Review The capacitance of a spherical capacitor is C 4 0 Physics for Scientists Engineers 2 r1r2 r2 r1 r1 is the radius of the inner sphere r2 is the radius of the outer sphere Spring Semester 2005 The capacitance of an isolated spherical conductor is Lecture 14 C 4 0 R R is the radius of the sphere February 2 2005 Physics for Scientists Engineers 2 1 February 2 2005 Review 2 2 Energy Stored in Capacitors The equivalent capacitance for n capacitors in parallel is A battery must do work to charge a capacitor We can think of this work as changing the electric potential energy of the capacitor n Ceq Ci The differential work dW done by a battery with voltage V to put a differential charge dq on a capacitor with capacitance C is i 1 dW Vdq The equivalent capacitance for n capacitors in series is q dq C The total work required to bring the capacitor to its full charge q is Wt dW n 1 1 Ceq i 1 Ci qt 0 q 1 qt2 dq C 2C This work is stored as electric potential energy U February 2 2005 Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 3 1 q2 1 1 CV 2 qV 2C 2 2 February 2 2005 Physics for Scientists Engineers 2 4 1 Energy Density in Capacitors Energy Density in Capacitors 2 Recognizing that V d is the magnitude of the electric field E we obtain an expression for the electric potential energy density for parallel plate capacitor We can define the energy density u as the electric potential energy per unit volume u U volume Taking the special case of a parallel plate capacitor that has no fringe field the volume between the plates is the area of each plate times the distance between the plates Ad u 1 U CV 2 CV 2 2 Ad Ad 2Ad 0 A 2 2 V 1 V d u 0 2Ad 2 d Physics for Scientists Engineers 2 1 0E 2 2 This result which is specific to the parallel plate capacitor is in fact much more general This equation holds for all electric fields produced in any way Inserting our formula for the capacitance of a parallel plate capacitor we get February 2 2005 u Can be used to describe the electric potential energy stored in an electric field per unit volume occupied by that field 5 February 2 2005 Example Thundercloud Physics for Scientists Engineers 2 6 Example Thundercloud 2 Question 1 Suppose a thundercloud with a width of 2 0 km and a length of 3 0 km hovers over a flat area at an altitude of 500 m and carries a charge of 160 C We can approximate the cloud ground system as a parallel plate capacitor whose capacitance is C Question 1 What is the potential difference between the cloud and the ground 0 A 8 85 10 12 F m 2000 m 3000 m 0 11 F d 500 m The charge carried by the cloud is 160 C which means that each plate of the capacitor has a charge of 80 C Question 2 Knowing that lightning strikes require electric field strengths of approximately 2 5 MV m are these conditions sufficient for a lightning strike V Question 3 q 80 C 7 2 10 8 V C 0 11 F 720 million volts What is the total electrical energy contained in this cloud February 2 2005 Physics for Scientists Engineers 2 7 February 2 2005 Physics for Scientists Engineers 2 8 2 Example Thundercloud 3 Capacitors with Dielectrics Question 2 We have been discussing capacitors with air or vacuum between the plates We know the potential difference between the cloud and ground so we can calculate the electric field E However most real life capacitors have an insulating material called a dielectric between the two plates V 7 2 10 8 V 1 5 MV m d 500 m The dielectric serves several purposes Which is lower than 2 5 MV m so no lightning cloud to ground May have lightning to radio tower or tree Provides a convenient way to maintain mechanical separation between the plates Question 3 Provides electrical insulation between the plates The total energy stored in a parallel plate capacitor is Allows the capacitor to hold a higher voltage Increases the capacitance of the capacitor 1 U qV 0 5 80 C 7 2 10 8 V 2 9 1010 J 2 Takes advantage of the molecular structure of the dielectric material Enough energy to run a 1500 W hair dryer for more than 5000 hours February 2 2005 Physics for Scientists Engineers 2 9 Capacitors with Dielectrics 2 10 Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as We can express the capacitance of a capacitor with a dielectric with dielectric constant between the plates as E C Cair Eair q q 0 A A The constant 0 is the electric permittivity of free space Where Cair is the capacitance of the capacitor without the dielectric The constant is the electric permittivity of the dielectric material Placing the dielectric between the plates of the capacitor has the effect of lowering the electric field between the plates and allowing more charge to be stored in the capacitor Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 Parallel Plate Capacitor with Dielectric Placing a dielectric between the plates of a capacitor increases the capacitance of the capacitor by a numerical factor called the dielectric constant February 2 2005 February 2 2005 0 11 February 2 2005 Physics for Scientists Engineers 2 12 3 Dielectric Strength Dielectric Constant The dielectric strength of a material measures the ability of that material to withstand voltage differences The dielectric constant of vacuum is defined to be one If the voltage across a dielectric exceeds the breakdown potential the dielectric will break down and begin to conduct charge between the plates The dielectric constants and dielectric strengths of common materials are The dielectric constant of is close to one and we will use the dielectric constant of air as one in our problems Real life dielectrics enable a capacitor provide a given capacitance and hold the required voltage without breaking down Capacitors are usually specified in terms of their capacitance and rated voltage February 2 2005 Physics for Scientists Engineers 2 13 Material Dielectric Constant Air 1 atm Polystyrene Mylar Paper Water 1 00059 2 6 3 1 3 0 80 4 February 2 2005 Capacitor with Dielectric Dielectric Strength kV mm 2 5 20 280 8 3 1 Physics for Scientists Engineers 2 Capacitor with Dielectric 2 Question 1 Question 3 Consider a parallel plate capacitor with capacitance C 2 00 F connected to a battery with voltage V 12 0 V as shown What is the charge stored in the capacitor We isolate the charged capacitor with a dielectric by disconnecting it from the battery We remove the dielectric keeping the capacitor isolated What happens to the charge and voltage on
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