3.capacitors

1

3. Capacitors by Sanjay Pandey

1. Capacitor Any two conductors separated by an insulator are said to form a capacitor. One of the conductors is given a positive charge and the other a negative charge. 2.

Capacitance - A measure of the capacity of an object to store charge for a given potential difference.

For a given capacitor, the charge 𝑄𝑄 on the capacitor is proportional to the potential difference 𝑉𝑉 between the two plates So

𝑄𝑄 ∝ 𝑉𝑉

Or

𝐶𝐶 is called the capacitance of the capacitor.

𝑄𝑄 = 𝐶𝐶𝐶𝐶

SI unit of capacitance is 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐/𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 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.

You cannot change the capacitance of a capacitor by changing the charge or voltage that you apply to it. You can only change the capacitance by changing the geometry or the materials of the capacitor. To find the capacitance of a system: 1. 2.

Assume a charge q on the plates Calculate the E-field with this charge using Gauss’ Law

3.

Knowing E, calculate the magnitude of V between the plates. To always ensure getting a positive potential difference (i.e. the magnitude of V), integrate E with respect to r from the positive plate to the negative plate. (you will no longer need the negative sign)

∫ E.dS =

qin

εο

−

V = ∫ E.dr +

Note that here you are only finding the magnitude of V. If you are interested in finding V, you need to go back to the original equation: b

V = − ∫ E.dr a

4.

3.

Calculate C from C =

Q . You should get Q to cancel out. V

Calculation of Capacitance

For parallel plate capacitor

2

𝐶𝐶 =

𝜀𝜀 0 𝐴𝐴 𝑑𝑑

𝐴𝐴 = area of the flat plates (each used in the capacitor)

𝑑𝑑 = distance between the plate  To make the field between the plates uniform, the dimensions of plates should be large as compared to distance between the plates.

Spherical capacitor

It consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor. If inner sphere radius is 𝑅𝑅1 and Outer sphere radius is 𝑅𝑅2

Inner sphere is given positive charge and outer sphere negative charge. 𝐶𝐶 = 4𝜋𝜋𝜀𝜀0 𝑅𝑅1 𝑅𝑅2 /[𝑅𝑅2 − 𝑅𝑅1 ]

If the capacitor is an isolated sphere (outer sphere is assumed to be at infinity, hence 𝑅𝑅2 is infinity and 𝐶𝐶 = 4𝜋𝜋𝜀𝜀0 𝑅𝑅1

3 𝑉𝑉 becomes

𝑄𝑄/𝐶𝐶 = 𝑄𝑄/4𝜋𝜋𝜀𝜀0 𝑅𝑅1

V = potential

Parallel limit: if both 𝑅𝑅1 and 𝑅𝑅2 are made large but 𝑅𝑅2 − 𝑅𝑅1 = 𝑑𝑑 is kept fixed, then we can write 4𝜋𝜋𝑅𝑅1 𝑅𝑅2 = 4𝜋𝜋𝜋𝜋² = 𝐴𝐴; where 𝑅𝑅 is approximately the radius of each sphere, and 𝐴𝐴 is the surface area of the sphere. 4.

𝐶𝐶 = 𝜀𝜀0 𝐴𝐴/𝑑𝑑; where 𝐴𝐴 = 4𝜋𝜋𝑅𝑅1 𝑅𝑅2 = 4𝜋𝜋𝜋𝜋²

Cylindrical Capacitor

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

If inner cylinder radius is 𝑅𝑅₁ and Outer cylinder radius is 𝑅𝑅₂ and length is 𝑙𝑙, Inner cylinder is given positive charge and outer cylinder negative charge 5. Combination of capacitors • Series combination

𝐶𝐶 = 2𝜋𝜋𝜋𝜋₀𝑙𝑙/𝑙𝑙𝑙𝑙(𝑅𝑅₂/𝑅𝑅₁)

1/𝐶𝐶 = 1/𝐶𝐶₁ + 1/𝐶𝐶₂ + 1/𝐶𝐶₃ . ..

4 •

6.

Parallel combination

𝐶𝐶 = 𝐶𝐶₁ + 𝐶𝐶₂ + 𝐶𝐶₃

Force between plates of a capacitor

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

𝐹𝐹 = 𝑄𝑄²/2𝐴𝐴𝐴𝐴₀

Energy stored in a capacitor

Capacitor of capacitance C has a stored energy

𝑈𝑈 = 𝑄𝑄²/2𝐶𝐶 = 𝐶𝐶𝐶𝐶²/2 = 𝑄𝑄𝑄𝑄/2

where 𝑄𝑄 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 𝑃𝑃 (which is dipole moment induced per unit volume - where is the dipole? in the dielectric slab as the two sides have opposite charges) If 𝜎𝜎𝑝𝑝 is the magnitude of the induced charge per unit area on the faces. The dipole moment of the slab

= 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 × (𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)

= (𝜎𝜎𝑝𝑝 𝐴𝐴)𝑙𝑙 = 𝜎𝜎𝑝𝑝 (𝐴𝐴𝐴𝐴).

Where, 𝐴𝐴 is area of cross section of the dielectric slab

5 As polarization is defined as dipole moment induced per unit volume, 𝑃𝑃 =

𝜎𝜎𝑝𝑝 (𝐴𝐴𝐴𝐴) 𝐴𝐴𝐴𝐴

= 𝜎𝜎𝑝𝑝

(𝐴𝐴𝐴𝐴 = volume of slab)

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

slab.

10. Dielectric constant

Because of induced charge, electric field is produced in the slab which is against the field applied on the 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑅𝑅𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓/𝐾𝐾

𝐾𝐾 is greater than 1 and is a constant for give materials. 𝐾𝐾 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 𝐶𝐶 = 𝐾𝐾𝐾𝐾₀

where 𝐶𝐶₀ is capacitance of a similar capacitor without dielectric.

Because 𝐾𝐾 > 1, the capacitance of a capacitor is increased by a factor of 𝐾𝐾 when the space between the parallel plates is filled with a dielectric. 13. Magnitude of induced charge in term of 𝑲𝑲

𝑄𝑄𝑝𝑝 = 𝑄𝑄[1 − (1/𝐾𝐾)]

𝑄𝑄𝑝𝑝 = induced charge in the dielectric 𝑄𝑄 = Applied charge

𝐾𝐾 = dielectric constant

14. Gauss's law when dielectric materials are involved ∮ 𝐾𝐾𝑬𝑬. 𝒅𝒅𝒅𝒅 = 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 /𝜀𝜀0

… (1)

Where integration is over the surface, 𝑬𝑬 and 𝒅𝒅𝒅𝒅 are vectors, 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 is the free charge given (charge due to polarisation is not considered) and 𝐾𝐾 is dielectric constant. •

The law can also be written as

∮ 𝑫𝑫. 𝒅𝒅𝒅𝒅 = 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

where 𝑫𝑫 = 𝑬𝑬𝜀𝜀0 + 𝑷𝑷; 𝑬𝑬 and 𝑷𝑷 are vectors.

𝑬𝑬 = electric field and 𝑷𝑷 is polarisation

15. Electric field due to a point charge placed inside a dielectric

... (2)

6 𝐸𝐸 = 𝑞𝑞/4𝜋𝜋𝜋𝜋₀𝐾𝐾𝐾𝐾² 16. Energy in the electric field in a dielectric 17. Corona discharge

1

𝑢𝑢 = 𝐾𝐾𝐾𝐾₀𝐸𝐸² 2

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 – Van de Graaff Generator

The apparatus transfers positive charge to a sphere continuously till the potential reaches to around 3 × 106 𝑉𝑉 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.