# RC Circuit Answers - Rockwood Staff Websites Staff Websites

```RC Circuits Question:
1. A 3.00 Mo. resistor and a 1.00 ,IF capacitor are connected in series with an ideal battery of
V = 4.00 V. At 1.0 seconds after the connection is made what are the rates at which
(a) charge on the capacitor is increasing? (0.955 microA)
t1
_
()-&) =
•.
cE..
"C-\.,"f/O
')<1
(_b/a..c.)
L :::<-(if -(.
l~
.:L~I,
l
- 3YfO
(-3 1,,)(1
I ,
?ltoS') ~~~q.-S5-Y.-/O-1'
.4-'
(,"--- __J
(b) energy is being stored in the capacitor? (1.08 mircoJ)
\J
l(
="
V
_
\}c~ -
I
_
b II
--- 4(1
P '-(3.ss"c'D-'t-X(.r3) 11.O~)</O-(oJ]
~
(,) <h,,~1 ~"gy ;,.liP"';'" ;. <h, W'"'"" (274J<1 J)
0
L,
c..,
V
?r" L, 1(2
'"
-
t/~c.)
e.
+ I ~ 5)
I.l;~
ti ,5SlIO - 1-)[6 x/O"') ::- p:1-4 y: JO-
(d) energy is being delivered by the battery? (3.82~
r~V
'L
:; ~.55'J/O-1-.xy)
J
(f~/t/D
(0
(
J
r
S1
35
.'
AI' Advanced Physics RC Circuit Problems
I.vnglh
0.02
2012
11I~
~I
Paper Su ir
#2
A physics slUdent wishes to measure the resistivity of slightly conductive paper that has a thickness of 1.0 x I O-'m. The
student cuts a sheet of the conductive paper into strips of width 0.02 m and varying lengths. making five resistors labeled
Rita R5. Using an ohmmeter, the student measures the resistance of each strip, as shown ab"",e. The dala are recorded
below.
Resistor
RI
R2
R2
R4
R5
Length (m)
0.020
0.040
0.060
0.080
0.100
80.000
180,000
260.000
370.000
440.000
Resistance
(a)
(n)
Use the grid below to plot a linear graph of the data points from which the resistivity of the paper can be determined.
labels and sC(lles for both axes. Draw the straight line that best represents the data.
~ \ +--- "--[-'i':-
----
C
-..l\~f\-:::.h::::
--::.::-:
~=i:::'-=-i-- -=- _.
_L; __ ...
~ __ i_-l_J __ .
I'
!
;
Include
- -- -~-'.'.-..;-i- ;--_.-----:~-
_
=-:::.:::." ~
==
:::..:=
_.,_1=:====.__ .~-~-~ __ - ---
I
;
,
;
-u=
_.-i-~~I
V
~:T-:-~-(iJ-':~='~~~~~~;
~-I"O
OOD':-
-- - - -.
-T.H-
-1--
- --- - - ~
_;__
~_1
__._ -- --
-;-t---
~j-=
.-~~
~
~
~
j0~
~
~
~
~~;
~
===j- ~f~i~l::::
.~=~=Lt
9-o
D
:.L~-=;= -=
I
!
I
!
J~_j~~.r
/'
..-~-
-'- -- - - --I--1-_A'-::L --
100":
;
_l-_
1_.
I
__
._
Ii!
-.1_1I
--1--
• -- -
.-------
_.-
_--i_L.
--
' ..'--1
,
1. ..J_
~_.
2V - _ J--;- L_ =- -= -= =-~=- == _~ _ =_:= =~=.:-=-~._~ =- = -t=!=
1 '
I-.r~i-_._ ..
_._-_.~
.--.-;-.j-------.,.-.-iI
;---
•
j
:
Using the graph. calculate the resistivity
/'V\-:. (S::l6-Ic"o)
I
I '2. -
I
3
Xlo
.
•
(.10)
(.w)
(.0'1)
(b)
_ _
.~-
...
~:t:\_l,-- - ---~~-+-- ------"
---"."--;-b"";l~------4-:--
-/---f-,:-l---L
Y
•
._._
of the parer.
J-
0:'&
~
f
&.2.CJ 'IIDa;)(:),O'lIt5~J
("')
~ ~ (.01 ~I 'l:IO~)
::-
Slo~ . A
,I~
t
-)
-= LC6/o+ {l' fY\
='
f
AP Advanced PhlSics RC Circnit Problems
R5
I./</O,DOO
/\.-
°
The student uses resistors R4 and R5 to build a circuit using wire, a 1.5 V battery, an uncharged I ~F capacitor, and an open
switch. as shown above.
