23. Electrostatic Energy and Capacitors

2 Topics Capacitors Electrostatic Energy Using Capacitors

Capacitors

4 A capacitor is a device that stores charge –Q on one conductor and charge +Q on the other conductor The stored charge creates an electric field, and therefore, a potential difference between the conductors –Q–Q+Q+Q

5 Capacitance The capacitance of a device is defined by The unit is the farad (F) = Coulomb/Volt But since the farad is such a huge unit, the more commonly used units are  F = F or pF = F Q is the charge stored on a conductor V is the potential difference between the conductors

6 Capacitance Spherical Conductor The potential on the surface of a spherical conductor of radius R is Therefore, its capacitance is

7 Capacitance Parallel Plate Capacitors If two conducting plates of area A are separated by a small distance d the electric field between them will be approximately constant and of magnitude

8 Capacitance Parallel Plate Capacitors Since the electric field is constant, the potential difference between the plates is simply so the capacitance is

9 Capacitance Cylindrical Capacitors A coaxial cable of length L is an example of a cylindrical capacitor R2R2 R1R1

Electrostatic Energy

11 Electrostatic Energy q1q1 q2q2 q3q3 Total work done a a a Work is required to assemble a charge distribution

12 Electrostatic Energy dq The work dW required to add an element of charge dq to an existing charge distribution is where V is the potential at the final location of the charge element. The total work required is therefore Since the electric is conservative, the work is stored as electrostatic energy, U.

13 Storage of Electrostatic Energy Work must be done to move positive charge from a negatively charged conductor to one that is positively charged. Or to move negative charge in the reverse direction.

14 Storage of Electrostatic Energy In moving charge dq, the electrostatic energy of the capacitor is increased by Therefore,

15 Energy Density of Electric Field Potential energy Electric field Electric potential

16 The energy density u E This expression holds true for any electric field Energy Density of Electric Field

17 Example – A Thunderstorm How much electrical energy is stored in a typical thundercloud? Assume a cloud of height h = 10 km, radius r = 10 km, with a uniform electric field E = 10 5 V/m.

18 Example – A Thunderstorm Narrative The problem is about stored electric energy. Since the electric field, E, is uniform, so to is the energy density u E = ½  0 E 2 in the cloud. Therefore, the electric energy stored in the cloud is just the electric energy density times the volume of the cloud.

19 Example – A Thunderstorm Diagram Thundercloud h = 10 km r = 10 km volume =  r 2 h

20 Example – A Thunderstorm Calculation The electric energy density in the cloud is u E = ½  0 E 2 = 4.4 x J/m 3. The volume of the cloud is, v =  r 2 h, that is, v = 3.1 x m 3. Therefore, the total electric energy stored in the cloud is U = u E v = 140 GJ.

Using Capacitors

22 The Effect of Dielectrics Michael Faraday 1791 – 1867 wikimedia Michael Faraday discovered that the capacitance increases when the space between conductors is replaced by a dielectric. Today, we understand this to be a consequence of the polarization of molecules.

23 The Effect of Dielectrics The polarized molecules of the dielectric tend to align themselves parallel to the electric field created by the charges on the conductors  b  b

24 The Effect of Dielectrics The bound charge  b induced on the surface of the dielectric creates an electric field opposed to the electric field of the free charge  f on the conductors, thereby reducing the field between them.

25 The Effect of Dielectrics The reduction in electric field strength from the initial field E 0 to the reduced field E is quantified by the dielectric constant  (kappa) The dielectric increases the capacitance by the same factor .

26 The Effect of Dielectrics For a parallel plate capacitor, with a dielectric between the plates, the electric field is is called the permittivity The product of the dielectric constant  and the permittivity of free space  0

27 Capacitors in Parallel At equilibrium, the potential across each capacitor is the same, namely, 12 V same potential

28 The two capacitors are equivalent to a single capacitor with capacitance

29 Capacitors in Series The sum of the potentials across both capacitors will be equal to 12 V

30 The potential V 1 across C 1 plus the potential V 2 across C 2 is equal to the potential difference V between points a and b: V = V 1 + V 2

31 Summary Capacitance C = Q / V (farad) Parallel plateC =  0 A/d Capacitors In parallel C = C 1 + C 2 In series 1/C = 1/C 1 + 1/C 2 Stored energy U = ½ QV Energy density u E = ½  0 E 2 Effect of dielectric E = E 0 / 