1 U qV 2 q CV - Purdue University

37

Energy Stored in Capacitors

U =1

2qV

q = CV

U =1

2CV 2

U =1

2

q2

C

or

Monday, February 10, 2014

38

We define the , u, as the electric potential energy per unit volume

Taking the ideal case of a parallel plate capacitor that has no fringe field, the volume between the plates is the area of each plate times the distance between the plates, Ad

Inserting our formula for the capacitance of a parallel plate capacitor we find

Energy Density in Capacitors (1)


39

Recognizing that V/d is the magnitude of the electric field, E, we obtain an expression for the electric potential energy density for parallel plate capacitor

This result, which we derived for the parallel plate capacitor, is in fact completely general.

This equation holds for all electric fields produced in any way• The formula gives the quantity of electric field energy per unit volume.

Energy Density in Capacitors (2)


40

An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC.

Question 1:

How much potential energy is stored in the electric field of the charged conductor?

Answer:

Key Idea: An isolated sphere has a capacitance of C=4πε0R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to

Example: Isolated Conducting Sphere (1)

… and substituting C=4πε0R gives


41

An isolated conducting sphere whose radius R is 6.85 cm has a charge of q = 1.25 nC.

Question 2: What is the field energy density at the surface of the sphere?Answer: Key Idea: The energy density u depends on the magnitude of the

electric field E according to

so we must first find the E field at the surface of the sphere. Recall:

Example: Isolated Conducting Sphere (2)

q


What is the total energy in E-field?

6

Utot =

� ∞

RudV =

4π

� ∞

R

1

2�0E

2r2dr =

2π�0

� ∞

R

�1

4π�0

�2 q2

r4r2dr =

1

2

q2

4π�0R=

1

2qV


What is the total energy in E-field?

6

Utot =

� ∞

RudV =

4π

� ∞

R

1

2�0E

2r2dr =

2π�0

� ∞

R

�1

4π�0

�2 q2

r4r2dr =

1

2

q2

4π�0R=

1

2qV Yes!


42

Example: Thundercloud (1)

Suppose a thundercloud with horizontal dimensions of 2.0 km by 3.0 km hovers over a flat area, at an altitude of 500 m and carries a charge of 160 C.

Question 1:• What is the potential difference

between the cloud and the ground?

Question 2:• Knowing that lightning strikes require

electric field strengths of approximately2.5 MV/m, are these conditions sufficientfor a lightning strike?

Question 3:• What is the total electrical energy contained in this cloud?


V =1

2

q

C= 7.2 108

September 21, 2010 43

Example: Thundercloud (2)Question 1: What is the potential difference between the cloud and

the ground?

Answer: We can approximate the cloud-ground system as a parallel plate

capacitor whose capacitance is

The charge carried by the cloud is 160 C

720 million volts

++++++++++++…++++++++++++ …



Example: Thundercloud (3)Question 2: Knowing that lightning strikes require electric field

strengths of approximately 2.5 MV/m, are these conditions sufficient for a lightning strike?

Answer: We know the potential difference between the cloud and ground so

we can calculate the electric field

E is lower than 2.5 MV/m, so no lightning cloud to ground• May have lightning to radio tower or tree….



Example: Thundercloud (4)

Question 3: What is the total electrical energy contained in this cloud?

Answer: The total energy stored in a parallel place capacitor is


Physics of a spark

11

+q -q

d

∆V

E ∼ ∆V/d


Physics of a spark

11

+q -q

d

∆V

E ∼ ∆V/d

∆V d1 E ∼ ∆V/d1 � E0


Physics of a spark

11

+q -q

d

∆V

E ∼ ∆V/d

∆V d1 E ∼ ∆V/d1 � E0

e E

Ek ∼ Eλ ∼ 1eV

λ


Electric curcuit

12


Circuit diagram

13

Lines represent conductorsThe battery or power supply is represented byThe capacitor is represented by the symbol

Battery provides (a DC) potential difference V


Illustrate the charging processing using a circuit diagram.

This circuit has a switch• (pos c) When the switch is in position c, the circuit is open (not connected).• (pos a) When the switch is in position a, the battery is connected across the capacitor. Fully charged, q = CV.• (pos b) When the switch is in position b, the two plates of the capacitor are connected. Electrons will move around the circuit--a current will flow--and the capacitor will discharge.

8

Charging/Discharging a Capacitor (2)

c

c


15

+

-


15

V+

-


15

V+

-


15

V

I

+

-


15

V

I

+

-


15

V

I

+

-

+

-


15

V

I

+

-

+

-


15

V V+

-

+

-


15

V V+

-

+

-


15

V V+

-

+

-


15

V V+

-

+

-I


15

V V+

-

+

-I


15

V V+

-

+

-I


25

Capacitors in Circuits

A circuit is a set of electrical devices connected with conducting wires

Capacitors can be wired together in circuits in parallel or series• Capacitors in circuits connected

by wires such that the positively charged plates are connected together and the negatively charged plates are connected together, are connected in parallel

• Capacitors wired together such that the positively charged plate of one capacitor is connected to the negatively charged plate of the next capacitor are connected in series

+ + +

+

+

+

- --

-

--


26

Capacitors in Parallel (1)

Consider an electrical circuit with three capacitors wired in parallel

Each of three capacitors has one plate connected to the positive terminal of a battery with voltage V and one plate connected to the negative terminal.

The potential difference V across each capacitor is the same.