(e) Calculate the time constant of the circuit.
.L
)lDixi>
r
~::.~/f«s3fl
(d) At time t = 0, the student closes the switch. On the axes below, sketch the magnitude of the voltage
ii, across
the capacitor
and the magnitudes of the voltage~-l and VR5 across each resistor as functio~s oftime t. Clearl)~ laGefeach curve
according to the circuit element it represenis:-bn the axes. explicitly label any intercepts. asymptotes, maxima, or minima
with values or expressions, as appropriate.
t.<;v
°
"
time (t)
.'
J:
R
T
E:
2007EJ.
S:,\h ~5°i
T c = 4000 pF
A student sets up the circuit above in the lab. The values of the resistance and capacitance are as shown, but the constant
voltage G delivered by the ideal battery is unknown. At time t = 0, the capacitor is uncharged and the student closes the
switch. The current as a function of time is measured using a computer system, and the follo\\ing graph is obtained.
_? .'
.)
- - -- -,.- -- -'--r - - -- -1'-- - - -"-1-I
.~
Z.O
.•..~.
1.5
1.0
"~ (l.S
U
+.
j
I
I
I
1_ •. _.
I
J
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
I
I
I
I
f
I
I
I
.:
- - ---...,.
- - - ---f- - - - -•
I
I
- ---I
1
'"__.L. __ .
I
1
•
<
"
-=E
!
. _.l.
I
!
t-- - - - ----1-- - - --1
_.---.i ------:------+---- - ~----I.
I
+.
•
J
•
j
~ c- ::[SSD)( 4tifJ '1-/01.)
-:: ;( ,'d-s
-~
I
I
-----t_-~--~
••~-.L:-----t-----~------1
1---1
••••
_,
,
I
o
,
!
••••••
4
2
10
6
'lime IS)
(a) Using the data above, calculate t7:-battery voltage G.
t><1o R :l?
~
~SM frJl~50):o~
(b) Calculate the voltage across the capacitor at time t = 4.0 s.
\/::\1,,0.>1 (l_it{l2C.j
::
V
\
::=\.~J,_()_'-I__
)
~
L;;;J4 (1_.i~/z;J ..
(c) Calculate the cha?,e on the capacit~\' at t = 4.0 s.
Q::C\l
-:.\,:!ODONO-~
II
\
]l1.0'-\\ =[O.oott C_
(d) On the axes below, sketch a graph of the charge on the capacitor as a function of time.
,605
'-
10
'lilll~(~)
(e) Calculate the power being dissipated as heat in the resistor at t = 4.0 s.
.
i
P -:: r; l- f....
(\..
~ ~
-5' -
-
L{~~2.lSEo)
=- '6 ,~).IO tY
(I) The capacitor is now discharged, its dielectric of constant K = 1 is replaced by a dielectric of constant K = 3, and the procedure is
repeated. Is the amounJ_of charge on one plate of the capacitor at t = 4.0 s now greater than. less than, or the same as before?
vi
Grealer than
. L
-.
Less thanT
.he same ~
,,--I'
et;J:;;J:!:u
Q~G-I.J Vs.U"~
\$0 1't... ~ l'Q
ww..
L4#
60 b!1
l?o.W
I
SJ
-.......:
11
E-=-
)
C
L
..-'
___
r
R1
~
S2
R2
2006E2. The circuit above contains a capacitor of capacitance C, a power supply of emf E , two resistors of resistances R, and R,. and
two switches. 5, and 5,. Initially, the capacitor is uncharged and both switches are open. Switch 5, then gets closed at time t ~
O.
a.
b.
~
a differential equation that can be solved to obtain the charge on the capacitor as a function of time t.
E-U<,- %:~O
L::
d~=
£ - CfJ
R
(Jt
G
--.1- Clf>- C£)
IZL
Solve the differential equation in art a. to detennine the charge on the capacitor as a function of time t.
(1u i{')
\n (O-U:
-
D
JAo Cb-Ct,
)\%
-:0
-
o
5- ~
(2.c..
%-cs:i
-c.~
b
\umerical values for the components are given as follows:
In j .:~~
:::-tiL
,j.
-tk.c.
t
IlC
1;-
Go:;
0
Cf.-C£.tJ
tt-= Q£C'-e,
E~12V
tJ
- /4,
C= 0.060 F
R, ~R,~4700n
c.
1.\
\-,1..
Determine the time at wh~h thetapacitor has a voltage 4.0 V across it.
