3.capacitors

  • Uploaded by: Sanjay Pandey
  • 0
  • 0
  • May 2020
  • PDF

This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA


Overview

Download & View 3.capacitors as PDF for free.

More details

  • Words: 1,072
  • Pages: 4
1

3. Capacitors

1. Capacitor A combination of two conductors placed close to each other is called a capacitor. Oneof the conductors is given a positive charge and the other a negative charge. 2.

Capacitance

For a given capacitor, the charge ๐‘„๐‘„ on the capacitor is proportional to the potential difference ๐‘‰๐‘‰ between the two plates So

๐‘„๐‘„ โˆ ๐‘‰๐‘‰

Or

๐ถ๐ถ is called the capacitance of the capacitor.

๐‘„๐‘„ = ๐ถ๐ถ๐ถ๐ถ

SI unit of capacitance is ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘/๐‘ฃ๐‘ฃ๐‘ฃ๐‘ฃ๐‘ฃ๐‘ฃ๐‘ฃ๐‘ฃ which is written as farad. The symbol F is used for it.

To put equal and opposite charges on the two conductors they may be connected to the terminals of a battery. 3.

Calculation of Capacitance

For parallel plate capacitor

๐ถ๐ถ =

๐œ€๐œ€0 ๐ด๐ด ๐‘‘๐‘‘

๐ด๐ด = area of the flat plates (each used in the capacitor) ๐‘‘๐‘‘ = distance between the plate

Spherical capacitor

It consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor. If inner sphere radius is ๐‘…๐‘…1 and Outer sphere radius is ๐‘…๐‘…2

Inner sphere is given positive charge and outer sphere negative charge. ๐ถ๐ถ = 4๐œ‹๐œ‹๐œ€๐œ€0 ๐‘…๐‘…1 ๐‘…๐‘…2 /[๐‘…๐‘…2 โˆ’ ๐‘…๐‘…1 ]

If the capacitor is an isolated sphere (outer sphere is assumed to be at infinity, hence ๐‘…๐‘…2 is infinity and ๐‘‰๐‘‰ becomes

V = potential

๐ถ๐ถ = 4๐œ‹๐œ‹๐œ€๐œ€0 ๐‘…๐‘…1

๐‘„๐‘„/๐ถ๐ถ = ๐‘„๐‘„/4๐œ‹๐œ‹๐œ€๐œ€0 ๐‘…๐‘…1

Parallel limit: if both ๐‘…๐‘…1 and ๐‘…๐‘…2 are made large but ๐‘…๐‘…2 โˆ’ ๐‘…๐‘…1 = ๐‘‘๐‘‘ is kept fixed, then we can write 4๐œ‹๐œ‹๐‘…๐‘…1 ๐‘…๐‘…2 = 4๐œ‹๐œ‹๐œ‹๐œ‹ยฒ = ๐ด๐ด; where ๐‘…๐‘… is approximately the radius of each sphere, and ๐ด๐ด is the surface area of the sphere. 4.

๐ถ๐ถ = ๐œ€๐œ€0 ๐ด๐ด/๐‘‘๐‘‘; where ๐ด๐ด = 4๐œ‹๐œ‹๐‘…๐‘…1 ๐‘…๐‘…2 = 4๐œ‹๐œ‹๐œ‹๐œ‹ยฒ

Cylindrical Capacitor

It consists of a solid or hollow cylindrical conductor surrounded by another concentric hollow cylindrical conductor.

2 If inner cylinder radius is ๐‘…๐‘…โ‚ and Outer cylinder radius is ๐‘…๐‘…โ‚‚ and length is ๐‘™๐‘™, Inner cylinder is given positive charge and outer cylinder negative charge ๐ถ๐ถ = 2๐œ‹๐œ‹๐œ‹๐œ‹โ‚€๐‘™๐‘™/๐‘™๐‘™๐‘™๐‘™(๐‘…๐‘…โ‚‚/๐‘…๐‘…โ‚)

5. Combination of capacitors โ€ข Series combination โ€ข

6.

Parallel combination

1/๐ถ๐ถ = 1/๐ถ๐ถโ‚ + 1/๐ถ๐ถโ‚‚ + 1/๐ถ๐ถโ‚ƒ . .. ๐ถ๐ถ = ๐ถ๐ถโ‚ + ๐ถ๐ถโ‚‚ + ๐ถ๐ถโ‚ƒ

Force between plates of a capacitor

Plates on a parallel capacitor attract each other with a force 7.

๐น๐น = ๐‘„๐‘„ยฒ/2๐ด๐ด๐ด๐ดโ‚€

Energy stored in a capacitor

Capacitor of capacitance C has a stored energy

๐‘ˆ๐‘ˆ = ๐‘„๐‘„ยฒ/2๐ถ๐ถ = ๐ถ๐ถ๐ถ๐ถยฒ/2 = ๐‘„๐‘„๐‘„๐‘„/2

where ๐‘„๐‘„ is the charge given to it. 8.

Dielectric material

9.

Change in capacitance of a capacitor with dielectric in it.

