1. Lecture 10 Capacitance and capacitors
General Physics II, Lec 10, By/ T.A. Eleyan

2. General Physics II, Lec 10, By/ T.A. Eleyan
Capacitors
Capacitors are devices that store energy in an electric field.
Capacitors are used in many every-day applications
Heart defibrillators
Camera flash units
Capacitors are an essential part of electronics.
Capacitors can be micro-sized on computer chips or super-sized for high power circuits such as FM radio transmitters.
General Physics II, Lec 10, By/ T.A. Eleyan

3. Definition of Capacitance
The definition of capacitance is
The units of capacitance are coulombs per volt.
The unit of capacitance has been given the name farad (abbreviated F) named after British physicist Michael Faraday ( )
A farad is a very large capacitance
Typically we deal with F (10-6 F), nF (10-9 F), or pF (10-12 F)
General Physics II, Lec 10, By/ T.A. Eleyan

4. The parallel-plate capacitor
The capacitance of a device depends on the area of the plates and the distance between the plates
where A is the area of one of the plates, d is the separation, e0 is a constant (permittivity of free space),
e0= 8.85×10-12 C2/N·m2 A +Q d A -Q
General Physics II, Lec 10, By/ T.A. Eleyan

5. General Physics II, Lec 10, By/ T.A. Eleyan
Example: A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine
the capacitance
the charge on each plate
Solution:
Since we are dealing with the parallel-plate capacitor, the capacitance can be found as
Given:
DV=10,000 V
A = 2.00 m2
d = 5.00 mm
Find:
C=?
Q=?
Once the capacitance is known, the charge can be found from the definition of a capacitance via charge and potential difference:
General Physics II, Lec 10, By/ T.A. Eleyan

6. Cylindrical Capacitor
Consider a capacitor constructed of two collinear conducting cylinders of length L.
The inner cylinder has radius r1 and the outer cylinder has radius r2.
Both cylinders have charge per unit length  with the inner cylinder having positive charge and the outer cylinder having negative charge.
We will assume an ideal cylindrical capacitor
The electric field points radially from the inner cylinder to the outer cylinder.
The electric field is zero outside the collinear cylinders.
General Physics II, Lec 10, By/ T.A. Eleyan

7. General Physics II, Lec 10, By/ T.A. Eleyan
We apply Gauss’ Law to get the electric field between
the two cylinder using a Gaussian surface with radius r
and length L as illustrated by the red lines
… which we can rewrite to get an expression for the electric field between the two cylinders
General Physics II, Lec 10, By/ T.A. Eleyan

8. General Physics II, Lec 10, By/ T.A. Eleyan
As we did for the parallel plate capacitor, we define the
voltage difference across the two cylinders to be V=V1 – V2.
The capacitance of a cylindrical capacitor is
General Physics II, Lec 10, By/ T.A. Eleyan

9. General Physics II, Lec 10, By/ T.A. Eleyan
Spherical Capacitor
Consider a spherical capacitor formed by two
concentric conducting spheres with radii r1 and r2
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
General Physics II, Lec 10, By/ T.A. Eleyan

10. General Physics II, Lec 10, By/ T.A. Eleyan
To calculate the electric field, we use a Gaussian surface consisting 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
General Physics II, Lec 10, By/ T.A. Eleyan

11. General Physics II, Lec 10, By/ T.A. Eleyan
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
General Physics II, Lec 10, By/ T.A. Eleyan

12. Combinations of capacitors
It is very often that more than one capacitor is
used in an electric circuit
We would have to learn how to compute the equivalent capacitance of certain combinations of capacitors C2 C5 C1 C2 C3 C4 C1 C3
General Physics II, Lec 10, By/ T.A. Eleyan

13. a. Parallel combination
Connecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference for both capacitors,
A total charge transferred to the system from the battery is the sum of charges of the two capacitors,
+Q1
-Q1 C1
V=Vab a b
+Q2
-Q2 C2
General Physics II, Lec 10, By/ T.A. Eleyan

14. Parallel combination:
Analogous formula is true for any number of
capacitors,
It follows that the equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors
(parallel combination)
General Physics II, Lec 10, By/ T.A. Eleyan

15. General Physics II, Lec 10, By/ T.A. Eleyan
Example: A 3 mF capacitor and a 6 mF capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.
+Q1
-Q1 C1
V=Vab a b
+Q2
-Q2 C2
Given:
V = 18 V
C1= 3 mF
C2= 6 mF
Find:
Ceq=?
Q=?
First determine equivalent capacitance of C1 and C2:
Next, determine the charge
General Physics II, Lec 10, By/ T.A. Eleyan

