lecture 8 - purdue university

Lecture 8

12

Remember the definition of capacitance…

… so the capacitance of a parallel plate capacitor is

Parallel Plate Capacitor (4)

Variables: A is the area of each plate d is the distance between the plates

Note that the capacitance depends only on the geometrical factors and not on the amount of charge or the voltage across the capacitor.

15

Example: Capacitance, Charge, and Electrons …

Question: A storage capacitor on a random access memory (RAM) chip has a capacitance of 55 nF. If the capacitor is charged to 5.3 V, how many excess electrons are on the negative plate?

Answer: Idea: We can find the number of excess electrons on the

negative plate if we know the total charge q on the plate. Then, the number of electrons n=q/e, where e is the electron charge in coulomb.

Second idea: The charge q of the plate is related to the voltage V to which the capacitor is charged: q=CV.

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§ Consider a capacitor constructed of two collinear conducting cylinders of length L• Example: coax cable

§ The inner cylinder has radius r1 andthe outer cylinder has radius r2

§ Both cylinders have charge perunit length λ with the inner cylinderhaving positive charge and the outercylinder having negative charge

Cylindrical Capacitor (1)

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§ We apply Gauss’ Law to get the electric field between the two cylinders using a Gaussian surface with radius r and length L as illustrated by the red lines

§ … which we can rewrite to get anexpression for the electric fieldbetween the two cylinders

r1< r < r2

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§ As we did for the parallel plate capacitor, we define the voltage difference across the two cylinders to be V = |V1 – V2|

§ Thus, the capacitance of a cylindrical capacitor is

Note that C depends on geometrical factors only.

19

Spherical Capacitor (1)

§ Consider a spherical capacitor formed by two concentric conducting spheres with radii r1 and r2

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§ Let’s assume that the inner sphere has charge +q and the outer sphere has charge –q

§ The electric field is perpendicular to the surface of both spheres and points radially outward

§ To calculate the electric field, we use a Gaussian surfaceconsisting of a concentric sphere of radius r such that r1 < r < r2

§ The electric field is always perpendicular to the Gaussian surface so

§ … which reduces to 21


…makes sense!

22


§ To get the electric potential we follow a method similar to the one we used for the cylindrical capacitor and integrate from the negatively charged sphere to the positively charged sphere

§ Using the definition of capacitance we find

§ The capacitance of a spherical capacitor is then

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Capacitance of an Isolated Sphere

§ We obtain the capacitance of a single conducting sphere by taking our result for a spherical capacitor and moving the outer spherical conductor infinitely far away

§ Using our result for a spherical capacitor…

§ …with r2 = ∞ and r1 = R we find

…meaning V = q/4πε0R (we already knew that!)

iClicker

§ A metal ball of radius R has a charge q. § Charge is changed q -> - 2q. How does it’s capacitance

changed?

12

q

A: C->2 C0B: C-> C0C: C-> C0/2 D: C->- C0E: C->-2 C0

Physics of a spark

13

+q -q

d

�V

E ⇠ �V/d

Physics of a spark

13

+q -q

d

�V

E ⇠ �V/d

�V d1 E ⇠ �V/d1 � E0

Physics of a spark

13

+q -q

d

�V

E ⇠ �V/d

�V d1 E ⇠ �V/d1 � E0

e E

Ek ⇠ E� ⇠ 1eV

�

� ⇠ 1µm

Espark ⇠ MeV/m

Physics of a spark

13

+q -q

d

�V

E ⇠ �V/d

�V d1 E ⇠ �V/d1 � E0

e E

Ek ⇠ E� ⇠ 1eV

�

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Energy Stored in Capacitors

U =1

2qV

q = CV

U =1

2CV 2

U =1

2

q2

C

or

(V created by “q”s, self-interaction)

§ Capacitors store electric energy

37


U =1

2qV

q = CV

U =1

2CV 2

U =1

2

q2

C

or

(V created by “q”s, self-interaction)

§ Capacitors store electric energy

We want small voltage, large energy: large C

38

§ We define the energy density, 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)

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§ 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

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§ 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?

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Utot

=

Z 1

R

udV =

4⇡

Z 1

R

1

2✏0E

2r2dr =

2⇡✏0

Z 1

R

✓1

4⇡✏0

◆2 q2

r4r2dr =

1

2

q2

4⇡✏0R=

1

2qV

dV = d⇤ sin �d�r2dr

= 4⇥r2dr


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Utot

=

Z 1

R

udV =

4⇡

Z 1

R

1

2✏0E

2r2dr =

2⇡✏0

Z 1

R

✓1

4⇡✏0

◆2 q2

r4r2dr =

1

2

q2

4⇡✏0R=

1

2qV


19

Utot

=

Z 1

R

udV =

4⇡

Z 1

R

1

2✏0E

2r2dr =

2⇡✏0

Z 1

R

✓1

4⇡✏0

◆2 q2

r4r2dr =

1

2

q2

4⇡✏0R=

1

2qV


19

Utot

=

Z 1

R

udV =

4⇡

Z 1

R

1

2✏0E

2r2dr =

2⇡✏0

Z 1

R

✓1

4⇡✏0

◆2 q2

r4r2dr =

1

2

q2

4⇡✏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

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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

++++++++++++…++++++++++++ …

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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….

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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

Electric circuits

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Circuit diagram

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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

demo

27

28

V+

-

28

V

I

+

-

28

V

I

+

-

+

-

28

V+

-

+

-

28

V V+

-

+

-

28

V V+

-

+

-

I

28

V V+

-

+

-

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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

+ + +

+

+

+

- --

-

--

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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

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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

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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

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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

32

Example: System of Capacitors (1)

Question: What is the capacitance of this system of capacitors?

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

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§ 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

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§ We can see that C4 and C123 are in series

§ We find for the equivalent capacitance:


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§ 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.

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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

54


§ 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

55




E

55




E

+-

Induced Electric field

48

E


48

E

E


48

EE

E


48

EE

Against the external field!

E

§ 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

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

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 - a spark - 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

59

Example: Dielectric Constant of Wax

§ An air-filled parallel plate capacitor has a capacitance of 1.3 pF. The separation of the plates is doubled, and wax is inserted between them. The new capacitance is 2.6pF.

Question: Find the dielectric constant of the wax.Answer:§ Key Ideas: The original capacitance is given by

§ Then the new capacitance is Thus

rearrange the equation:

60

Example: Dielectric Material§ Given a 7.4 pF air-filled capacitor. You are asked to convert

it to a capacitor that can store up to 7.4 µJ with a maximum voltage of 652 V.

Question: What dielectric constant should the material have that you insert to achieve

these requirements? Answer: § Key Idea: The capacitance with the dielectric in place is given by C=κCair

§ and the energy stored is given by So,

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Review - So Far …

§ The capacitance of a spherical capacitor is

• r1 is the radius of the inner sphere

• r2 is the radius of the outer sphere

§ The capacitance of an isolated spherical conductor is

• R is the radius of the sphere

37

§ A battery must do work to charge a capacitor.§ We can think of this work as changing the electric potential energy

of the capacitor.§ The differential work dW done by a battery with voltage V to put a

differential charge dq on a capacitor with capacitance C is

§ The total work required to bring the capacitor to its full charge q is

§ This work is stored as electric potential energy


49

Review - So Far …

§ 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

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Review (2)

§ Placing a dielectric between the plates of a capacitor increases the capacitance by κ (dielectric constant)

§ The dielectric has the effect of lowering the electric field between the plates (for given charge q)

§ We also define the electric permitivity of the dielectric material as

lecture 8 - purdue university

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