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Announcements
I need paperwork/notification for any special needs by the end of lectures today.
If possible, Test Preparation Homework for Exam 1 will be available on the Physics 2135 web site (under Handouts) late Friday (otherwise Sunday). It will be handed out in lecture next Monday. This is next Tuesday’s homework. Do not ignore it!
Test 1 Room Assignments (next slide) are available on the Physics 2135 web site (under Course Information). This information will be included with Test Preparation Homework 1.
Exam 1 is Tuesday, February 17, 5:00-6:00 pm.
Know the exam time!
Find your room ahead of time!
If at 5:00 on test day you are lost, go to 104 Physics and check the exam room schedule, then go to the appropriate room and take
the exam there.
Exam is from 5:00-6:00 pm!
Physics 2135 Test Rooms, Spring 2015:
Instructor Sections RoomDr. Kurter F, H 104 PhysicsDr. Madison E, G 199 ToomeyDr. Parris K, M 125 Butler-Carlton (Civil Eng.)Mr. Upshaw A, C, J, L G-3 SchrenkDr. Waddill B, D G-31 EECH (Electrical Eng.)
Special Accommodations Testing Center
More Announcements Exam 1 special arrangements:
Test Center students. The Test Center e-mails you confirming your appointment. The Testing Center will tell you in that email when your exam starts, and no one will be admitted after 5:15 pm, unless you have made other arrangements.
If you have not received a confirming e-mail, you are NOT on the Test Center list!
Other special cases: you should already have been in correspondence with me or your recitation instructor.
Anybody else must let me know by the end of the 1:00 lecture today about special needs for the exam.
Today’s agenda:
Capacitance.You must be able to apply the equation C=Q/V.
Capacitors: parallel plate, cylindrical, spherical.You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors.
Circuits containing capacitors in series and parallel.You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.
Capacitors and Dielectrics
Capacitance
A capacitor is basically two parallel conducting plates with air or insulating material in between.
V0 V1
E
L
A capacitor doesn’t have to look like metal plates.
Capacitor for use in high-performance audio systems.
When a capacitor is connected to an external potential, charges flow onto the plates and create a potential difference between the plates.
Capacitor plates build up charge.
The battery in this circuit has some voltage V. We haven’t discussed what that means yet.
The symbol representing a capacitor in an electric circuit looks like parallel plates. Here’s the symbol for a battery, or an external potential. +-
-
-V
+-
-+
If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy).
Capacitors are also very good at releasing their stored charge all at once. The capacitors in your tube-type TV are so good at storing energy that touching the two terminals at the same time can be fatal, even though the TV may not have been used for months.High-voltage TV capacitors are supposed to have “bleeder resistors” that drain the charge away after the circuit is turned off. I wouldn’t bet my life on it.
Graphic from http://www.feebleminds-gifs.com/.
+
+
-
-V
conducting wires
On-line “toy” here.
assortment of capacitors
The magnitude of charge acquired by each plate of a capacitor is Q=CV where C is the capacitance of the capacitor.
The unit of C is the farad but most capacitors have values of C ranging from picofarads to microfarads (pF to F).
micro 10-6, nano 10-9, pico 10-12 (Know for exam!)
QC
V C is always
positive.
+Q
+
-Q
-V
CHere’s this V thing again. It is the potential difference provided by the “external potential.” For example, the voltage of a battery. V is really a V.
V is really V.
Today’s agenda:
Capacitance.You must be able to apply the equation C=Q/V.
Capacitors: parallel plate, cylindrical, spherical.You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors.
Circuits containing capacitors in series and parallel.You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.
Parallel Plate Capacitance
V0 V1
E
d
We previously calculated the electric field between two parallel charged plates:
0 0
QE .
A
This is valid when the separation is small compared with the plate dimensions. We also showed that E and V are related:
+Q-Q
A
d d
0 0V E d E dx Ed .
0
0
AQ Q QC
V Ed dQd
A
This lets us calculate C for a parallel plate capacitor.
Reminders:Q
CV
Q is the magnitude of the charge on either plate.
V is actually the magnitude of the potential difference between the plates. V is really |V|. Your book calls it Vab.
C is always positive.
