capacitors - michigan state universityfebruary 5, 2014 physics for scientists & engineers 2,...

Capacitors

February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

September 17, 2008 Physics for Scientists & Engineers 2, Lecture 14 2

Review !  The electric potential energy stored in a capacitor

is given by

!  The field energy density stored in a parallel plate capacitor is given by

!  The field energy density in general is

212

U CV=

20

12

u Eε=

2

012

Vud

ε ⎛ ⎞= ⎜ ⎟⎝ ⎠


Capacitors with Dielectrics !  Placing a dielectric between the plates of a capacitor

increases the capacitance of the capacitor by a numerical factor called the dielectric constant, κ

!  We can express the capacitance of a capacitor with a dielectric with dielectric constant κ between the plates as

!  Where Cair is the capacitance of the capacitor without the dielectric

!  Placing the dielectric between the plates of the capacitor has the effect of lowering the electric field between the plates and allowing more charge to be stored in the capacitor

C =κCair

C =

qΔV


Parallel Plate Capacitor with Dielectric !  Placing a dielectric between the plates of a parallel plate

capacitor modifies the electric field as

!  ε0 is the electric permittivity of free space !  ε is the electric permittivity of the dielectric material

!  Note that the replacement of ε0 by ε is all that is needed to generalize our expressions for the capacitance

!  The potential difference across a parallel plate capacitor is

!  The capacitance is then

E = Eair

κ= qκε0A

= qεA

ε =κε0

ΔV = Ed = qd

κε0A

C = q

ΔV= κε0A

d


Dielectric Strength !  The dielectric strength of a material measures the ability of

that material to withstand potential difference !  If the electric field strength in the dielectric exceeds the

dielectric strength, the dielectric will break down and begin to conduct charge between the plates via a spark, which usually destroys the capacitor

!  A useful capacitor must contain a dielectric that not only provides a given capacitance but also enables the device to hold the required potential difference without breaking down

!  Capacitors are usually specified in terms of their capacitance and by the maximum potential difference that they are designed to handle


Dielectric Constant !  The dielectric constant of vacuum is defined to be 1 !  The dielectric constant of air is close to 1 and we will use

the dielectric constant of air as 1 in our problems !  The dielectric constants of common materials are listed

below (more are listed in the book in Table 24.1)


Microscopic Perspective on Dielectrics

!  Let’s consider what happens at the atomic and molecular level when a dielectric is placed in an electric field

!  A polar dielectric material is composed of molecules that have a permanent electric dipole moment

!  A nonpolar dielectric material is composed of atoms or molecules that have no inherent electric dipole moment Dipole moment is induced by external electric field


Microscopic Perspective on Dielectrics

!  In both polar and non-polar dielectrics, the fields resulting from aligned electric dipole moments tend to partially cancel the original electric field

!  The resulting electric field inside the capacitor then is the original field minus the induced field Er =E −Ed

- +

February 5, 2014 Physics for Scientists&Engineers 2 10

Example I !  A parallel plate capacitor whose capacitance C is 13.5pF is charged by

a battery to a potential difference of V =12.5V between its plates. The battery is now disconnected and material with κ=6.5 is slipped between the plates.

(a) What is the potential energy before the material is inserted? (b) What is U after the material has been inserted? (a) (b) Key Idea: Because the battery is disconnected, the charge on the

capacitor cannot change!


Example II !  A parallel plate capacitor whose capacitance C is 13.5pF is charged by


(a) What is the potential energy before the material is inserted? (b) What is U after the material has been inserted? (b) Key Idea: Because the battery is disconnected, the charge on the

capacitor cannot change, but the capacitance does change (C--> κC)!


