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Exam 1: Tuesday, Feb 14, 5: 00 -6: 00 PM Test rooms: • Instructor Exam 1: Tuesday, Feb 14, 5: 00 -6: 00 PM Test rooms: • Instructor • Dr. Hale • Dr. Kurter • Dr. Madison • Dr. Parris • Mr. Upshaw • Dr. Waddill Sections F, H B, N K, M J, L A, C, E, G D • Special Accommodations Room 104 Physics 125 BCH 199 Toomey St Pat’s Ballroom* G-3 Schrenk 120 BCH Testing Center *exam 1 only If at 5: 00 on test day you are lost, go to 104 Physics and check the exam room schedule, then go to the appropriate room and take the exam there.

Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Capacitors: parallel plate, cylindrical, spherical. You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors. Circuits containing capacitors in series and parallel. You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.

Capacitors: the basics What is a capacitor? • device for storing charge • simplest Capacitors: the basics What is a capacitor? • device for storing charge • simplest example: two parallel conducting plates separated by air A E V 0 assortment of capacitors d V 1

Capacitors in circuits symbol for capacitor (think parallel plates) symbol for battery, or external Capacitors in circuits symbol for capacitor (think parallel plates) symbol for battery, or external potential battery voltage V is actually potential difference between the terminals + V • when capacitor is connected to battery, charges flow onto the plates Capacitor plates build - up charges +Q and -Q conducting wires + • when battery is disconnected, charge remains on plates V

Capacitance How much charge can a capacitor store? Better question: How much charge can Capacitance How much charge can a capacitor store? Better question: How much charge can a capacitor store per voltage? Capacitance: V is really | V|, the potential difference across the capacitor capacitance C is a device property, it is always positive unit of C: farad (F) 1 F is a large unit, most capacitors have values of C ranging from picofarads to microfarads (p. F to F). micro 10 -6, nano 10 -9, pico 10 -12 (Know for exam!)

Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Capacitors: parallel plate, cylindrical, spherical. You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors. Circuits containing capacitors in series and parallel. You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.

Capacitance of parallel plate capacitor -Q electric field between two parallel charged plates: +Q Capacitance of parallel plate capacitor -Q electric field between two parallel charged plates: +Q E Q is magnitude of charge on either plate. potential difference: capacitance: V 0 d V 1 A

Parallel plate capacitance depends “only” on geometry. -Q +Q E This expression is approximate, Parallel plate capacitance depends “only” on geometry. -Q +Q E This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum (lecture 9). V 0 Greek letter Kappa. For today’s lecture (and for exam 1), use =1. is NOT the same as k=9 x 109! d V 1 A

Capacitance of coaxial cylinder • capacitors do not have to consist of parallel plates, Capacitance of coaxial cylinder • capacitors do not have to consist of parallel plates, other geometries are possible • capacitor made of two coaxial cylinders: L

from Gauss law: (see lectures 4 and 6) Gaussian surface b r a Q from Gauss law: (see lectures 4 and 6) Gaussian surface b r a Q E dl capacitance per unit length: -Q

Example application: coaxial cables, capacitance per length is a critical part of the specifications. Example application: coaxial cables, capacitance per length is a critical part of the specifications. L

Isolated Sphere Capacitance isolated sphere can be thought of as concentric spheres with the Isolated Sphere Capacitance isolated sphere can be thought of as concentric spheres with the outer sphere at an infinite distance and zero potential. We already know the potential outside a conducting sphere: The potential at the surface of a charged sphere of radius R is so the capacitance at the surface of an isolated sphere is

Capacitance of Concentric Spheres If you have to calculate the capacitance of a concentric Capacitance of Concentric Spheres If you have to calculate the capacitance of a concentric spherical capacitor of charge Q… In between the spheres (Gauss’ Law) b a +Q -Q You need to do this derivation if you have a problem on spherical capacitors!

Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 cm and are separated by a 1. 0 mm air gap. d = 0. 001 m area = 0. 2 m x 0. 03 m If you keep everything in SI (mks) units, the result is “automatically” in SI units.

Example: what is the charge on each plate if the capacitor is connected to Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery? 0 V V= 12 V +12 V *Remember, it’s the potential difference that matters.

