2, and V3. Which of these are true? q1 = q2 q2 = q3 V2 = V3 E = V1 V1 < V2 Ceq > C1 C 2 3 1 E + - -q2 +q1 -q1 +q3 -q3 +q2 V1 V2 V3

ACT/Preflight 4.4: Which is true? A circuit consists of three initially uncharged capacitors C1, C2, and C3, which are then connected to a battery of emf E. The capacitors obtain charges q1, q2,q3, and have voltages across their plates V1, V2, and V3. C 2 3 1 E + - -q2 +q1 -q1 +q3 -q3 +q2 V1 V2 V3 1) q1 = q2 Not necessarily C1 and C2 are NOT in series. 2) q2 = q3 Yes! C2 and C3 are in series.

ACT/Preflight 4.4: Which is true? A circuit consists of three initially uncharged capacitors C1, C2, and C3, which are then connected to a battery of emf E. The capacitors obtain charges q1, q2,q3, and have voltages across their plates V1, V2, and V3. C 2 3 1 E + - -q2 +q1 -q1 +q3 -q3 +q2 V1 V2 V3 10V 0V 7V?? 1) V2 = V3 Not necessarily, only if C1 = C2 2) E = V1 Yes! Both ends are connected by wires

ACT/Preflight 4.4: Which is true? A circuit consists of three initially uncharged capacitors C1, C2, and C3, which are then connected to a battery of emf E. The capacitors obtain charges q1, q2,q3, and have voltages across their plates V1, V2, and V3. C 2 3 1 E + - -q2 +q1 -q1 +q3 -q3 +q2 V1 V2 V3 10V 0V 7V?? 1) V1 < V2 Nope, V1 > V2. (E.g. V1 = 10-0, V2 =10-7) 2) Ceq > C1 Yes! C1 is in parallel with C23

Recap of Today’s Lecture Capacitance C = Q/V Parallel Plate: C = 0A/d Capacitors in parallel: Ceq = C1+C2 Capacitors in series: Ceq = 1/(1/C1+1/C2) Batteries provide fixed potential difference