Download Capacitor, capacitance, energy stored

Ph272(A) General Physics II Fall 2005
Instructor: Dr. Lam
Reading #4: Ch.24-2 to 24-6
HW#4: Capacitor and capacitance
1. (20 points) Capacitor and dielectric breakdown
A parallel plate capacitor has an area of 100 cm2 and a separation of 1 mm between the plates.
The space between the plates is filled with air. Given that the dielectric strength of air is
3kV/mm and dielectric constant of 1.00059 (see Table 29-2 on P. 671).
(a) What is the maximum voltage difference can be applied to the capacitor before dielectric
breakdown occurs?
(b) What is the maximum charge stored?
(c) What is the maximum energy stored?
(d) If the separation is increased, state whether the maximum voltage, maximum charge, and
maximum energy increase, decrease, or stay the same.
2. (15 points) Capacitor and dielectric materials
Suppose the capacitor of Q.1 is connected to a 50 V battery.
(a) When the capacitor is fully, how much charge is stored? How much energy stored?
(b) With the battery remain connected (i.e. voltage across the capacitor remains 50 V), insert a 1
mm thick dielectric material with dielectric constant = 4 in between the plates. What is the new
charge on the capacitor? Where do the charges come from? How much energy is stored?
[Note: This result implies that by inserting or removing a dielectric material within a capacitor,
one can cause charges flows to or from the capacitor. I am sure a good engineer can think of a
way of making this into a sensor.]
(c) Suppose we disconnect the battery first before we insert the dielectric material. After the
dielectric material is inserted, what is the charge stored, the voltage difference and the energy
stored? [Note: The energy stored in part c should be less than that from part a. Since systems
like to be in a lower energy state, it implies that if you place a dielectric material near a charged
capacitor, the capacitor would “suck” the dielectric in! The dielectric constant of the material
must be greater than that of air.)
3. (20 points) Force between charged capacitor plates.
Consider a parallel plate capacitor with area A and separation x. Assume x << A . The plates
are charged with +q and -q, see figure.
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+q
-q
x
Obviously, there is a force of attraction between the two plates. Derive an expression for the
force using the following two methods:
(a) Method 1: Find the electric field generated by the positive plate then calculate the force on
the negative plate due to this E-field. (Do you remember the relationship between F and E?)
(b) Method 2: Write down the expression for the energy stored by a capacitor; it should depend
1
r
dU ˆ
i to find the force. Compare this result with part a, are
on x. Use the relationship that F = −
dx
they the same?
(c) Re-express your results in (a) and (b) in terms of voltage across the capacitor. [Note: This
result implies that you can control the force between the plates by varying the voltage. Any
engineer out there who
€ can think of an application using this fact?!]
4. (15 points) Series and parallel circuit with capacitors
Consider the following circuit. The ideal battery has an emf=10 V. The capacitances of the
capacitors are: C1 = 10µF, C2 = 5µF, C3 = 4 µF .
C1
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10 V
C3
C2
(a) Find the charge stored and voltage across each capacitor.
(b) Calculate the total energy stored by adding the energy stored in each capacitor.
(c) Calculate the total energy stored by using equivalent capacitance. Compare with result in
part b.
____________
5. (20 poitns) (Optional/Extra Credit) In the text, the formula for a spherical capacitor with
ab
inner radius a and outer radius b is given by C = 4 πεo
(Eq. 30-8). The formula for a
b−a
cylindrical capacitor with inner radius a and outer radius b and length L is given by
L
C = 2πεo
(Eq. 30-11).
ln(b / a)
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ε A
The formula for a parallel plate capacitor is given by C = o (Eq. 30-5).
d
If the difference between the outer and inner radius (b-a) is much smaller than either radius, then
both the spherical and cylindrical capacitor should look like a parallel plate capacitor.
Show that when (b-a) << a and b, then both Eq. 30-8 and Eq. 30-11 are approximately equal to
Eq. 30-5.
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6. (20 points0 (Optional/Extra Credit) Energy can be thought of stored in the field, however
we must be careful when using it. Here is an illustration.
(a) Consider a point charge q. Find the electric field everywhere and calculate the energy stored
1 r r
εo E (r ) dV .
using U =
2
∫
all space
In class, I mentioned that there is no electrostatic potential energy when there is only one point
charge. Here you get a nonzero result (in fact infinite!). It turns out that the expression
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2
U=
∫
all space
1 r r
εo E (r ) dV contains not only the potential energy of interaction between charges
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but also the energy to create the charge. The energy to create a point charge is infinite!
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(b) Instead of a point charge, suppose the charge is spread out over a solid sphere of radius R
with uniform charge density ρ . Now Find the electric field everywhere and calculate the
1 r r
εo E (r ) dV . Now the result is not infinite.
energy stored using U =
2
∫
all space
[Note: A debatable€question is whether an electron is a point charge or a spread out charge
sphere!]
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