16. General Physics II, Lec 10, By/ T.A. Eleyan
Series combination
Connecting a battery to the serial combination of capacitors is equivalent to introducing the same charge for both capacitors,
A voltage induced in the system from the battery is the sum of potential differences across the individual capacitors,
+Q1
-Q1 C1
+Q2
-Q2 C2
V=Vab a c b
General Physics II, Lec 10, By/ T.A. Eleyan

17. General Physics II, Lec 10, By/ T.A. Eleyan
Series combination:
Analogous formula is true for any number of
capacitors,
It follows that the equivalent capacitance of a series combination of capacitors is always less than any of the individual capacitance in the combination
(series combination)
General Physics II, Lec 10, By/ T.A. Eleyan

18. General Physics II, Lec 10, By/ T.A. Eleyan
Example: A 3 mF capacitor and a 6 mF capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance and total charge deposited.
+Q1
-Q1 C1
+Q2
-Q2 C2
V=Vab a c b
Given:
V = 18 V
C1= 3 mF
C2= 6 mF
Find:
Ceq=?
Q=?
First determine equivalent capacitance of C1 and C2:
Next, determine the charge
General Physics II, Lec 10, By/ T.A. Eleyan

19. Example: Capacitors in Series and Parallel
Three capacitors are connected as shown.
(a) Find the equivalent capacitance of the 3-capacitor combination.
(b) The capacitors, initially uncharged, are connected across a 6.0 V battery. Find the charge and voltage drop for each capacitor.
General Physics II, Lec 10, By/ T.A. Eleyan

20. General Physics II, Lec 10, By/ T.A. Eleyan

21. Energy stored in a charged capacitor
Consider a battery connected to a capacitor
A battery must do work to move electrons from one plate to the other. The work done to move a small charge q across a voltage V is
W = V q.
As the charge increases, V increases so the work to bring q increases. Using calculus we find that the energy (U) stored on a capacitor is given by: V V q Q
General Physics II, Lec 10, By/ T.A. Eleyan

22. Example: electric field energy in parallel-plate capacitor
Find electric field energy density (energy per unit volume) in a parallel-plate capacitor
Recall
Thus,
and so, the energy density is + -
General Physics II, Lec 10, By/ T.A. Eleyan

23. General Physics II, Lec 10, By/ T.A. Eleyan
Example: In the circuit shown V = 48V, C1 = 9mF, C2 = 4mF and C3 = 8mF.
(a) determine the equivalent capacitance of the circuit,
(b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance, C1 C2 V C3
The energy stored in the capacitor C123 is then
General Physics II, Lec 10, By/ T.A. Eleyan

24. Capacitors with dielectrics
A dielectrics is an insulating material (rubber, glass, etc.)
Consider an insolated, charged capacitor
Notice that the potential difference decreases (k = V0/V)
Since charge stayed the same (Q=Q0) → capacitance increases
dielectric constant: k = C/C0
Dielectric constant is a material property +Q -Q
Insert a dielectric V0 V
General Physics II, Lec 10, By/ T.A. Eleyan

25. General Physics II, Lec 10, By/ T.A. Eleyan
Capacitance is multiplied by a factor k when the
dielectric fills the region between the plates completely
E.g., for a parallel-plate capacitor
The capacitance is limited from above by the electric discharge that can occur through the dielectric material separating the plates
In other words, there exists a maximum of the electric field, sometimes called dielectric strength, that can be produced in the dielectric before it breaks down
General Physics II, Lec 10, By/ T.A. Eleyan

26. General Physics II, Lec 10, By/ T.A. Eleyan
Dielectric constants and dielectric strengths of various materials at room temperature
Material
Dielectric constant, k
Dielectric strength (V/m)
Vacuum
1.00 --
Air
3 × 106
Water 80
Fused quartz
3.78
9× 106
General Physics II, Lec 10, By/ T.A. Eleyan

27. General Physics II, Lec 10, By/ T.A. Eleyan
Example: Take a parallel plate capacitor whose plates have an area of 2 m2 and are separated by a distance of 1cm. The capacitor is charged to an initial voltage of 3 kV and then disconnected from the charging source. An insulating material is placed between the plates, completely filling the space, resulting in a decrease in the capacitors voltage to 1 kV. Determine the original and new capacitance, the charge on the capacitor, and the dielectric constant of the material.
Since we are dealing with the parallel-plate capacitor, the original capacitance can be found as
The dielectric constant and the new capacitance are
The charge on the capacitor can be found to be
General Physics II, Lec 10, By/ T.A. Eleyan