We can write the charge on each capacitor as …

.. key point for capacitors in parallel


27

Capacitors in Parallel (2)

We can consider the three capacitors as one equivalent capacitor Ceq that holds a total charge q given by

We can now define Ceq by

A general result for n capacitors in parallel is

If we can identify capacitors in a circuit that are wired in parallel, we can replace them with an equivalent capacitance


28

Capacitors in Series (1) Consider a circuit with three capacitors wired in series

The positively charged plate of C1 is connected to the positive terminal of the battery

The negatively charge plate of C1 is connected to the positively charged plate of C2

The negatively charged plate of C2 is connected to the positively charge plate of C3

The negatively charge plate of C3 is connected to thenegative terminal of the battery

The battery produces an equal charge q on each capacitor because the battery induces a positive charge on the positive place of C1, which induces a negative

charge on the opposite plate of C1, which induces a positive charge on C2, etc.

.. key point for capacitors in series


29

Capacitors in Series (2) Knowing that the charge is the same on all three capacitors

we can write

We can express an equivalent capacitance Ceq as

We can generalize to n capacitors in series

If we can identify capacitors in a circuit that are wired in series, we can replace them with an equivalent capacitance


31

Review

The equivalent capacitance for n capacitors in parallel is

The equivalent capacitance for n capacitors in series is

=

=


iClicker

Three capacitors, each with capacitance C, are connected as shown in the figure. What is the equivalent capacitance for this arrangement of capacitors?

a) C/3b) 3Cc) C/9d) 9Ce) none of the above


iClicker

Three capacitors, each with capacitance C, are connected as shown in the figure. What is the equivalent capacitance for this arrangement of capacitors?

a) C/3b) 3Cc) C/9d) 9Ce) none of the above


32

Example: System of Capacitors (1)

Question: If each capacitor has a capacitance of 5 nF, what is the capacitance of this system of capacitors?

Answer:Find the equivalent capacitanceAnalyze each piece of the circuit individually, replacing pairs in series or in parallel by one capacitor with equivalent capacitance


33


We can see that C1 and C2 are in parallel,

and that C3 is also in parallel with C1 and C2

We find C123 = C1 + C2 + C3

… and make a new drawing


34


We can see that C4 and C123 are in series

We find for the equivalent capacitance:



35


We can see that C5 and C1234 are in parallel

We find for the equivalent capacitance



36


So the equivalent capacitance of our system of capacitors


46

Capacitors with Dielectrics (1)

So far, we have discussed capacitors with air or vacuum between the plates.

However, most real-life capacitors have an insulating material, called a dielectric, between the two plates.

The dielectric serves several purposes:• Provides a convenient way to maintain mechanical separation between

the plates (plates attract!)• Provides electrical insulation between the plates• Allows the capacitor to hold a higher voltage

• Increases the capacitance of the capacitor• Takes advantage of the molecular structure of the dielectric material


47

Capacitors with Dielectrics (2)

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.


48

Parallel Plate Capacitor with Dielectric

Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as

The constant ε0 is the electric permittivity of free space

The constant ε is the electric permittivity of the dielectric material


53

Microscopic Perspective on Dielectrics (1)

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

There are two types of dielectric materials• Polar dielectric

• Non-polar dielectric

Polar dielectric material is composed of molecules that have a permanent electric dipole moment due to their molecular structure• e.g., water molecules

Normally the directions of the

electric dipoles are randomly

distributed:


54


When an electric field is applied to these polar molecules, they tend to align with the electric field

Non-polar dielectric material is composed of atoms or molecules that have no electric dipole moment


55


These atoms or molecules can be induced to have a dipole moment under the influence of an external electric field

This induction is caused by the opposite direction of the electric force on the negative and positive charges of the atom or molecule, which displaces the center of the relative charge distributions and produces an induced electric dipole moment - +


In both the case of the polar and non-polar dielectric materials, the resulting aligned electric dipole moments tend to partially cancel the original electric field

The electric field inside the capacitor then is the original field minus the induced field

56


=E

κ

E0


In both the case of the polar and non-polar dielectric materials, the resulting aligned electric dipole moments tend to partially cancel the original electric field

The electric field inside the capacitor then is the original field minus the induced field

56


=E

κ

E0Ed


51

Dielectric Strength

The “dielectric strength” of a material measures the ability of that material to withstand voltage differences

If the voltage across a dielectric exceeds the breakdown potential, the dielectric will break down and begin to conduct charge between the plates

Real-life dielectrics enable a capacitor to provide a given capacitance and withstand the required voltage without breaking down

Capacitors are usually specified in terms of their capacitance and rated (i.e., maximum) voltage


52

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


57

Capacitor with Dielectric (1)Question 1:

Consider a parallel plate capacitor with capacitance C = 2.00 µF connected to a battery with voltage V = 12.0 V as shown. What is the charge stored in the capacitor?

Question 2:Now insert a dielectric with dielectric constant κ = 2.5 between the plates of the capacitor. What is the charge on the capacitor?

The additional charge is provided by the battery.


58

Capacitor with Dielectric (2) We isolate the charged capacitor with a dielectric by

disconnecting it from the battery. We remove the dielectric, keeping the capacitor isolated.

Question 3:

What happens to the charge and voltage on the capacitor?

The charge on the isolated capacitor cannot change because there is nowhere for the charge to flow. Q remains constant.

The voltage on the capacitor will be

The voltage went up because removing the dielectric increased the electric field and the resulting potential difference between the plates.

V increases


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