1February 2, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 12February 2, 2005 Physics for Scientists&Engineers 2 2CapacitorsCapacitors Capacitors are devices that can store electricalenergy Capacitors are used in many every-day applications• Heart defibrillators• Camera flash units Capacitors are an integral part of modern dayelectronics• Capacitors and be micro-sized on computer chips orsuper-sized for high power circuits such as FM radiotransmittersFebruary 2, 2005 Physics for Scientists&Engineers 2 3CapacitanceCapacitance Capacitors come in a variety of sizes andshapes However, in general, a capacitor consists oftwo separated conductors, usually calledplates, even if these conductors are notsimple planes We will start our study of capacitors by defining a convenientgeometry and generalize from there We will start with a capacitor consisting of two parallel conductingplanes, each with area A separated by a distance d We assume that these plates are in a vacuum (air is very close to avacuum) We call this device a parallel plate capacitorFebruary 2, 2005 Physics for Scientists&Engineers 2 4Parallel Plate CapacitorParallel 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 itselfevenly over the surface of the conducting plates The electric potential, V, is proportional to the amount of charge onthe plates+q!q2February 2, 2005 Physics for Scientists&Engineers 2 5Parallel Plate Capacitor (2)Parallel Plate Capacitor (2) The proportionality constant between the charge q and the electricpotential difference V is the capacitance C We will call the electric potential difference V the potential or thevoltage between the plates The capacitance of a device depends on the area of the plates and thedistance between the plates, but does not depend on the voltageacross the plates or the charge on the plates The capacitance of a device tells us how much charge is required toproduce a given voltage across the plates+q!qq = CVFebruary 2, 2005 Physics for Scientists&Engineers 2 6Definition of CapacitanceDefinition 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 MichaelFaraday (1791 - 1867) A farad is a very large capacitance• Typically we deal with µF (10-6 F), nF (10-9 F), or pF (10-12 F)C =qV1 F =1 C1 VFebruary 2, 2005 Physics for Scientists&Engineers 2 7Charging/Discharging a CapacitorCharging/Discharging a Capacitor We can charge a capacitor by connecting the capacitor to abattery or to a power supply A battery or power supply is designed to supply charge at agiven voltage When we connect a capacitor to a battery, charge flowsfrom the battery until the capacitor is fully charged If we then disconnect the battery or power supply, thecapacitor will retain its charge and voltage A real-life capacitor will leak charge, but here we willassume ideal capacitors that hold their charge and voltageindefinitelyFebruary 2, 2005 Physics for Scientists&Engineers 2 8Charging/Discharging a Capacitor (2)Charging/Discharging a Capacitor (2) Let’s illustrate the charging processing usinga circuit diagram In the 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• When the switch is between positions a and b, the circuit is open(not connected)• When the switch is in position a, the battery is connected acrossthe capacitor• When the switch is in position b, the two plates of the capacitor areconnected• Charge will flow between the plates and the capacitor will discharge3February 2, 2005 Physics for Scientists&Engineers 2 9Parallel Plate CapacitorParallel 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 chargeplate to the negatively charged plate We will assume ideal parallel plate capacitors in which the electric fieldis constant between the plates and zero elsewhere Real-life capacitors have fringe field near the edgesFebruary 2, 2005 Physics for Scientists&Engineers 2 10Parallel Plate Capacitor (2)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 The electric field is zero inside the conductor so the top part of the Gaussiansurface does not contribute to the integral The vertical parts of the Gaussian surface do not contribute because theelectric field is zero outside the capacitor The only contribution to the integral comes from the Gaussian surface insidethe constant electric field of the capacitor !0!Eid!A = q""February 2, 2005 Physics for Scientists&Engineers 2 11Parallel Plate Capacitor (3)Parallel Plate Capacitor (3) We then get the electric field E inside the parallel plate capacitor to be Now we calculate the electric potential across the plates of thecapacitor in terms of the electric field We define the electric potential across the capacitor to be V and wecarry out the integral in the direction of the blue arrow The integral gives us!0EA = q !V = "!Eid!sif#V = EdFebruary 2, 2005 Physics for Scientists&Engineers 2 12Parallel Plate Capacitor (4)Parallel Plate Capacitor (4) Remembering the definition of capacitance We get the result for the capacitance of a parallel plate capacitor Where•A is the area of each plate•d is the distance between the plates Note that this result for the capacitance of a parallel plate capacitordepends only on the geometry of the capacitor and not on the amountof charge or the voltage across the capacitorC =qVC =!0Ad4February 2, 2005 Physics for Scientists&Engineers 2 13Example - Capacitance of a Parallel Plate CapacitorExample - Capacitance of a Parallel Plate Capacitor We have a parallel plate capacitorconstructed of two parallel plates,each with area 625 cm2 separatedby a distance of 1.00 mm. What is the capacitance of
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