PHY 184

1
PHY 184

• Spring 2007
• Lecture 13

Title Capacitors
2
Notes

• Homework Set 3 is done!
• Homework Set 4 is open and Set 5 opens Thursday
  morning
• Midterm 1 will take place in class Thursday,
  February 8.
• One 8.5 x 11 inch equation sheet (front and back)
  is allowed.
• The exam will cover
• Chapters 16 - 19
• Homework Sets 1 - 4

3
Review

• 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

4
Review (2)

• The equivalent capacitance for n capacitors in
  parallel is
• The equivalent capacitance for n capacitors in
  series is

5
Example - System of Capacitors

• Lets analyze a system of five capacitors
• If each capacitor has a capacitance of 5 nF, what
  is the capacitance of this system of capacitors?

6
System of Capacitors (2)

• We can see that C1 and C2 are in parallel and
  that C3 is also in parallel with C1 and C2
• We can define C123 C1 C2 C3 15 nF
• and make a new drawing

7
System of Capacitors (3)

• We can see that C4 and C123 are in series
• We can define
• and make a new drawing

8
System of Capacitors (4)

• We can see that C5 and C1234 are in parallel
• We can define
• And make a new drawing

9
System of Capacitors (5)

• So the equivalent capacitance of our system of
  capacitors
• More than one half of the total capacitance of
  this arrangement is provided by C5 alone.
• This result makes it clear that one has to be
  careful how one arranges capacitors in circuits.

10
Clicker Question

• Find the equivalent capacitance Ceq
• A)
• B)
• C)

11
Clicker Question

• Find the equivalent capacitance Ceq
• C)

First Step C1 and C2 are in series
Second Step C12 and C3 are in parallel
12
A capacitor stores energy. Field Theory
The energy belongs to the electric field.
13
Energy Stored in Capacitors

• 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

14
Energy Density in Capacitors

• 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

15
Energy Density in Capacitors (2)

• 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.

16
Example - isolated conducting sphere

• An isolated conducting sphere whose radius R is
  6.85 cm has a charge of q1.25 nC.
• a) How much potential energy is stored in the
  electric field of the charged conductor?
• Key Idea An isolated sphere has a
  capacitance of C4?e0R (see previous lecture).
  The energy U stored in a capacitor depends on the
  charge and the capacitance according to

and substituting C4pe0R gives
17
Example - isolated conducting sphere

• An isolated conducting sphere whose radius R is
  6.85 cm has a charge of q1.25 nC.
• b) What is the field energy density at the
  surface of the sphere?
• 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

18
Example Thundercloud

• 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 betweenthe
  cloud and the ground?
• Question 2
• Knowing that lightning strikes requireelectric
  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?

19
Example Thundercloud (2)

• Question 1
• We can approximate the cloud-ground system as a
  parallel plate capacitor whose capacitance is
• The charge carried by the cloud is 160 C, which
  means that the plate surface facing the earth
  has a charge of 80 C
• 720 million volts

20
Example Thundercloud (3)

• Question 2
• 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.
• Question 3
• The total energy stored in a parallel place
  capacitor is
• Enough energy to run a 1500 W hair dryer for more
  than 5000 hours

21
Clicker Question

• A 1.0 ?F capacitor and a 3.0 ?F capacitor are
  connected in parallel across a 500 V potential
  difference V. What is the total energy stored in
  the capacitors?
• A) U0.5 J
• B) U0.27 J
• C) U1.5 J
• D) U0.02 J

Hint Use
22
Clicker Question

• A 1.0 ?F capacitor and a 3.0 ?F capacitor are
  connected in parallel across a 500 V potential
  difference V. What is the total energy stored in
  the capacitors?
• A) U0.5 J

U 0.5 J
23
Capacitors with Dielectrics

• We have only 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.
• 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

24
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, ?
• We can express the capacitance of a capacitor
  with a dielectric with dielectric constant ?
  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.

25
Parallel Plate Capacitor with Dielectric

• Placing a dielectric between the plates of a
  parallel plate capacitor modifies the electric
  field as
• The constant ?0 is the electric permittivity of
  free space
• The constant ? is the electric permittivity of
  the dielectric material