28. General Physics II, Lec 10, By/ T.A. Eleyan
How does an insulating dielectric material reduce electric fields by
producing effective surface charge densities?
Reorientation of polar molecules + - + -
Induced polarization of non-polar molecules + -
Dielectric Breakdown: breaking of molecular bonds/ionization of molecules.
General Physics II, Lec 10, By/ T.A. Eleyan

29. (Capacitance and capacitors)
Additional Questions
(Capacitance and capacitors)
General Physics II, Lec 10, By/ T.A. Eleyan

30. General Physics II, Lec 10, By/ T.A. Eleyan
Two capacitors, C1=2mF and C2=16mF, are connected in parallel.  What is the value of the equivalent capacitance of the combination?
(2) Calculate the equivalent capacitance of the two capacitors in the previous exercise if they are connected in series.
(3) A 100pF capacitor is charged to a potential difference of 50V, the charging battery then being disconnected.  The capacitor is then connected in parallel with a second (initially uncharged) capacitor.  If the measured potential difference drops to 35V, what is the capacitance of this second capacitor?
(4) A parallel-plate capacitor has circular plates of 8.0cm radius and 1.0mm separation.  What charge will appear on the plates if a potential difference of 100V is applied?
General Physics II, Lec 10, By/ T.A. Eleyan

31. General Physics II, Lec 10, By/ T.A. Eleyan
(5) In figure the battery supplies 12V.  (a) Find the charge on each capacitor when switch S1 is closed, and (b) when later switch S2 is also closed.  Assume C1=1mF, C2=2mF, C3=3mF, and C4=4mF.
(6) A 16pF parallel-plate capacitor is charged by a 10V battery.  If each plate of the capacitor has an area of 5cm2, what is the energy stored in the capacitor?  What is the energy density (energy per unit volume) in the electric field of the capacitor if the plates are separated by air?
(7) The energy density in a parallel plate capacitor is given as 2.1×l0-9.  What is the value of the electric field in the region between the plates?
General Physics II, Lec 10, By/ T.A. Eleyan

32. General Physics II, Lec 10, By/ T.A. Eleyan
(8) a- Determine the equivalent capacitance for the capacitors shown in figure.   b- If they are connected to 12V battery, calculate the potential difference across each capacitor and the charge on each capacitor
                                                                          
(9) A 6.0mF capacitor is connected in series with a 4.0mF capacitor and a potential difference of 200 V is applied across the pair. (a) What is the charge on each capacitor? (b) What is the potential difference across each capacitor?
(10) Repeat the previous problem for the same two capacitors connected in parallel.
General Physics II, Lec 10, By/ T.A. Eleyan

33. General Physics II, Lec 10, By/ T.A. Eleyan
[10] In a parallel plate capacitor an electron moves from the positively charged plate to the negatively charged plate. Which of the following are true for the electron?
The potential energy increases, and the electrical potential increases.
The potential energy increases, and the electrical potential decreases.
The potential energy decreases, and the electrical potential increases.
The potential energy decreases, and the electrical potential decreases.
The potential energy of the electron is unchanged.
[11] In Fig. the charge on the 3 micro Farad capacitor is 1 micro Coulomb. What is the voltage across the 2 micro Farad capacitor?
General Physics II, Lec 10, By/ T.A. Eleyan

34. General Physics II, Lec 10, By/ T.A. Eleyan
[12] A parallel plate capacitor is held at constant voltage. Initially there is only air between the plates. If a dielectric with a dielectric constant of 2 is inserted into the capacitor, what happens to the energy stored in the capacitor?
The energy will go up by a factor of 4.
The energy will go up by a factor of 2.
The energy will be unchanged.
The energy will go down by a factor of 2.
The energy will go down by a factor of 4.
[13] Evaluate the equivalent capacitance of the configuration shown in Figure. All the capacitors are identical, and each has capacitance C.
General Physics II, Lec 10, By/ T.A. Eleyan

35. General Physics II, Lec 10, By/ T.A. Eleyan
[14] For the system of capacitors shown in Figure, find (a) the equivalent capacitance of the system, (b) the potential across each capacitor, (c) the charge on each capacitor, and (d) the total energy stored by the group
[15] Find the equivalent capacitance between points a and b in the combination of capacitors shown in Figure.
General Physics II, Lec 10, By/ T.A. Eleyan

36. General Physics II, Lec 10, By/ T.A. Eleyan
[15] Two capacitors when connected in parallel give an equivalent capacitance of 9.00 pF and give an equivalent capacitance of 2.00 pF when connected in series. What is the capacitance of each capacitor?
[16] Two capacitors when connected in parallel give an equivalent capacitance of Cp and an equivalent capacitance of Cs when connected in series. What is the capacitance of each capacitor?
General Physics II, Lec 10, By/ T.A. Eleyan