Example: what is the electric field between the plates? 0 V E d = Example: what is the electric field between the plates? 0 V E d = 0. 001 +12 V V= 12 V

Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Today’s agenda: Capacitors and Capacitance. You must be able to apply the equation C=Q/V. Capacitors: parallel plate, cylindrical, spherical. You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors. Circuits containing capacitors in series and parallel. You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.

Circuits Containing Capacitors in Parallel Vab Capacitors connected in parallel: C 1 a C Circuits Containing Capacitors in Parallel Vab Capacitors connected in parallel: C 1 a C 2 b C 3 + V all three capacitors must have the same potential difference (voltage drop) Vab = V General concept: When circuit components are connected in parallel, then the voltage drops across these components are all the same.

C 1 Q 1 = C 1 V & Q 2 = C 2 C 1 Q 1 = C 1 V & Q 2 = C 2 V & Q 3 = C 3 V a + Q 2 C 3 Q 3 + V Imagine replacing the parallel combination of capacitors by a single equivalent capacitor a “equivalent” means “stores the same total charge if the voltage is the same. ” Ceq Q + V Qtotal = Ceq V = Q 1 + Q 2 + Q 3 Important!

Summarizing the equations on the last slide: Q 1 = C 1 V Q Summarizing the equations on the last slide: Q 1 = C 1 V Q 2 = C 2 V Q 3 = C 3 V C 1 C 2 a Q 1 + Q 2 + Q 3 = Ceq V C 3 + - Using Q 1 = C 1 V, etc. , gives C 1 V + C 2 V + C 3 V = Ceq V C 1 + C 2 + C 3 = Ceq Generalizing: Ceq = i Ci V (after dividing both sides by V) (capacitances in parallel add up) b

Circuits Containing Capacitors in Series Capacitors connected in series: C 1 C 2 C Circuits Containing Capacitors in Series Capacitors connected in series: C 1 C 2 C 3 + +Q V -Q charge +Q flows from the battery to the left plate of C 1 charge -Q flows from the battery to the right plate of C 3 (+Q and –Q: the same in magnitude but of opposite sign)

Charges +Q and –Q attract equal and opposite charges to the other plates of Charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors: C 1 +Q -Q A C 2 +Q -Q B C 3 +Q -Q + V These equal and opposite charges came from the originally neutral circuit regions A and B. Because region A must be neutral, there must be a charge +Q on the left plate of C 2. Because region B must be neutral, there must be a charge -Q on the right plate of C 2.

Vab a C 1 +Q -Q V 1 C 2 A +Q -Q V Vab a C 1 +Q -Q V 1 C 2 A +Q -Q V 2 B C 3 +Q -Q V 3 b + V The charges on C 1, C 2, and C 3 are the same, and are Q = C 1 V 1 Q = C 2 V 2 Q = C 3 V 3 The voltage drops across C 1, C 2, and C 3 add up Vab = V 1 + V 2 + V 3. General concept: When circuit components are connected in series, then the voltage drops across these components add up to the total voltage drop.

replace three capacitors by a single equivalent capacitor Ceq +Q -Q V + V replace three capacitors by a single equivalent capacitor Ceq +Q -Q V + V “equivalent” means it has the same charge Q and the same voltage drop V as the three capacitors Q = Ceq V

Collecting equations: Q = C 1 V 1 Q = C 2 Vab = Collecting equations: Q = C 1 V 1 Q = C 2 Vab = V 1 + V 2 + V 3. Q = Ceq V Substituting for V 1, V 2, and V 3: Substituting for V: Dividing both sides by Q: Q = C 3 V 3 Important!

Generalizing: OSE: (capacitors in series) Generalizing: OSE: (capacitors in series)

Summary (know for exam!): Parallel Series C 1 C 2 C 3 equivalent capacitance Summary (know for exam!): Parallel Series C 1 C 2 C 3 equivalent capacitance charge Q’s add V’s add voltage same V same Q C 3

Example: determine the capacitance of a single capacitor that will have the same effect Example: determine the capacitance of a single capacitor that will have the same effect as the combination shown. Use C 1 = C 2 = C 3 = C. C 2 C 1 C 3 Start by combining parallel combination of C 2 and C 3 C 23 = C 2 + C 3 = C + C = 2 C

Now I see a series combination. C 23 = 2 C C 1 = Now I see a series combination. C 23 = 2 C C 1 = C

Example: for the capacitor circuit shown, C 1 = 3 F, C 2 = Example: for the capacitor circuit shown, C 1 = 3 F, C 2 = 6 F, C 3 = 2 F, and C 4 =4 F. (a) Find the equivalent capacitance. (b) if V=12 V, find the potential difference across C 4. C 1 C 2 C 3 V I’ll work this at the blackboard. C 4 Homework Hint: each capacitor has associated with it a Q, C, and V. If you don’t know what to do next, near each capacitor, write down Q= , C= , and V=. Next to the = sign record the known value or a “? ” if you don’t know the value. As soon as you know any two of Q, C, and V, you can determine third. This technique often provides visual clues about what to do next.