Vc..~VlJqlf(f--i
/rl")
~'"
I '2.
1- e.:t-/
it "/IY
20'Z.
.
5\
After switch 51has been closed for a long time, switch 5, gets closed at a new time t '" O.
d.
On the axes b~low, sketch graphs ofth~ current 11in R1 versus time and of the c;rrent J.:. in R2 versus time, beginning when switch
5, is closed at new time t ~ O. Clearly label which gniph is I, and which is I, .
Current
TlIIle
)
R2
S
R,
20V=
:-l
I
~ 12
/
8
/
~
~
E 6
'w~ 4
""-
U
I
....--
~10
.'!
20~F
15 Jill
I
I
14
/
I
1/
2
0
o
5
15
10
Time
(5)
2004E2. In the circuit shown above left. the switch S is initially in the open position and the capacitor C is initially uncharged. A
voltage probe and a computer (not shown) are used to measure the potential difference across the ~apacitor as a function-of...time.after
the switch is closed. The graph produced by the computer is shown above right. The battery has an emf 0.QQ..Y and negligible internal
resistance. Resistor .Kt.has a resistance of Jil91 and the capacitor C has a capacitance of~F.
a. Determine the voltage across resistor R2 immediately after the switch is closed.
'dfJ V
'0 Ie
I
Cap ~
tJiJJ ukfe
(JJJ
b. Determine the voltage across resistor R2 a l?ng time after the switch is closed.
20
'6Y
-{L-;;
c. Calculate the value of the resistor R,.
'6 V
-= ~
12'L
q;/'?,'f./Ti<{ .;; ~2- "'"
d. Calculate the energy stored in the capacitor a long time after the switch is closed.
,
lh= \(
c V 'I...
-::..
~I
e.
20 y<
F(
J
0, 601Y'/
12-)1.:
On the axes below, graph the current rn R, as a function of time from 0 to 15 s. Label the vertical axis with appropriate values.
'J.
"'--'-_ L __ J. __
_
20
T"
'-1'-'
J. __ J__
,,'-'-'T""
-1'-'
_ _ -1
L. __ L __ l.__
I~ __ •
1 __ •1 __ ~I
I
1
~1
1
1 __ ~
1 __ +
1
~
I
1
I
',' -,_.
r '-1'-'
__ J __ J
1
L __
I'
1~ __
4__
~ __ ~1
I
••
I
1
-- --r--T--'-- --,---r--r--T-- --~--,--"'T---~--
--r-
IO,XO
•••
_J
1.--t--,---~---t--r--+---1--1--~---r-.L.._I._.
_•. _1_ . .L.._1._ .. ..J.._I_ .•.. __I._.
1
I
1
1
I
I
1
1
1
I
1
I
1
1
I
1
1
I
1
I
I
--r--~--T -,-- --"'T---r--~--T----'--~--"'T---r-1
--r--r--T--'
__ L.__ L __ J. __
- ---l---r--r--r-J__
_~
~ __ ~ __ L
--'--'---I---r-J __ ~
L __L.__
--r--~--+--~-----r--~--T--__~--~--~--r--,'_'-'T"'-l'-',-. --1'-' ','-l-'r-'-I--'
Current
I
I
I
1
I
1
I
I
1
1
__ L __ l. __ J. __
in{~~\
J
1
~
I
1
1
1
I
I
1
1
1
1
I
I
__J __ J __~ __L. __
L __ ~__
1
1
1
I
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
•
I
1
1
I
I
1
1
I
I
•
1
1
I
-~--+--+--~----~---~--~--~----~---r--r--T--'-- --,---r--~--T-- --,--~---r--r--.•.-f-- _._
- t- -.•..
+--,----:---:--r-- t-- --1-~---:---:_-_1._.
. ...•._1_ ..•... _1._.
...J.. _1_ .•.. _1._.
--r--r--T--'-- --"'T---r--~--T-- --,--~---r--r--
-__-}---tt-- --4---1-- -:---r-L __ L __-4--4
~ __ -- - --:---tL.__-~-~ __ L
r
__~--+--+--~--__~---~--~
__
--~--~--~---r-J
1
I
•
1
1
1
1
1
o
~
J __ ~__ ~
1
1
1
1
1
1
T-1
1
1
I'
~ __
I
1
1
1
15
10
5
Time (s)
Resistor R2 is removed and replaced \vith another resistor of lesser resistance. Switch S remains closed for a long time.
