Capacitors Capacitors are devices that can store electrical energy Capacitors are used in many every day applications Physics for Scientists Engineers 2 Heart defibrillators Camera flash units Capacitors are an integral part of modern day electronics Spring Semester 2005 Lecture 12 Capacitors and be micro sized on computer chips or super sized for high power circuits such as FM radio transmitters February 2 2005 Physics for Scientists Engineers 2 1 February 2 2005 Capacitance q q However in general a capacitor consists of two separated conductors usually called plates even if these conductors are not simple planes We charge the capacitor by placing We will start our study of capacitors by defining a convenient geometry and generalize from there a charge q on the top plate a charge q on the bottom plate We will start with a capacitor consisting of two parallel conducting planes each with area A separated by a distance d Because the plates are conductors the charge will distribute itself evenly over the surface of the conducting plates We assume that these plates are in a vacuum air is very close to a vacuum The electric potential V is proportional to the amount of charge on the plates We call this device a parallel plate capacitor Physics for Scientists Engineers 2 2 Parallel Plate Capacitor Capacitors come in a variety of sizes and shapes February 2 2005 Physics for Scientists Engineers 2 3 February 2 2005 Physics for Scientists Engineers 2 4 1 Parallel Plate Capacitor 2 Definition of Capacitance The definition of capacitance is q q C The units of capacitance are coulombs per volt The proportionality constant between the charge q and the electric potential difference V is the capacitance C The unit of capacitance has been given the name farad abbreviated F named after British physicist Michael Faraday 1791 1867 q CV We will call the electric potential difference V the potential or the voltage between 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 1F Physics for Scientists Engineers 2 1C 1V A farad is a very large capacitance The capacitance of a device tells us how much charge is required to produce a given voltage across the plates February 2 2005 q V Typically we deal with F 10 6 F nF 10 9 F or pF 10 12 F 5 Charging Discharging a Capacitor February 2 2005 Physics for Scientists Engineers 2 6 Charging Discharging a Capacitor 2 Let s illustrate the charging processing using a circuit diagram We can charge a capacitor by connecting the capacitor to a battery or to a power supply In the circuit diagram A battery or power supply is designed to supply charge at a given voltage Lines represent conductors The battery or power supply is represented by When we connect a capacitor to a battery charge flows from the battery until the capacitor is fully charged The capacitor is represented by the symbol This circuit has a switch If we then disconnect the battery or power supply the capacitor will retain its charge and voltage 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 across the capacitor A real life capacitor will leak charge but here we will assume ideal capacitors that hold their charge and voltage indefinitely When the switch is in position b the two plates of the capacitor are connected Charge will flow between the plates and the capacitor will discharge February 2 2005 Physics for Scientists Engineers 2 7 February 2 2005 Physics for Scientists Engineers 2 8 2 Parallel Plate Capacitor Parallel Plate Capacitor 2 Consider two parallel conducting plates separated by a distance d 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 Gaussian surface does not contribute to the integral The vertical parts of the Gaussian surface do not contribute because the electric field is zero outside the capacitor The only contribution to the integral comes from the Gaussian surface inside the constant electric field of the capacitor 0 EidA q 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 charge 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 February 2 2005 Physics for Scientists Engineers 2 9 February 2 2005 Parallel Plate Capacitor 3 0 EA q i Remembering the definition of capacitance C Now we calculate the electric potential across the plates of the capacitor in terms of the electric field V 10 Parallel Plate Capacitor 4 We then get the electric field E inside the parallel plate capacitor to be f Physics for Scientists Engineers 2 q V We get the result for the capacitance of a parallel plate capacitor Eids C We define the electric potential across the capacitor to be V and we carry out the integral in the direction of the blue arrow 0 A d 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 capacitor depends only on the geometry of the capacitor and not on the amount of charge or the voltage across the capacitor The integral gives us V Ed February 2 2005 Physics for Scientists Engineers 2 11 February 2 2005 Physics for Scientists Engineers 2 12 3 Example Capacitance of a Parallel Plate Capacitor We have a parallel plate capacitor constructed of two parallel plates each with area 625 cm2 separated by a distance of 1 00 mm What is the capacitance of this parallel plate capacitor Example 2 Capacitance of a Parallel Plate Capacitor F m 0 0625 m 2 1 00 10 C 0 553 nF 3 m 5 53 10 What area is required to produce a capacitance of 1 00 F A 10 F Physics for Scientists Engineers 2 13 February 2 2005 Cylindrical Capacitor Physics for Scientists Engineers 2 14 We can apply Gauss Law to get the electric field between the two cylinder using a Gaussian surface in the for of a cylinder with radius r and length L as illustrated by the red lines The inner cylinder has radius r1 and the outer cylinder has radius r2 0 EidA 0 EA 0 E 2 r L q L Both cylinders have charge per unit length
View Full Document
Unlocking...