V0 V1
E
d
+Q-Q
A
0ACd
Parallel plate capacitance depends “only” on geometry.
This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum (lecture 9).
0ACd
Greek letter Kappa. For today’s lecture (and for exam 1), use Kappa=1.Do not use =9x109! Because it isn’t!
We can also calculate the capacitance of a cylindrical capacitor (made of coaxial cylinders).
L
Coaxial Cylinder Capacitance
The next slide shows a cross-section view of the cylinders.
Q
-Q
br
a
E
d
Gaussian surface
Q λ L λ LC = = =
bΔV ΔV2k λ ln
a
02πε LLC = =
b b2k ln ln
a a
Lowercase c is capacitance per unit length: 02πεCc = =
bLln
a
2kλE =
r
This derivation is sometimes needed for homework problems! (Hint: 4.10, 11, 12.)
Some necessary details are not shown on this slide! See lectures 4 and 6.
b b
b a r
a a
ΔV = V -V = - E d = - E dr
b
a
dr bΔV = - 2k λ = - 2k λ ln
r a
Isolated Sphere Capacitance
An isolated sphere can be thought of as concentric spheres with the outer sphere at an infinite distance and zero potential.We already know the potential outside a conducting sphere:
0
QV .
4 r
The potential at the surface of a charged sphere of radius R is
0
QV
4 R
so the capacitance at the surface of an isolated sphere is
0
QC 4 R.
V
Capacitance of Concentric Spheres
If you have to calculate the capacitance of a concentric spherical capacitor of charge Q…
In between the spheres (Gauss’ Law)
20
QE
4 r
b
2a0 0
Q dr Q 1 1V
4 r 4 a b
04QC
1 1Va b
You need to do this derivation if you have a problem on spherical capacitors! (not this semester)
+Q
-Q
b
a
If there is spherical capacitor homework, details will be provided in lecture!
Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 cm and are separated by a 1.0 mm air gap.
d = 0.001area = 0.2 x 0.03
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
0ACd
128.85 10 0.2 0.03C
0.001
12C 53 10 F
C 53 pF
Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery?
0 V
+12 V
V= 12V
Q CV
12Q 53 10 12
10Q 6.4 10 C
*Remember, it’s the potential difference that matters.
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
Example: what is the electric field between the plates?
0 V
+12 V
V= 12V
d = 0.001
E
VE
d
12VE
0.001 m
VE 12000 ,"up."
m
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
Demo: Professor Tries to AvoidSpot-Welding His Fingers
to the Terminals of a CapacitorWhile Demonstrating Energy
Storage
Asynchronous lecture students: we’ll try to make a video of this.
Today’s agenda:
Capacitance.You must be able to apply the equation C=Q/V.
Capacitors: parallel plate, cylindrical, spherical.You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors.
Circuits containing capacitors in series and parallel.You must understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.
Capacitors in Circuits
Recall: this is the symbol representing a capacitor in an electric circuit.And this is the symbol for a battery… +-
…or this…
…or this.
Capacitors connected in parallel:C1
C2
C3
+ -
V
The potential difference (voltage drop) from a to b must equal V.
a b
Vab = V = voltage drop across each individual capacitor.
Vab
Circuits Containing Capacitors in Parallel
Note how I have introduced the idea that when circuit components are connected in parallel, then the voltage drops across the components are all the same. You may use this fact in homework solutions.
C2
C3
+ -
C1
C2
C3
+ -
V
a
Q = C V
Q1 = C1 V
& Q2 = C2 V
& Q3 = C3 V
Now imagine replacing the parallel combination of capacitors by a single equivalent capacitor.
By “equivalent,” we mean “stores the same total charge if the voltage is the same.”
Ceq
+ -
V
a
Qtotal = Ceq V = Q1 + Q2 + Q3
Q3
Q2
Q1
+ -
Q
Important!
Q1 = C1 V Q2 = C2 V Q3 = C3 V
Q1 + Q2 + Q3 = Ceq V
Summarizing the equations on the last slide:
Using Q1 = C1V, etc., gives
C1V + C2V + C3V = Ceq V
C1 + C2 + C3 = Ceq (after dividing both sides by V)
Generalizing: Ceq = Ci (capacitors in parallel)
C1
C2
C3
+ -
V
a b
Capacitors connected in series:
C1 C2
+ -
V
C3
An amount of charge +Q flows from the battery to the left plate of C1. (Of course, the charge doesn’t all flow at once).