Example III !  A parallel plate capacitor whose capacitance C is 13.5pF is charged by


(b) What is U after the material has been inserted? The potential energy decreased by a factor κ. The “missing” energy, in

principle, would be apparent to the person inserting the material. The capacitor would exert a tiny tug on the material and would do work on it, in amount

Capacitance of a Coaxial Cable !  Coaxial cables are used to transport signals between devices

with minimum interference !  A 20.0 m long coaxial cable is composed of a conductor and

a coaxial conducting shield around the conductor !  The space between the conductor and the shield is filled with

polystyrene !  The radius of the conductor is

0.250 mm and the radius of the shield is 2.00 mm

!  PROBLEM !  What is the capacitance of the coaxial

cable? February 5, 2014 Physics for Scientists & Engineers 2, Chapter 24 14

Capacitance of a Coaxial Cable SOLUTION

!  We can think of the coaxial cable as a cylinder !  The dielectric constant of polystyrene is 2.6 !  We can treat the coaxial cable as a cylindrical capacitor with

r1 = 0.250 mm and r2 = 2.00 mm, filled with a dielectric with κ = 2.6

!  The capacitance of the coaxial cable is


C =κ 2πε0L

ln r2

r1

⎛⎝⎜

⎞⎠⎟

= 2.6( )2π 8.85 ⋅10−12 F/m( ) 20.0 m( )

ln 2.00 ⋅10−3 m2.50 ⋅10−4 m

⎛⎝⎜

⎞⎠⎟

C =1.39 ⋅10−9 F=1.39 nF

Capacitor Partially Filled with a Dielectric PROBLEM:

!  A parallel plate capacitor is constructed of two square conducting plates with side length L = 10.0 cm.

!  The distance between the plates is d = 0.250 cm. !  A dielectric with dielectric constant κ =15.0 and thickness

0.250 cm is inserted between the plates. !  The dielectric is L = 10.0 cm wide and L/2 = 5.00 cm long. !  What is the capacitance of this capacitor?

February 5, 2014 Chapter 24 17

Capacitor Partially Filled with a Dielectric SOLUTION:

!  We have a capacitor partially filled with a dielectric. !  We can treat this capacitor as two capacitors in parallel. !  One capacitor is a parallel plate capacitor with plate area

A = L(L/2) and air between the plates. !  The second capacitor is a parallel plate capacitor with plate

area A = L(L/2) and a dielectric between the plates. !  Sketch


Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Capacitor Partially Filled with a Dielectric Research

!  The capacitance of a parallel place capacitor is: !  If a dielectric is placed between the plates we have:

!  The capacitance of two capacitors in parallel is:

Simplify

!  Putting our equations together gives us:


C1 =

ε0 Ad

C2 =κ

ε0 Ad

C12 = C1 + C2

C12 = C1 =

ε0 Ad

+κε0 Ad

= κ+1( )ε0 Ad

Capacitor Partially Filled with a Dielectric

!  The area of the plates for each capacitor is:

!  Putting our expressions together gives us:

!  Calculate

!  Putting in our numerical values:


A = L L / 2( )= L2 / 2

C12 = κ+1( )

ε0 L2 / 2( )d

=κ+1( )ε0L2

2d

C12 =

15.0+1( ) 8.85⋅10−12 F/m( ) 0.100 m( )2

2 0.00250 m( )= 2.832 ⋅10−10 F

Capacitor Partially Filled with a Dielectric Round

!  Double-check

!  To double-check our answer, we calculate the capacitance of the capacitor without any dielectric:

!  Calculate the capacitance of the capacitor with dielectric:

!  Our result for the partially filled capacitor is half of the sum of these two results, so it seems reasonable.


C12 = 2.83⋅10−10 F = 283 pF

C1 =

8.85⋅10−12 F/m( ) 0.100 m( )2

0.00250 m( )= 35.4 pF

C1 = 15( )35.4 pF = 531 pF


Supercapacitor / Ultracapacitor !  Supercapacitors (ultracapacitors) are made using a material

with a very large surface area between the capacitor plates !  Two layers of activated charcoal are given opposite charge

and are separated by an insulating material

!  This produces a capacitor with capacitance millions of times larger than ordinary capacitors

!  However, the potential difference can only be 2 to 3 V

capacitors - michigan state universityfebruary 5, 2014 physics for scientists & engineers 2,...

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