1/30/07184 Lecture 131 PHY 184 Spring 2007 Lecture 13 Title: Capacitors

1/30/07184 Lecture 132NotesNotes  Homework Set 3 is done!  Homework Set 4 is open and Set 5 opens Thursday morning  Midterm 1 will take place in class Thursday, February 8.  One 8.5 x 11 inch equation sheet (front and back) is allowed.  The exam will cover Chapters Homework Sets 1 - 4

1/30/07184 Lecture 133ReviewReview  The capacitance of a spherical capacitor is r 1 is the radius of the inner sphere r 2 is the radius of the outer sphere  The capacitance of an isolated spherical conductor is R is the radius of the sphere

1/30/07184 Lecture 134 Review (2)  The equivalent capacitance for n capacitors in parallel is  The equivalent capacitance for n capacitors in series is

1/30/07184 Lecture 135 Example - System of Capacitors  Let’s analyze a system of five capacitors  If each capacitor has a capacitance of 5 nF, what is the capacitance of this system of capacitors?

1/30/07184 Lecture 136 System of Capacitors (2)  We can see that C 1 and C 2 are in parallel and that C 3 is also in parallel with C 1 and C 2  We can define C 123 = C 1 + C 2 + C 3 = 15 nF  … and make a new drawing

1/30/07184 Lecture 137 System of Capacitors (3)  We can see that C 4 and C 123 are in series  We can define  … and make a new drawing = 3.75 nF

1/30/07184 Lecture 138 System of Capacitors (4)  We can see that C 5 and C 1234 are in parallel  We can define  And make a new drawing = 8.75 nF

1/30/07184 Lecture 139 System of Capacitors (5)  So the equivalent capacitance of our system of capacitors  More than one half of the total capacitance of this arrangement is provided by C 5 alone.  This result makes it clear that one has to be careful how one arranges capacitors in circuits.

1/30/07184 Lecture 1310 Clicker Question  Find the equivalent capacitance C eq A) B) C)

1/30/07184 Lecture 1311 Clicker Question  Find the equivalent capacitance C eq C) First Step: C 1 and C 2 are in series Second Step: C 12 and C 3 are in parallel

1/30/07184 Lecture 1312 A capacitor stores energy. Field Theory: The energy belongs to the electric field.

1/30/07184 Lecture 1313  A battery must do work to charge a capacitor.  We can think of this work as changing the electric potential energy of the capacitor.  The differential work dW done by a battery with voltage V to put a differential charge dq on a capacitor with capacitance C is  The total work required to bring the capacitor to its full charge q is  This work is stored as electric potential energy Energy Stored in Capacitors

1/30/07184 Lecture 1314  We define the energy density, u, as the electric potential energy per unit volume  Taking the ideal 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  Inserting our formula for the capacitance of a parallel plate capacitor we find Energy Density in Capacitors

1/30/07184 Lecture 1315  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  This result, which we derived for the parallel plate capacitor, is in fact completely general.  This equation holds for all electric fields produced in any way The formula gives the quantity of electric field energy per unit volume. Energy Density in Capacitors (2)

1/30/07184 Lecture 1316  An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. a) How much potential energy is stored in the electric field of the charged conductor? Key Idea: An isolated sphere has a capacitance of C=4  0 R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to Example - isolated conducting sphere … and substituting C=4  0 R gives

1/30/07184 Lecture 1317  An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. b) What is the field energy density at the surface of the sphere? Key Idea: The energy density u depends on the magnitude of the electric field E according to so we must first find the E field at the surface of the sphere. Recall: Example - isolated conducting sphere (Why?)

1/30/07184 Lecture 1318 Example: Thundercloud  Suppose a thundercloud with horizontal dimensions of 2.0 km by 3.0 km hovers over a flat area, at an altitude of 500 m and carries a charge of 160 C.  Question 1: What is the potential difference between the cloud and the ground?  Question 2: Knowing that lightning strikes require electric field strengths of approximately 2.5 MV/m, are these conditions sufficient for a lightning strike?  Question 3: What is the total electrical energy contained in this cloud?

1/30/07184 Lecture 1319 Example: Thundercloud (2)  Question 1  We can approximate the cloud-ground system as a parallel plate capacitor whose capacitance is  The charge carried by the cloud is 160 C, which means that the “plate surface” facing the earth has a charge of 80 C  720 million volts … …

1/30/07184 Lecture 1320 Example: Thundercloud (3)  Question 2  We know the potential difference between the cloud and ground so we can calculate the electric field  E is lower than 2.5 MV/m, so no lightning cloud to ground May have lightning to radio tower or tree….  Question 3  The total energy stored in a parallel place capacitor is Enough energy to run a 1500 W hair dryer for more than 5000 hours

1/30/07184 Lecture 1321 Clicker Question  A 1.0  F capacitor and a 3.0  F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors? A) U=0.5 J B) U=0.27 J C) U=1.5 J D) U=0.02 J Hint: Use

1/30/07184 Lecture 1322 Clicker Question  A 1.0  F capacitor and a 3.0  F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors? A) U=0.5 J U = 0.5 J

1/30/07184 Lecture 1323 Capacitors with Dielectrics  We have only discussed capacitors with air or vacuum between the plates.  However, most real-life capacitors have an insulating material, called a dielectric, between the two plates.  The dielectric serves several purposes: Provides a convenient way to maintain mechanical separation between the plates. Provides electrical insulation between the plates. Allows the capacitor to hold a higher voltage Increases the capacitance of the capacitor Takes advantage of the molecular structure of the dielectric material

1/30/07184 Lecture 1324 Capacitors with Dielectrics (2)  Placing a dielectric between the plates of a capacitor increases the capacitance of the capacitor by a numerical factor called the dielectric constant,   We can express the capacitance of a capacitor with a dielectric with dielectric constant  between the plates as  … where C air is the capacitance of the capacitor without the dielectric  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.

1/30/07184 Lecture 1325 Parallel Plate Capacitor with Dielectric  Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as  The constant  0 is the electric permittivity of free space  The constant  is the electric permittivity of the dielectric material