In dielectric materials, there are no free electrons. Electrons are bound to the nucleus in atoms. Basically they are insulators. But when a charge is applied, in these materials also atoms or molecules are oriented in a such way that there is an induced. For example, in the case of rectangular slab of a dielectric, if an electric field is applied from left to right, the left surface of the slab gets a negative charge, and the right surface gets positive charge. The surface charge density of the induced charge can be related to a measure called Polarization ๐‘ƒ๐‘ƒ (which is dipole moment induced per unit volume - where is the dipole? in the dielectric slab as the two sides have opposite charges) If ๐œŽ๐œŽ๐‘๐‘ is the magnitude of the induced charge per unit area on the faces.

The dipole moment of the slab

= ๐‘๐‘โ„Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ร— (๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“)

= (๐œŽ๐œŽ๐‘๐‘ ๐ด๐ด)๐‘™๐‘™ = ๐œŽ๐œŽ๐‘๐‘ (๐ด๐ด๐ด๐ด).

Where, ๐ด๐ด is area of cross section of the dielectric slab

As polarization is defined as dipole moment induced per unit volume, ๐‘ƒ๐‘ƒ =

๐œŽ๐œŽ๐‘๐‘ (๐ด๐ด๐ด๐ด ) ๐ด๐ด๐ด๐ด

= ๐œŽ๐œŽ๐‘๐‘

(๐ด๐ด๐ด๐ด = volume of slab)

Thus the induced surface charge density is equal in magnitude to the polarization P. 10. Dielectric constant

3

slab.

Because of induced charge, electric field is produced in the slab which is against the field applied on the ๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘… ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ = ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ โˆ’ ๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘–๐‘– ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ ๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘…๐‘… ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ = ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“/๐พ๐พ

๐พ๐พ is greater than 1 and is a constant for give materials. ๐พ๐พ is called the dielectric constant or relative permittivity of the dielectric. 11. Dielectric strength

If a very high electric field is created in a dielectric, electrons in valence shell may get detached from their parent atoms and move freely like in a conductor. This phenomenon is called is dielectric breakdown. The electric field at which breakdown occurs is called the dielectric strength of the material. 12. Capacitance of a parallel plate capacitor with dielectric ๐ถ๐ถ = ๐พ๐พ๐พ๐พโ‚€

where ๐ถ๐ถโ‚€ is capacitance of a similar capacitor without dielectric.

Because ๐พ๐พ > 1, the capacitance of a capacitor is increased by a factor of ๐พ๐พ when the space between the parallel plates is filled with a dielectric. 13. Magnitude of induced charge in term of ๐‘ฒ๐‘ฒ

๐‘„๐‘„๐‘๐‘ = ๐‘„๐‘„[1 โˆ’ (1/๐พ๐พ)]

๐‘„๐‘„๐‘๐‘ = induced charge in the dielectric ๐‘„๐‘„ = Applied charge

๐พ๐พ = dielectric constant

14. Gauss's law when dielectric materials are involved โˆฎ ๐พ๐พ๐‘ฌ๐‘ฌ. ๐’…๐’…๐’…๐’… = ๐‘„๐‘„๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ /๐œ€๐œ€0

โ€ฆ (1)

Where integration is over the surface, ๐‘ฌ๐‘ฌ and ๐’…๐’…๐’…๐’… are vectors, ๐‘„๐‘„๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ is the free charge given (charge due to polarisation is not considered) and ๐พ๐พ is dielectric constant. โ€ข

The law can also be written as

โˆฎ ๐‘ซ๐‘ซ. ๐’…๐’…๐’…๐’… = ๐‘„๐‘„๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“

... (2)

where ๐‘ซ๐‘ซ = ๐‘ฌ๐‘ฌ๐œ€๐œ€0 + ๐‘ท๐‘ท; ๐‘ฌ๐‘ฌ and ๐‘ท๐‘ท are vectors. ๐‘ฌ๐‘ฌ = electric field and ๐‘ท๐‘ท is polarisation

15. Electric field due to a point charge placed inside a dielectric ๐ธ๐ธ = ๐‘ž๐‘ž/4๐œ‹๐œ‹๐œ‹๐œ‹โ‚€๐พ๐พ๐พ๐พยฒ

16. Energy in the electric field in a dielectric 17. Corona discharge

1

๐‘ข๐‘ข = ๐พ๐พ๐พ๐พโ‚€๐ธ๐ธยฒ 2

4 If a conductor has a pointed shape like a needle and a charge given to it, the charge density at the pointed end will be very high. Correspondingly, the electric field near these pointed ends will be very high which may cause dielectric breakdown in air. The charge may jump from the conductor to the air. Often this discharge of charge inot air is accompanied by a visible glow surrounding the pointed end and this phenomenon is called corona discharge. 18. High voltage generator โ€“ Van de Graaff Generator

The apparatus transfers positive charge to a sphere continuously till the potential reaches to around 3 ร— 106 ๐‘‰๐‘‰ at which point corona discharge takes place and hence no further charge can be transferred. The charge of course can be increased by enclosing the sphere in a highly evacuated chamber.

More Documents from "Sanjay Pandey"