III B. Conductors, Capacitors, Dielectrics              

 

AP Exam Objectives

 

B. Conductors, Capacitors, Dielectrics

1. Electrostatics with Conductors

     a. Students should understand the nature of electric fields in and around conductors so they

        can:

        (1) Explain the mechanics responsible for the absence of electric field inside a conductor,   and why all excess charge must reside on the surface of the conductor.

        (2) Explain why a conductor must be an equipotential, and apply this principle in analyzing   

             what happens when conductors are joined by wires.

        (3) Determine the direction of the force on a charged particle brought near an uncharged or  

             grounded conductor

 

     b. Students should be able to describe and sketch a graph of the electric field and potential

         inside and outside a charged conducting sphere so they can:

        (1) Describe qualitatively the process of charging by induction.

        (2) Determine the direction of the force on a charged particle brought near an uncharged or  

             grounded conductor

 

2. Capacitors and Dielectrics

     a. Students should know the definition of capacitance so they can relate stored charge and

        voltage for a capacitor.

     b. Students should understand energy storage in capacitors so they can:

        (1) Relate voltage, charge, and stored energy for a capacitor.

        (2) Recognize situations in which energy stored in a capacitor is converted to other forms.

     c. Students should understand the physics of the parallel-plate capacitor so they can:

        (1) Describe the electric field inside the capacitor, and relate the strength of this field to the potential difference between the plates and the plate separation.

        (4) Determine how changes in dimension will affect the value of capacitance.

 

Sharing of Charge                                                                                                                                                                                                                                     

·        All systems, mechanical and electrical, come to equilibrium when the energy of the system is at a minimum. When a conductor is in equilibrium, the electric field everywhere inside the conductor is zero. This is fairly obvious for a neutral conductor where the electric field emanating from all of the positive charges end on the negative charges. But, what happens when an excess charge is placed on a conductor. Since the charges within a conductor are free to move about, the excess charge moves to the outside surface of the conductor, reestablishing equilibrium and the internal field reduces to zero

·        Charges move until all parts of a conductor are at the same potential.

·        A charged sphere shares charge equally with a neutral sphere of equal size. Charges flow until all parts of a conducting body, two touching spheres in this case, are at the same potential.

Which way will the charge flow when each of the following conducting spheres are connected?   

 


- Q

 

- Q

 
                                                               

- Q

 

- Q

 

- Q

 
                                                           

 

 

Which way will the charge flow when each of the above conducting spheres are connected to the Earth?

·        The potential on the larger sphere is lower than the potential on the smaller sphere because the charges are farther apart thus the repulsive force between them is reduced. If the two spheres are touched together, charges will move to the sphere with the lower potential; that is from the smaller to the larger sphere. The result is a greater charge on the larger sphere when the two different-sized spheres are at the same potential.

grounding- touching an object to Earth to eliminate excess charge.

·        Earth is a very large sphere so a charged body that touches it will result in practically any amount of charge to flow from the body to the Earth, allowing the body to become neutral.

 

Electric Fields Near Conductors

·        The electric field around a conducting body depends on the structure and shape of the body.

·        The charges on a conductor are spread apart as far apart as possible in order to make the energy of the system as low as possible. The result is that all charges are on the surface of a solid conductor. If the conductor is hollow, excess charges will move to the outside surface. In this way, a closed metal container shields the inside from electric fields. (Faraday cage)

·        The electric field around the outside of a conductor depends on the shape of the body as well as its potential. The charges are closer together at the sharp points of a conductor, therefore, the field lines are closer together; the field is stronger.

·        Application: To reduce corona and sparking, conductors that are highly charged or operate at high potentials are made smooth in shape. Lightning rods are pointed so that the electric field will be strong near the end of the rod thus attracting lightning to it rather than the building it is protecting.


Storing Electric Energy- The Capacitor                                                      P 59                         

1746- Musschenbroek- invented a device that could store electric charge.

