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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 ๐๐ on the capacitor is proportional to the potential difference ๐๐ between the two plates So
๐๐ โ ๐๐
Or
๐ถ๐ถ is called the capacitance of the capacitor.
๐๐ = ๐ถ๐ถ๐ถ๐ถ
SI unit of capacitance is ๐๐๐๐๐๐๐๐๐๐๐๐๐๐/๐ฃ๐ฃ๐ฃ๐ฃ๐ฃ๐ฃ๐ฃ๐ฃ 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
๐ถ๐ถ =
๐๐0 ๐ด๐ด ๐๐
๐ด๐ด = area of the flat plates (each used in the capacitor) ๐๐ = 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. ๐ถ๐ถ = 4๐๐๐๐0 ๐
๐
1 ๐
๐
2 /[๐
๐
2 โ 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 ๐๐ becomes
V = potential
๐ถ๐ถ = 4๐๐๐๐0 ๐
๐
1
๐๐/๐ถ๐ถ = ๐๐/4๐๐๐๐0 ๐
๐
1
Parallel limit: if both ๐
๐
1 and ๐
๐
2 are made large but ๐
๐
2 โ ๐
๐
1 = ๐๐ is kept fixed, then we can write 4๐๐๐
๐
1 ๐
๐
2 = 4๐๐๐๐ยฒ = ๐ด๐ด; where ๐
๐
is approximately the radius of each sphere, and ๐ด๐ด is the surface area of the sphere. 4.
๐ถ๐ถ = ๐๐0 ๐ด๐ด/๐๐; where ๐ด๐ด = 4๐๐๐
๐
1 ๐
๐
2 = 4๐๐๐๐ยฒ
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 ๐๐, Inner cylinder is given positive charge and outer cylinder negative charge ๐ถ๐ถ = 2๐๐๐๐โ๐๐/๐๐๐๐(๐
๐
โ/๐
๐
โ)
5. Combination of capacitors โข Series combination โข
6.
Parallel combination
1/๐ถ๐ถ = 1/๐ถ๐ถโ + 1/๐ถ๐ถโ + 1/๐ถ๐ถโ . .. ๐ถ๐ถ = ๐ถ๐ถโ + ๐ถ๐ถโ + ๐ถ๐ถโ
Force between plates of a capacitor
Plates on a parallel capacitor attract each other with a force 7.
๐น๐น = ๐๐ยฒ/2๐ด๐ด๐ด๐ดโ
Energy stored in a capacitor
Capacitor of capacitance C has a stored energy
๐๐ = ๐๐ยฒ/2๐ถ๐ถ = ๐ถ๐ถ๐ถ๐ถยฒ/2 = ๐๐๐๐/2
where ๐๐ 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 ๐๐ (which is dipole moment induced per unit volume - where is the dipole? in the dielectric slab as the two sides have opposite charges) If ๐๐๐๐ is the magnitude of the induced charge per unit area on the faces.
The dipole moment of the slab
= ๐๐โ๐๐๐๐๐๐๐๐ ร (๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐)
= (๐๐๐๐ ๐ด๐ด)๐๐ = ๐๐๐๐ (๐ด๐ด๐ด๐ด).
Where, ๐ด๐ด is area of cross section of the dielectric slab
As polarization is defined as dipole moment induced per unit volume, ๐๐ =
๐๐๐๐ (๐ด๐ด๐ด๐ด ) ๐ด๐ด๐ด๐ด
= ๐๐๐๐
(๐ด๐ด๐ด๐ด = volume of slab)
Thus the induced surface charge density is equal in magnitude to the polarization P. 10. Dielectric constant
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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
๐๐๐๐๐๐๐๐๐๐ = ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด ๐๐๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐๐๐๐๐๐๐๐๐๐ = ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด ๐๐๐๐๐๐๐๐๐๐/๐พ๐พ
๐พ๐พ is greater than 1 and is a constant for give materials. ๐พ๐พ 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 ๐ถ๐ถ = ๐พ๐พ๐พ๐พโ
where ๐ถ๐ถโ is capacitance of a similar capacitor without dielectric.
Because ๐พ๐พ > 1, the capacitance of a capacitor is increased by a factor of ๐พ๐พ when the space between the parallel plates is filled with a dielectric. 13. Magnitude of induced charge in term of ๐ฒ๐ฒ
๐๐๐๐ = ๐๐[1 โ (1/๐พ๐พ)]
๐๐๐๐ = induced charge in the dielectric ๐๐ = Applied charge
๐พ๐พ = dielectric constant
14. Gauss's law when dielectric materials are involved โฎ ๐พ๐พ๐ฌ๐ฌ. ๐
๐
๐
๐
= ๐๐๐๐๐๐๐๐๐๐ /๐๐0
โฆ (1)
Where integration is over the surface, ๐ฌ๐ฌ and ๐
๐
๐
๐
are vectors, ๐๐๐๐๐๐๐๐๐๐ is the free charge given (charge due to polarisation is not considered) and ๐พ๐พ is dielectric constant. โข
The law can also be written as
โฎ ๐ซ๐ซ. ๐
๐
๐
๐
= ๐๐๐๐๐๐๐๐๐๐
... (2)
where ๐ซ๐ซ = ๐ฌ๐ฌ๐๐0 + ๐ท๐ท; ๐ฌ๐ฌ and ๐ท๐ท are vectors. ๐ฌ๐ฌ = electric field and ๐ท๐ท is polarisation
15. Electric field due to a point charge placed inside a dielectric ๐ธ๐ธ = ๐๐/4๐๐๐๐โ๐พ๐พ๐พ๐พยฒ
16. Energy in the electric field in a dielectric 17. Corona discharge
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๐ข๐ข = ๐พ๐พ๐พ๐พโ๐ธ๐ธยฒ 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 ๐๐ 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.