+Q -Q
An amount of charge -Q flows from the battery to the right plate of C3. Note that +Q and –Q must be the same in magnitude but of opposite sign.
Circuits Containing Capacitors in Series
C1 C2
+ -
V
C3
+QA
-QB
The charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors:
-Q +Q
These equal and opposite charges came from the originally neutral circuit regions A and B.
Because region A must be neutral, there must be a charge +Q on the left plate of C2.
Because region B must be neutral, there must be a charge -Q on the right plate of C2.
+Q -Q
C1 C2
+ -
V
C3
+QA
-QB
-Q +Q+Q -Q
Q = C1 V1 Q = C2 V2 Q = C3 V3
The charges on C1, C2, and C3 are the same, and are
But we don’t know V1, V2, and V3 yet.
a b
We do know that Vab = V and also Vab = V1 + V2 + V3.
V3V2V1
Vab
Note how I have introduced the idea that when circuit components are connected in series, then the voltage drop across all the components is the sum of the voltage drops across the individual components. This is actually a consequence of the conservation of energy. You may use this fact in homework solutions.
Ceq
+ -
V
+Q -QV
Let’s replace the three capacitors by a single equivalent capacitor.
By “equivalent” we mean V is the same as the total voltage drop across the three capacitors, and the amount of charge Q that flowed out of the battery is the same as when there were three capacitors.
Q = Ceq V
Collecting equations:
Q = C1 V1 Q = C2 V2 Q = C3 V3
Vab = V = V1 + V2 + V3.
Q = Ceq V
Substituting for V1, V2, and V3:1 2 3
Q Q QV = + +
C C C
Substituting for V:eq 1 2 3
Q Q Q Q = + +
C C C C
Dividing both sides by Q:eq 1 2 3
1 1 1 1 = + +
C C C C
Important!
Generalizing:
OSE: (capacitors in series)ieq i
1 1 =
C C
Summary (know for exam!):
Series
eq ii
C C
same Q, V’s add
Parallel
same V, Q’s add
ieq i
1 1
C C
C1 C2 C3
C1
C2
C3
C3
C2
C1
I don’t see a series combination of capacitors, but I do see a parallel combination.
C23 = C2 + C3 = C + C = 2C
Example: determine the capacitance of a single capacitor that will have the same effect as the combination shown. Use C1 = C2 = C3 = C.
C1= CC23 = 2C
Now I see a series combination.
eq 1 23
1 1 1 = +
C C C
eq
1 1 1 2 1 3 = + = + =
C C 2C 2C 2C 2C
eq
2C = C
3
Example: for the capacitor circuit shown, C1 = 3F, C2 = 6F, C3 = 2F, and C4 =4F. (a) Find the equivalent capacitance. (b) if V=12 V, find the potential difference across C4.
I’ll work this at the blackboard.
C3
C2C1 C4
V
Homework Hint: each capacitor has associated with it a Q, C, and V. If you don’t know what to do next, near each capacitor, write down Q= , C= , and V= . Next to the = sign record the known value or a “?” if you don’t know the value. As soon as you know any two of Q, C, and V, you can determine the third. This technique often provides visual clues about what to do next.
You really need to know this:
Capacitors in series…all have the same chargeadd the voltages to get the total voltage
Capacitors in parallel…all have the same voltageadd the charges to get the total charge
(and it would be nice if you could explain why)
Homework Hint!
What does our text mean by Vab?
C3
C2C1 C4
V
a b
Our text’s convention is Vab = Va – Vb. This is explained on page 84. This is in contrast to Physics 23 notation, where Vab = Vb – Va.
In the figure on this slide, if Vab = 100 V then point a is at a potential 100 volts higher than point b, and Vab = -100 V; there is a 100 volt drop on going from a to b.
A “toy” to play with…
http://phet.colorado.edu/en/simulation/capacitor-lab
(You might even learn something.)
For now, select
“multiple capacitors.”
Pick a circuit.