(a) Find Ceq. (b) if V=12 V, find V 4. C 1=3 F C (a) Find Ceq. (b) if V=12 V, find V 4. C 1=3 F C 2=6 F C 4=4 F C 3=2 F C 1 and C 3 are not in parallel. Make sure you understand why! C 2 and C 4 are not in series. Make sure you understand why! V=12 V C 1 and C 2 are in series. Make sure you use the correct equation! Don’t forget to invert: C 12 = 2 F.

(a) Find Ceq. (b) if V=12 V, find V 4. C 12=2 F C (a) Find Ceq. (b) if V=12 V, find V 4. C 12=2 F C 4=4 F C 12 and C 4 are not in series. Make sure you understand why! C 3=2 F V=12 V C 12 and C 3 are in parallel. Make sure you use the correct equation!

(a) Find Ceq. (b) if V=12 V, find V 4. C 123=4 F C (a) Find Ceq. (b) if V=12 V, find V 4. C 123=4 F C 4=4 F C 123 and C 4 are in series. Make sure you understand why! Combined, they make give Ceq. V=12 V Make sure you use the correct equation! Don’t forget to invert: Ceq = 2 F.

(a) Find Ceq. (b) if V=12 V, find V 4. Ceq=2 F Ceq = (a) Find Ceq. (b) if V=12 V, find V 4. Ceq=2 F Ceq = 2 F. V=12 V If you see a capacitor circuit on the test, read the problem first. Don’t go rushing off to calculate Ceq. Sometimes you are asked to do other things. Truth in advertising: there’s a high probability you will need to calculate C eq at some point in the problem.

(a) Find Ceq. (b) if V=12 V, find V 4. Q 1=? C 1=3 (a) Find Ceq. (b) if V=12 V, find V 4. Q 1=? C 1=3 F V 1=? Q 2=? C 2=6 F Q =? 4 V 2=? C =4 F 4 V 4=? Q 3=? C 3=2 F V 3=? Homework Hint: each capacitor has associated with it a Q, C, and V. If you don’t know what to do next, near each capacitor, write down Q= , C= , and V=. Next to the = sign record the known value or a “? ” if you don’t know the value. As soon as you know any two of Q, C, and V, you can determine third. This technique often provides visual clues about what to do next. V=12 V We know C 4 and want to find V 4. If we know Q 4 we can calculate V 4. Maybe that is a good way to proceed.

(a) Find Ceq. (b) if V=12 V, find V 4. Q 123=? C 123=4 (a) Find Ceq. (b) if V=12 V, find V 4. Q 123=? C 123=4 F V 123=? Q 4=? C 4=4 F V 4=? C 4 is in series with C 123 and together they form Ceq. Therefore Q 4 = Q 123 = Qeq. V=12 V

You really need to know this: Capacitors in series… all have the same charge You really need to know this: Capacitors in series… all have the same charge add the voltages to get the total voltage Capacitors in parallel… all have the same voltage add the charges to get the total charge (and it would be nice if you could explain why)

Homework Hint! C 1 What does our text mean by Vab? C 2 a Homework Hint! C 1 What does our text mean by Vab? C 2 a Our text’s convention is Vab = Va – Vb. This is explained on page 759. This is in contrast to Physics 1135 notation, where Va b = Vb – Va. C 4 b C 3 V In the figure on this slide, if Vab = 100 V then point a is at a potential 100 volts higher than point b, and Va b = -100 V; there is a 100 volt drop on going from a to b.

A “toy” to play with… http: //phet. colorado. edu/en/simulation/capacitor-lab (You might even learn something. A “toy” to play with… http: //phet. colorado. edu/en/simulation/capacitor-lab (You might even learn something. ) For now, select “multiple capacitors. ” Pick a circuit.