Capacitance Capacitance of Sphere Spherical Capacitor Parallel plate cylinders, Energy in dielectrics. PPT No. 14

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1 Capacitance Capacitance of Sphere Spherical Capacitor Parallel plate cylinders, Energy in dielectrics PPT No. 14

2 Capacitor Capacitor is a device used to store energy as potential energy in an electric field. Capacitors also have many other uses in the present electronic and microelectronic age. The basic elements of any capacitortwo isolated conductors of arbitrary shape which are close to each other and have equal and opposite charge.

3 Capacitor Symbol: The symbols of capacitor are shown below:- variable capacitor

4 Capacitor Charge on a Capacitor When a capacitor is charged, its plates have equal but opposite charge of +Q and Q. Net charge on a capacitor is always zero, The term charge Q on a capacitor, refers to the absolute value i.e. the magnitude of the charge on each plate.

5 Capacitor Capacitance When an isolated conductor is given a charge Q, it acquires certain potential V. The plates in a capacitor are conductors, they are equipotential surfaces: All points on a plate are at the same electric potential V.

6 Capacitance The charge Q and the potential difference V for a capacitor are directly proportional to each other, i.e. Q= CV. The proportionality constant C is called the capacitance of the capacitor. The capacitance of a capacitor is defined as the magnitude of the charge Q on the +ve plate divided by the magnitude of the potential difference V between plates C = Q/V

7 Capacitance Capacitance of a conductor is equal to the total charge in coulomb required to raise the potential by 1 volt. SI unit of capacitance is farad. 1 farad = 1 coulomb/ 1 volt Dielectrics are used in capacitors as polarizable media, to reduce dangerously strong electric fields, to increase the capacitance by factor k.

8 Capacitance of a Cylindrical Capacitor An excellent example of a cylindrical capacitor is the coaxial cable used in cable TV systems. Suppose that capacitor is composed of an inner cylinder with radius a enclosed by an outer cylinder with radius b. It has a uniform charge per unit length

9 Capacitance of a Cylindrical Capacitor Fig. (a) A cylindrical capacitor (b) Horizontal cross-section

10 Capacitance of a Cylindrical Capacitor First the electric field E between the plates is evaluated, using Gauss' Law. Consider a Gaussian surface that is cylindrical in shape having radius r and length L

11 Using the relation between V and E Capacitance of a Cylindrical Capacitor

12 Capacitance of a Cylindrical Capacitor This expression gives the potential difference across the cylinders in terms of dimensions of capacitor a, b and charge/ unit length λ Since the outer plate is negative, its voltage can be set equal to 0, Hence the potential difference V across the capacitor plates is given by

13 Capacitance of a Cylindrical Capacitor Capacitance C is calculated using the relation Q = CV This expression represents the capacitance per unit length of cylindrical capacitor.

14 Capacitance of A Parallel-plate Capacitor Consider the configuration of a parallel plate capacitor: Two parallel conducting plates of area A separated by a distance d. Surface density of charge of each plate is σ = Q/A. Magnitude of total charge Q on each plate is given by Q = σ A Electric field between the plates is E directed from the positive plate to negative plate is E = σ / ε0 (by applying Gauss Law)

15 Capacitance of A Parallel-plate Capacitor Fig. A Parallel Plate Capacitor

16 Capacitance of A Parallel-plate Capacitor The potential difference across the plates is equal to However, plate B is defined to be at V = 0,

17 Capacitance of A Parallel-plate Capacitor Substituting into Q = CV yields

18 Capacitance of A Parallel-plate Capacitor If a dielectric medium of dielectric constant k is filled completely between the plates then capacitance increases by k times i.e. C = kε o A / d The capacitance is maximized if the dielectric constant is maximized (k Maximum) the capacitor plates have large area (A Maximum) and are placed as close together as possible (d Minimum)

19 Capacitance of a Spherical Conductor Consider an isolated, initially uncharged, metal conductor. If a small amount of charge, q, is placed on the conductor, its voltage becomes as compared to V = 0 at infinity

20 Capacitance of a Spherical conductor Fig. A charged Spherical Conductor

21 Capacitance of a Spherical conductor Capacitance of a Spherical conductor is totally dependent on the sphere's radius.

22 Capacitance of A Spherical Capacitor A Spherical Capacitor consists of a conducting sphere of radius a surrounded by a concentric spherical shell of radius b. The inner sphere is given charge + Q, then charge induced on the inner surface of spherical shell is Q.

23 Capacitance of A Spherical Capacitor Fig. A Spherical Capacitor

24 Capacitance of A Spherical Capacitor Consider the Gaussian surface of radius r such that a < r < b. The electric field intensity E at any point on the Gaussian surface is given by Gauss law as (ε 0 is the permittivity of free space) E is contributed by the inner sphere alone and it is directed radially outwards

25 Capacitance of A Spherical Capacitor The potential difference between the concentric sphere and shell can be found by integrating the electric field along a radial line: From the definition of capacitance, the capacitance is

26 Capacitance of A Spherical Capacitor If the space between the sphere and shell is filled with a material with dielectric constant k, then the capacitance of the above arrangement is multiplied by k.

27 Energy in Dielectrics The energy density on capacitor with air as dielectric During charging of a capacitor electrons are removed from the positive plate and carried to the negative plate. Work is done to deposit charge Q to acquire potential V. It is stored in the form of potential energy. The relation for energy per unit volume stored in a capacitor with air as dielectric is derived in terms of the magnitude of its electric field as follows

28 Energy in Dielectrics

29 B) The Energy Density within a Dielectric Medium The capacitance C of a capacitor filled with a dielectric having dielectric constant k is increased by k times of its value in vacuum C0. The work requird to charge a capacitor filled with a dielectric increases by the factor k. The formula for energy stored in an electrostatic system within a dielectric medium can be derived as follows

30 B) The Energy Density within a Dielectric Medium Consider a system of free charges embedded in a dielectric medium. It is assumed that the original charges and the dielectric are held fixed in position, so that no mechanical work is performed. When a small amount of free charge is added to the system having electrostatic potential the polarization and bound charge distribution change.

31 B) The Energy Density within a Dielectric Medium For only the work done on the incremental free charge, the increase in the total energy U is calculated by taking the integral over all space is the change in the electric displacement vector D associated with the charge increment

32 B) The Energy Density within a Dielectric Medium This equation can be written using integration by parts surface integral is written using divergence theorem. It vanishes if the dielectric medium is of finite spatial extent. Incremental increase in energy is given by

33 B) The Energy Density within a Dielectric Medium It is assumed that dielectric medium is linear. Hence E can be expressed as a function of D by where the dielectric constant is independent of the electric field. Then energy increment can be integrated.

34 B) The Energy Density within a Dielectric Medium The change in energy associated with taking the displacement field from zero to D(r) At all points in space is It reduces to The electrostatic energy density inside a dielectric is =>

35 B) The Energy Density within a Dielectric Medium The electrostatic energy density inside a dielectric is An important note- This standard result is valid only for linear dielectrics where electric displacement D varies linearly with the electric field E.

36 The Energy Density within a Dielectric Medium

HW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.

HW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 22.P.053 The figure below shows a portion of an infinitely