(f) Indicate below whether the energy stored in the capacit~eater
than, less than, or the same as it was with resistor R1 in the
circuit;
"'y"'Greater
-I
than
__
Less than
__
The same as
An.
v~ 1\.1 \')
I .." ~
v-ou IJ
(MV{W:Ji ~
~
IS
~D.-:C'J1
~1J.i
\~, A
vc,'\
2003E2. In the laboratory, you connect a resistor and a capacitor ,,;th unkno"n values in series with a battery of emfE ~ 12 V . You
include a switch in the circuit. \\'hen the switch is closed at time t = 0, the circuit is completed, and you measure the current tTiroli'gh the
resistor as a function of time as plotted beloVo'.
i(A)
0.010
t (5)
8
4
16
12
A data-fitting program finds that the current decays according to the equation i(l) = E: e -1/4
R
a.
Using common symbols for the battery, the resistor, the capacitor, and the switch, draw the circuit that you constructed. Show the
circuit before the s\\"itch is closed and include whatever other devices
need to measure the current through the resistor to
obtain the above plot. label each component in your diagram.
\
.
you
CO
e:.
b.
Having obtained the curve shown above, determine the value of the resistor that you placed in this circuit.
-8.
12-
((
c.
I 2..00 J\-
,01
What capacitance did you insert in the circuit to give the result above?
Vc(V)
Switch at A
Charging
y::;:.,e. c.
Switch at B
Discharging
I
---------------------~---------------------~
\2.00 G
I
t (,)
4
8
J2
16
20
24
28
32
You are now asked to reconnect the circuit with a new switch in such a way as to charge and discharge the capacitor. When the
switch in the circuit is in position A, the capacitor is charging; and when the switch is in position B. the capacitor is discharging, as
represented by the graph below of voltage V c across the capacitor as a function of time.
d.
Draw a schematic diagram of the RC circuit that you constructed that would produce the graph above. Clearly indicate switch
positions A and B on your circuit diagram and include whatever other devices you need to measure the voltage across the
capacitor to obtain the above plot. Label each component in your diagram.
..
vI"
o
1
T
R~::;'ill
c
1
Q L <: 10
/y ~~~
2002E2. Your engineering firm has built the RC circuit shown abov~e
currentis measmedu:r'the time t after the switch is closed
at t ~ 0 and the best-fit curve is represented by the equation I(t) ~ 5.20 .-"10, where I is in milliamperes and t is in seconds.
·D.~;"'."~:C;'''~:~.~(';;;;:;5)"e
V l
(PO
b. Determine the value of the capacitance C predicted by the equation.
c. The charging voltage is measured in the laboratory and .fu!!.l!d.Jobe greater than predicted in part a.
i. Give one possible expl~nation for this finding.
_L..
'
• ~
_' k
ii.
Op.e..",ed-t''Jed
a-v. ~.
(h,a..O 0-
.J\.LA-.L~1>\
l,.l) ,{
-&?
'9(~((
\/l
10 ~
(f'-I¥f' _
~
V\A
.
~~
•
U(e.u-d
Explain the implications that your answer to part i has for the predicted value of the capacitance.
-~
c..0J'7IU_dO~
\..0Z)\"J
d ~
~
d. Your laboratory supervisor tells you that the charging time must be decreased. You may add resistors or capacitors to the
original components and reconnect the RC circuit. In parts i and ii below, show how to reconnect the circuit, using either an
additional resistor or a capacitor to decrease the charging time.
i.
Indicate how a resistor may be added to gecrease the charging time. Add the necessary resistor and connections to the
following diagram.
••
II.
Instead of a resistor, use a capacitor. Indicate how the capacitor may be added to decrease the charging time. Add the
necessary capacitor and connections to the following diagram.
l'
WV
~
\__ - __ 3
'\1
~
>
~
"
~
~
~
"
/~
ij
"5
I
8
I
1
!
I"
'\.
6
1
1
1
1
1"'-
I
I"
4
I
I
!
2
0..
o
I I
1
,
\.
1
"-
I
-
•..•••
1
o
20
40
60
1
1
80
•.•.•
100
•••••
1
1
120
140 160 180 200
Time (min)
2001E2.
You have been hired to detennine the internal resistance of8.0 "F capacitors for an electronic component manufacturer.
(Ideal capacitors have an infinite internal resistance - that IS, the matenaroetween their plates is a perfect insulator. In practice,
however, the material has a very small, but nonzero. conductivity.) You cannot simply connect the capacitors to an ohmmeter,
because
their resistance
is too large for an ohmmeter
to measure. Therefore you charge the capacitor to a potential difference
of 10 V
with a battery, disconnect it from the battery and measure the potential difference across the capacitor every 20,mmutes with an ideal
voltmeter, obtaining the graph shown above.
