Capacitors and Capacitance

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CHAPTER 25: CAPACITANCE

Dr Reem M. Altuwirqi

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•

What is Capacitors and Capacitance?

•

The Capacitance of different types of capacitors:

• Parallel plates – Cylindrical – Spherical – Isolated sphere

•

How do we add the effect of capacitors?

•

How do we find out the energy stored in a capacitor?

•

Capacitors with dielectric materials.

What we will learn

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Capacitors and Capacitance

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Capacitance



Capacitors are important because they can provide a large supply of energy at a fast rate.



A capacitor consists of two conducting objects (+Q,-Q)



We refer to charge of the capacitor with Q (Q

_net

=0)



The symbol of a capacitor in electric circuits



There are different types of capacitors:

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Capacitance



When a capacitor is charged, the plates have equal and opposite charge (+Q,-Q)



The plates of the conductors are equipotential surfaces.



There is a potential difference between the plates V.



Capacitance: How much charge can be stored in a capacitor to produce V. Q  V Q  CV

C depends on the geometry of the plates and the material between

them and NOT on Q or V Units (C/V  Farad)

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Charging a capacitor



The battery maintains V between its terminals. (+ , -)



When the switch is closed, charge (electrons) flow under the influence of E set by the battery.



e moves from h  + (h loses electrons  + charged)



e moves from -  l (l gains electrons  - charged)



V on capacitor goes from 0  V

_battery

(fully charged E=0 )

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Charging a capacitor

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Calculating the Capacitance



Assumption 1:

 E and dA are parallel

 E.dA = EA How to calculate C?

1. Assume a charge q on the plates.

2. Find E between the plates in terms of q using Gauss’ law 3. Calculate V from E

4. Calculate C from C=q/V





 ^f

V i E.ds _o

qenc

d 

^E^^ ^A^ ^

S



Assumption 2:

 Path starts from -

 + along E

 E and ds are anti-parallel

 E.ds = -E ds

^

 Eds V

.) (E const q

EA _enc

o 



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Parallel Plate Capacitor

Cylindrical Capacitor

Spherical Capacitor

An Isolated Sphere

How to calculate C?

4. Calculate C from C=q/V





 ^f

V i E.ds _o

qenc

d 

^E^^ ^A^ ^

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Parallel

S

Plate Capacitor

How to calculate C?

4. Calculate C from C=q/V V ^^i^f E.ds o

qenc

d 

^E^ ^ ^A^ ^

o

qenc

d 



Ê^ ^ Â^ ^ ^qênc ^ ^ôÊA

d E E

V 







^ 



 E ds ds

d A Ed

EA V

C _ q _ ^o _ ^o d

C _ ^oA

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How to calculate C?

4. Calculate C from C=q/V V ^ ^i^f E.ds ^o qenc

d 

^E^^ ^A^ ^

o

qenc

d 



Ê^ ^ Â^ ^ ^qênc ^ ^ôÊ⁽²^^rL⁾



 



 











^

 

a b L

L E Lr

V ^a

b a

b

2 ln q

r dr 1 2

dr q 2

ds q

0

0 0





_b _a

 

^b ^L^a



L q V

C q

/ ln

2 /

2 ln q

0





dr ds  

b a

C L

/ ln

2₀



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Spherical Capacitor

How to calculate C?

d 

^E^^ ^A^ ^

o

qenc

d 



Ê^ ^ Â^ ^ ^qênc ^ ^ôÊ⁽⁴^^r²⁾

ab a b b

a E r

V ^a

b

 



 

 











 

^







0 0

2 0

4 q 1

1 4

q

r dr 1 4

dr q 4

ds q



a b

ab a

b ab q

V C q

 

 ₀

0

4 4

q 



dr ds  

a b C ab

 4₀ 

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Isolated Sphere

How to calculate C?

d 

^E^^ ^A^ ^

a b

ab a

b ab q

V C q

 

 ₀

0

4 4

q 



R a

b







R C  4₀

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Calculating the Capacitance

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Parallel Plate Capacitor

Spherical Capacitor

An Isolated Sphere

d C _ ^oA

^b ^L^a

C ln / 2₀

 b a

C ab

 4₀  C  4₀R

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Equivalent Capacitance

Parallel Series

3 ...

2

1   

 C C C

C_eq ^...

1 1

3 2

1



 C C C

C_eq

2

1 V

V

V    



,...

,

, ₂ ₃

1 C C

C C_eq 

2

1 Q

Q Q  

,...

,

, ₂ ₃

1 C C

C C_eq 

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Charging a capacitor

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Energy Stored in an Electric Field

Suppose that, at a given instant, a charge q’ has been transferred from one plate of a capacitor to the other. The potential difference V’ between the plates at that instant will be q’/C. If an extra increment of charge dq’ is then transferred, the increment of work required will be,

The work required to bring the total capacitor charge up to a final value q is

This work is stored as potential energy U in the capacitor, so that,

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Energy Density

In a parallel-plate capacitor, neglecting fringing, the electric field has the same value at all points between the plates. Thus, the energy density u—that is, the potential energy per unit volume between the plates—should also be uniform.

We can find u by dividing the total potential energy by the volume Ad of the space between the plates.

But since(C = ₀A/d), this result becomes

However, (E=-^V/s), V/d equals the electric field magnitude E. Therefore.

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Energy Density

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Energy Density

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Capacitors with Dielectric

d C   

^o

A d

C  

^o

A

4

2

1

r E q



o

 4

2

1

r E q



o



o

E 

 

o

E 

 

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A dielectric, is an insulating material such as mineral oil or plastic, and is characterized by a numerical factor , called the dielectric constant of the material.

The introduction of a dielectric also limits the potential difference that can be applied between the plates to a certain value Vmax, called the breakdown potential.

Every dielectric material has a characteristic dielectric strength, which is the maximum value of the electric field that it can tolerate without breakdown.

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Capacitors with Dielectric

V C q₁  ₁

V C V

C

q₂  ₂  ( ₁)

1

2 q

q 

1

1 C

V  q



 ₁ ¹

2 2

V C

q C

V  q  

) (q₂  q₁

) (V₂ V₁



1 2

V  V

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What have we learnt

•

What is Capacitors and Capacitance?

•

The Capacitance of different types of capacitors:

• Parallel plates – Cylindrical – Spherical – Isolated sphere

•

How do we add the effect of capacitors?

•

How do we find out the energy stored in a capacitor?

•