CAPACITANCE   EXERCISE
COMBINATION   OF   CAPACITORS
1.
In   this   exercise   you   are   to   find   the   voltage   across   each   capacitor   and   the   charge   on   each   capacitor   in   the   circuit   below.
Use   V   =   12   volts,   C
1
=   3
μ
F,   C
2
=   2
μ
F,   and   C
3
=   4
μ
F.
1.1.
Before   solving   the   problem,   answer   the   following:
(a)   Do   you   expect   the   voltages   across   each   capacitor   to   be   the   same?
Are   there   any   capacitors   that   you   know   will   have   the   same   voltage   across   them?
Solution:
Parallel   capacitors   have   the   same   voltage   V   –   here   C
2
and   C
3
are   parallel.
(b)   Do   you   expect   the   charges   on   each   capacitor   to   be   the   same?
Are   there   any   capacitors   that   you   know   will   have   the   same   charge?
Solution:
Series   capacitors   have   the   same   charge   q   –   here   C
1
and   C p
,   the   equivalent   capacitor   for   C
2
and   C
3
,   are   in   series.
1.2.
Draw   a   simplification   of   the   circuit   by   reducing   the   parallel   combination   of   C
2
and   C
3
to   a   single   capacitor   C p
.
What   is   the   value   of   C p
?
Solution:
2 3
6
1.3.
Now,   are   the   charges   the   same   on   C
1
and   C p
?
What   about   the   voltages?
Solution:

As   stated   above   –   the   charges   are,   but   not   the   voltages.
1.4.
Now   reduce   the   series   combination   C
1
and   C p
to   the   equivalent   C s
.
Find   C s
and   redraw   the   simplified   circuit.
Solution:
1
1
1
thus
1
·
1
or
1
·
1
3·6
3 6
18
9
2
1.5.
Find   the   charge   on   C s
using   Q   =   CV.
Explain   why   this   is   the   same   as   the   charge   on   C
1
and   C p
.
Solution:   capacitor   C s
.
24
‐  the   charge   on   the   series   capacitors   C
1
and   C p
is   the   same   as   on   the   equivalent
1.6.
Use   the   relationship   V   =   Q/C   to   find   the   voltages   across   C
1
and   C p
.
Is   V
1
+   V p
=   12   volts?
Solution:
8   and   4
Yes,   V
1
+   V p
=   12   V.
1.7.
Use   the   relationship   Q   =   CV   to   find   the   charges   on   C
2
and   C
3
.
Does   Q
2
+   Q
3
=   Q s
?
Solution:
2
2
2 · 4
Yes,   Q
2
+   Q
3
=   Q s
.
8
and
4 · 4 16
CAPACITOR   CONNECTED   TO   A   SPRING
2.
Consider   an   air ‐ filled   parallel ‐ plate   capacitor   with   one   plate   (labeled   a )   connected   to   a   spring   having   a   force   constant   k ,   and   another   plate   ( b )   held   fixed   as   shown   in   the   figure .
If   the   charges   placed   on   plates   a   and   b   are   + Q and   ‐ Q ,   respectively,   how   much   does   the   spring   expand   from   the   position
( x 0 )   with   no   charges   present?
Hint:   What   is   the   electric   field   generated   by   the   charges   on   plate   b ?
Solution:
The   spring   force   F    acting   on   plate   a   is   given   by
F

The   electrostatic   force F due   to   the   electric   field   created   by   plate   b   is:
F   for   the   total   electric   field   in   the   capacitor   we   found   E
=
σ
ε
0 this   can   be   viewed   as   the   sum   of   the   fields   from   plate   a   and   b :
=
Q
ε
0
A
(where   A   is   the   area   of   the   plate)
as   both   plates   contribute   equally   we   have   2   or   result   as   one   would   get   for   a   single   plate   with   charge   Q   and   area   A).
With   this   we   get:   F
Q
2
ε
0
A
(which   is   the   same
In   equilibrium   the   two   forces   are   equal   in   magnitude   but   opposite   in   direction   thus:
or
2
CAPACITOR   NETWORK
3.
Evaluate   the   equivalent   capacitance   of   the   configuration   shown   in   the   figure.
All   the   capacitors   are   identical,   each   with   a   capacitance   C.
Solution:
First   calculate   the   equivalent   capacitance   of   the   capacitors   connected   in   series:
C
C
C
C
C
C
,         and
2
those   are   connected   in   parallel   to   give   the   total   equivalent   capacitance
3
6 3 2
6
11
6
IMPOSSIBLE   NETWORK
C 2C
4.
Determine   the   equivalent   capacitance   of   the   combination   of   capacitors   shown   in   the   figure.
Hint:   You   will   have   to   take   advantage   of   the   symmetry   of   the   problem!
3C
C 2C
Solution:
Based   on   the   symmetry   of   the   problem   the   voltage   drop   over   the   two   capacitors   with   capacitance   C   is   the   same.
Similarly   the   voltage   drop   over   the   two   capacitors   with   capacitance   2C   is   the   same.
This   means   that   the   potential   at   both   ends   of   the   capacitor   with   capacitance   3C   is   the   same   –   or   in   other   words,   there   is   no   potential   difference   across   the   capacitor   with   capacitance   3C!
Therefore   this   capacitor   does   not   add   to

the   total   capacitance   of   the   circuit   –   there   are   no   charges   on   it.
Therefore   the   circuit   is   equivalent   to   the   following   circuit:
C 2C
C 2C
For   which   one   can   easily   calculate   the   equivalent   capacitance:
or
These   two   equivalent   capacitances   are   connected   in   parallel   to   give   the   overall   equivalent   capacitance:
4
3
SPHERICAL   CAPACITOR
5.
A   conducting   solid   sphere   of   radius   a,   carrying   a   charge   +Q   is   surrounded   by   a   thin   conducting   spherical   shell   with   radius   b   (b>a)   carrying   a   charge   of   –
Q.
This   is   a   spherical   capacitor.
a b a)   What   is   the   direction   and   magnitude   of   the   electric   field   in   the   following   regions:
+Q
1.)   r<a   2.)   a<r<b   and   3.)   r>b
Solution:   ‐ Q
1.)   For   r<a   there   is   no   electric   field   (inside   conductor).
2.)   For   a<r<b   one   has   a   spherical   charge   distribution   with   a   total   charge   of   +Q,   so   the   electric   field   is:
1
4
3.)   For   r>b   the   net   charge   enclosed   is   zero,   thus   there   is   no   electric   field   outside   the   spherical   capacitor.
b)   What   is   the   potential   difference    ∆   between   the   spherical   shell   and   the   solid   sphere?
What   is   the   capacitance   of   the   spherical   capacitor?
Solution:
For   a<r<b   we   have   for   the   potential   V(r):
∆   (which   is   a   positive   quantity   because   b>a)
Thus   the   capacitance   of   a   spherical   capacitor   is:
∆
4