- -- ~a:--I5efennine the internal resistance of the capacitor.
{} r
I, l-
VJ oJ
~i%Vo
::
1-0 ()\Ir-
~.1,
1""
l..Q
/..f\
'>c'(vu
1f/ 1lLIl 'YJ0
.s ----- . 'f,
)
~ 'f./O-11J
rhe capacitor can be approximated as a parallel-plate capacitor separated by a 0.10 mm thick dielectric with K ~ 5.6.
b. Determine the approximate surface area of one of the capacitor "plates."
"" (?c'o)(~.gSYlo-ll)
4
.1 x/0-3
d. Determine the magnitude of the charge leaving the positive plate of the capacitor in the first
1;-C£ LI_e--t1u.. )
(f)xIi);Il)( {- ~~)
_
-
Q _
_)".
~X ID
"-
(
I - e..
~
\,I.i~\
C. (,/11S)
)
J GOmin.
Physics Circuits
~<t
Cir.l;1'il?
~
Name:
_
1. When lighted. a 100-watt light bulb operating on a
110-volt household circuit has a resistance closest to
A. 10" 0
B. 10.10
f
C. 10
D. 100
E. 1000
2.
The five resistors shown below have the lengths and
cross-sectional areas indicated and are made of
material with the same resistivity. Which has the
greatest resistance?
( BI
f--1-J
(AI
A
(Al
IC)!-1-j
f------ 21-----1
\..1)
(Al
10)
II
7. Two resistors of the same length, both made of the
same material, are connected in a series to a battery as
shown above. Resistor II has a greater cross. sectional
area than resistor I. Which of the following quantities
has the same value for each resistor?
A. Potential difference between the two ends
B. Electric field strength within the resistor
C. Resistance
D. Current per unit area
E. Current
f--- 21----j
H {~1 GA)
lA
B
X
IA
10 V
-: 10 v
10
y
8.
The batteries in each of the circuits shown above are
identical and the wires have negligible resistance.
3.
In which circuit is the current furnished by the battery
the greatest?
(A)
(B)
(C)
(D)
(E)
4.
In which circuit is the equivalent resistance connected
to the battery the greatest?
(A)
(B)
(C)
(D)
(E)
5.
Which circuit dissipates the least power?
(A) .
(B)
(C)
(D)
(E)
In the circuit shown above, the emfs of the batteries
are given, as well as the currents in the outside
branches and the resistance in the middle branch. What
is the magnitude of the potential difference between X
and Y?
A. 4 V
B. 8 V
C. 10 V
D. 12 V
E. 16 V
R
2A
fA
40
60
R
6.
9.
,~ 24 v
In the circuit shown above, what is the resistance R ?
A. 30
B. 40
C. 60
D. 120
E. 180
In the circuit shown above, the capacitor C is first
charged by throwing switch S to the left, then
discharged by throwing S to the right. The time
constant for discharge could be increased by which of
the following?
A. Placing another capacitor in parallel with C
B. Placing another capacitor in series with C
C. Placing another resistor in parallel with the resistor
R
D. Increasing battery emf E
E. Decreasing battery emf E
Physics Circuits Test
Questions 10- 12 refer to the circuit shown below.
Assume the capacitor C is initially uncharged. The
following graphs may represent different quantities related
to the circuit as functions of time t after the switch S is
closed
10. Which graph best represents the voltage versus time
across the resistor R ?
(A)
(B)
(C)
(D)
(E)
11. Which graph best represents the current versus time in
the circuit?
(A)
(B)
(C)
(D)
(E)
12. Which graph best represents the voltage across the
capacitor versus time?
(A)
(B)
(C)
(D)
(E)
\
Name:
_
Av0lA.,~
~
KL Uy~
P(O~)e-n'\O
.?bl1-tf.2 u..)S\F"Y?~ o!' Q l)<IO')J) v ~ L-LW\\
C, 'Y~;},o\s
:lCDlof:l
---
Q)\.
;:).'-1 vb")
).0'-1 V
d) ~
q,o fL''', (j:; 2-)
(2C-
r:, t,
,
v'~'i~ iIi's-
C) D. 0 ali 2- (.
'PI G'V~
a.') J.~ ~ _ {(1- z c)
J1
Lf+P)
d)\~'J\~.VC.
~
c:tD'1 b'
1::;)
'\
.•
J)
IL
b}~o.tc (I-e
.
-s
.eo) 7-.J..? 'IIi)
--%G)
\'1/
c)t-;./N.35
```