Text Box: + + + + + + + + + +Leyden jar  (see our Leyden jar)

·        Two isolated conductors having a potential difference of DV

Text Box: - - - - - - - - - - - - - - -between them with equal and opposite charges + Q and - Q ,

constitute what is called a capacitor because as long as they

are isolated they have the capacity for storing charge.

·        The difference in potential is directly proportional to the charge of the objects. This should be rather obvious since doubling the charge also doubles the electric field in the region about the objects and hence, the work-per-unit charge (that is, the potential difference, DV) required to move a test charge between two points in the field also doubles.

Q µ DV      or       Q = CDV               Where C is a constant of proportionality

·        For a given size and shape of a capacitor, the ratio of stored charge to potential difference, Q/V, is a constant called the capacitance  (C).

·        The capacitance is independent of the charge on it. It can be measured by placing charge +q on one plate and -q on the other, and measuring the potential difference, DV, that results. The AP equation is...

  C = Q/V                unit: = 1C/1V  = farad (F)                                         (Eq ***)

·        A capacitor in electronics lingo is- a device designed to have a specific capacitance.

·        All capacitors used in circuitry are made up of two conductors of equal and opposite charges, separated by an insulator called a dielectric. Capacitors have become very important devices in electronic circuits, electrical machinery, etc. since they are devices which store charge and electrical energy.

Applications: Tuners in radio receivers, “condensers” in ignition systems of

                        cars, camera flash, strobe light   

Demo

 

Practice Problem 10  The Charge on the Plates of a Capacitor

Text Box: + + + + + + + + + +

A 3.0-mF capacitor is connected to a 12-V battery. What is the magnitude of the charge on each plate of the capacitor?

 
Text Box: - - - - - - - - - - - - - - -
 

 

 

 

 

 


C = Q /V

Q = C V

    = (3 x 10-6 C)(12 V)

    = 36 mC

 

 

 

The Parallel-Plate Capacitor

·        Recall that a constant electric field can be made by placing two flat plates with equal and opposite charges parallel to each other.  Represent this electric field on the diagram above.

·        For the parallel plate capacitor shown in the figure above, the voltage V is given by Ed where E is the electric field strength between the plates and d is the plate separation.

·        The electric field E depends on the amount of charge on the plates                     which in turn depends on the area of the plates (more area can hold more charge) with the proportionality constant = 1/ eo . Thus E = Q/(eo A). The capacitance of a parallel-plate capacitor, therefore, is directly proportional to the area of one of the plates and inversely proportion to their separation.

    C = Q/V         V = Ed           C = Q/Ed           

                                                                          C =  eo A/d                  (Eq) ***

A = the area of one of the plates.

eo = constant = permittivity of free space = 8.85 x 10-12 C2/N m2 = 1/4 p k 

 

Practice Problem 11  Calculating Capacitance for a Parallel Plate Capacitor

C =  eo A/d

   = (8.85 x 10-12 C2/N m2)(.0002 m2) / .001 m

   = 1.77 x 10-10 F

 

Energy Stored in a Charged Capacitor

·        The energy stored in a charged capacitor is equal to the work required to charge it. The work required to transfer a small amount of charge DQ from the negative to the positive plate at potential V is DW = VDQ. Since V increases linearly as more charge is added, the total work must be the average V, which is (1/2) (0 + Vf), times Q. So, the total work done and thus the electric potential energy (Uc) stored in the capacitor is

                       Uc = 1/2 QV = 1/2 CV2                                                                                                  (Eq) ***

 

Practice Problem 12  Energy Stored in a Charged Capacitor

Uc = ˝ CV2

     = ˝ (1.77 x 10-10 F)((120 V) 2  = .036 J

 

·        Thus the usefulness of a capacitor lies in its ability to store electrical energy which can, upon discharging the capacitor, then be converted to other useful forms such as sound, light, heat, and mechanical energy.