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1/25/07184 Lecture 111 PHY 184 Spring 2007 Lecture 11 Title: Capacitors
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1/25/07184 Lecture 112AnnouncementsAnnouncements Homework Set 3 is due Tuesday, Jan. 30 at 8:00 am. Homework Set 4 opened this morning. Today we will finish up the electric potential and start with capacitors.
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1/25/07184 Lecture 113 Review - Electric Potential, V(x) ► If a charge q moves in an electric field… ► For reference V=0 at infinity… ► E is the gradient of V…
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1/25/07184 Lecture 114 Review - Electric Potential (2) ► For a point source Q … ► For many sources … (superposition)
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1/25/07184 Lecture 115 Problem-Solving Strategies Given a charge distribution, calculate the electric field and the electric potential: (i) Use Gauss Law to derive the electric field E: (ii) For the potential (V=0 at infinity) use superposition principle For an arrangement of charges, remember the superposition principle! This holds for the electric force, the electric field, the electric potential and the electric potential energy. Note:
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1/25/07184 Lecture 116 Capacitors
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1/25/07184 Lecture 117CapacitorsCapacitors Capacitors are devices that store energy in an electric field. Capacitors are used in many every-day applications Heart defibrillators Camera flash units Capacitors are an essential part of electronics. Capacitors can be micro-sized on computer chips or super-sized for high power circuits such as FM radio transmitters.
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1/25/07184 Lecture 118CapacitanceCapacitance Capacitors come in a variety of sizes and shapes. Concept: A capacitor consists of two separated conductors, usually called plates, even if these conductors are not simple planes. We will define a simple geometry and generalize from there. We will start with a capacitor consisting of two parallel conducting plates, each with area A separated by a distance d. We assume that these plates are in vacuum (air is very close to a vacuum).
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1/25/07184 Lecture 119 Parallel Plate Capacitor We charge the capacitor by placing a charge +q on the top plate a charge -q on the bottom plate Because the plates are conductors, the charge will distribute itself evenly over the surface of the conducting plates. The electric potential, V, is proportional to the amount of charge on the plates. e.g., using a battery More precisely, potential difference V(+)-V(-) = V
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1/25/07184 Lecture 1110 The proportionality constant between the charge q and the electric potential difference V is the capacitance C. We will call the electric potential difference V the “potential” or the “voltage” across the plates. The capacitance of a device depends on the area of the plates and the distance between the plates, but does not depend on the voltage across the plates or the charge on the plates. The capacitance of a device tells us how much charge is required to produce a given voltage across the plates. Parallel Plate Capacitor (2)
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1/25/07184 Lecture 1111 Clicker Question What is the NET CHARGE on the charged capacitor? A: +q+(-q)=0 B: |+q|+|-q|=2q C: q D: none of the above
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1/25/07184 Lecture 1112 Clicker Question What is the NET CHARGE on the charged capacitor? A: +q+(-q)=0 Charges are added with their signs. However, we refer to the charge of a capacitor as being q (the charge of a capacitor is not the net charge!)
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1/25/07184 Lecture 1113 Definition of Capacitance The definition of capacitance is The units of capacitance are coulombs per volt. The unit of capacitance has been given the name farad (abbreviated F) named after British physicist Michael Faraday (1791 - 1867) A farad is a very large capacitance Typically we deal with F (10 -6 F), nF (10 -9 F), or pF (10 -12 F)
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1/25/07184 Lecture 1114 Charging/Discharging a Capacitor We can charge a capacitor by connecting the capacitor to a battery or to a DC power supply. A battery or DC power supply is designed to supply charge at a given voltage. When we connect a capacitor to a battery, charge flows from the battery until the capacitor is fully charged. If we then disconnect the battery or power supply, the capacitor will retain its charge and voltage. A real-life capacitor will leak charge slowly, but here we will assume ideal capacitors that hold their charge and voltage indefinitely.
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1/25/07184 Lecture 1115 Charging/Discharging a Capacitor (2) Illustrate the charging processing using a circuit diagram. Lines represent conductors The battery or power supply is represented by The capacitor is represented by the symbol This circuit has a switch (ab) (open) When the switch is between positions a and b, the circuit is open (not connected). (pos a) When the switch is in position a, the battery is connected across the capacitor. Fully charged, q = CV. (pos b) When the switch is in position b, the two plates of the capacitor are connected. Electrons will move around the circuit – a current will flow -- and the capacitor will discharge.
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1/25/07184 Lecture 1116 Demo: Big Spark Energy stored in this particular capacitor: 90 J This is equivalent to the kinetic energy of a mass of 1 kg moving at a velocity of 13.4 m/s!
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1/25/07184 Lecture 1117 Parallel plate capacitor – -- a simple ideal model
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1/25/07184 Lecture 1118 Parallel Plate Capacitor Consider two parallel conducting plates separated by a distance d This arrangement is called a parallel plate capacitor. The upper plate has +q and the lower plate has –q. The electric field between the plates points from the positively charged plate to the negatively charged plate. We will assume ideal parallel plate capacitors in which the electric field is constant between the plates and zero elsewhere. Real-life capacitors have fringe field near the edges.
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1/25/07184 Lecture 1119 Parallel Plate Capacitor (2) We can calculate the electric field between the plates using Gauss’ Law We take a Gaussian surface shown by the red dashed line Flux (through bottom surface of the top plate only!) = EA Enclosed charge = q
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1/25/07184 Lecture 1120 Parallel Plate Capacitor (3) Now we calculate the electric potential across the plates of the capacitor in terms of the electric field. We define the electric potential across the capacitor to be V and we carry out the integral in the direction of the blue arrow. V = Ed Note: V = V(+) V( )
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1/25/07184 Lecture 1121 Remember the definition of capacitance… … so the capacitance of a parallel plate capacitor is Parallel Plate Capacitor (4) Variables: A is the area of each plate d is the distance between the plates Note that the capacitance depends only on the geometrical factors and not on the amount of charge or the voltage across the capacitor.
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1/25/07184 Lecture 1122 Demo: Parallel Plate Capacitor The voltage of a charged capacitor as function of the distance between the plates For constant charge, the voltage is proportional to the distance between the plates
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1/25/07184 Lecture 1123 Example - Capacitance of a Parallel Plate Capacitor We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm 2 separated by a distance of 1.00 mm. What is the capacitance of this parallel plate capacitor? Result: A parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm separated by 1 mm produces a capacitor with a capacitance of about 0.5 nF. C = 0.553 nF
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1/25/07184 Lecture 1124 Example 2 - Capacitance of a Parallel Plate Capacitor We have a parallel plate capacitor constructed of two parallel plates separated by a distance of 1.00 mm. What area is required to produce a capacitance of 1.00 F? Result: A parallel plate capacitor constructed out of square conducting plates 10.6 km x 10.6 km (6 miles x 6 miles) separated by 1 mm produces a capacitor with a capacitance of 1 F.
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1/25/07184 Lecture 1125 Example - Capacitance, Charge and Electrons … A storage capacitor on a random access memory (RAM) chip has a capacitance of 55 nF. If the capacitor is charged to 5.3 V, how many excess electrons are on the negative plate? Idea: We can find the number of excess electrons on the negative plate if we know the total charge q on the plate. Then, the number of electrons n=q/e, where e is the electron charge in coulomb. Second idea: The charge q of the plate is related to the voltage V to which the capacitor is charged: q=CV.
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