Parallel and series capacitors—summary
Î
Capacitors in parallel
C eq = C1 + C 2 + ⋅ ⋅ ⋅
Î
Capacitors in series
1 1 1
= + +⋅⋅⋅
Ceq C1 C2
It is foolish to connect capacitors in series.
Example: 100 μF in series with 10 μF is 9 μF (check yourself).
The 100 μF capacitor will be totally wasted.
PHY2049: Chapter 25
1
(continued)
Î
smaller of two
Two capacitor in series
1
1
1
=
+
Ceq C1 C 2
C1C2
C2
=
< C2
Ceq =
C1 + C 2 1 + C 2 / C1
less than 1
‹
Î
Ceq even smaller than smaller of the two
n capacitors in series
1
1
1
1
=
+
+
+L
Ceq C1 C 2 C3
‹
Ceq
1
1
>
Ceq C1
Ceq < C1
smallest
even smaller than smallest of all
PHY2049: Chapter 25
2
Examples
Î
Four 1 μF in parallel. Find Ceq.
4 μF
Î
Four 1 μF in series. Find Ceq.
0.25 μF
Î
1.3 μF and 2.0 μF in series. Ceq is:
(a) 0.79 μF
‹ (b) 1.65 μF
‹ (c) 3.3 μF
‹
PHY2049: Chapter 25
3
Example: parallel-series combo
Î
Equivalent capacitance?
1.0 μF
2.0 μF
3.0 μF
‹
1 and 2 in parallel
1.0 + 2.0 = 3.0 μF
‹
Together, in series with 3
1
1
1
2
1
=
+
=
=
Ceq 3.0 3.0 3.0 1.5
1.5 μF
PHY2049: Chapter 25
4
(continued)
Î
Charge on C1?
2.0 μF
1.0 μF
10 V
3.0 μF
‹
Total charge
10 V x 1.5 μF = 15 μC
‹
Charge on 1 and 2
Same as the total!
‹
Charge on 1
q=q1 + q2
Potential differences across 1 and 2 are the
same.
q1 and q2 in proportion with C1 and
q1=q x 1.0/(1.0+2.0) = 5.0 μC
C2.
PHY2049: Chapter 25
+q
–q
+q
–q
+q
–q
5
Energy stored in capacitor
Î
In terms of charge
‹
Derived by considering work dW’ done by a fictitious process
which moves infinitesimally small amount of charge +dq’ from
conductor 1 to conductor 2 of capacitor, leaving behind –dq’ on
conductor 1:
q2
U=
2C
Î
In terms of potential
‹
Î
Since q=CV (definition of C)
Compare with
1
K = mv 2 (kinetic energy)
2
PHY2049: Chapter 25
1
U = CV 2
2
U=
1 2
kx (spring)
2
6
Energy stored in electric field
Î
Two alternative views
Energy is stored in charge configuration in capacitor
‹ Energy is stored in E field in capacitor
‹
Î
Second view (will be important later in dealing with
electromagnetic waves)
Define energy density
‹ Show for parallel-plate capacitor
‹
Î
u=
U
volume
u=
1
ε0 E 2
2
This equation holds for any E field produced at any point
in space by any source
‹
Derivation requires vector calculus
PHY2049: Chapter 25
7
Equivalence of two views (by example)
Spherical conductor
ÎView
1
+
+
+
+
+
+
+
‹ Energy
is stored in capacitor’s charge configuration
Q2
U=
2C
‹ Generalize definition of capacitance to single conductor
and find (see page 661)
2
C = 4πε0 R
ÎView
2
‹ Energy
‹ Inside
is stored in E field
E=0
+
+
Q
+
R+
+
+
+
+
+
+
Q
8πε0 R
1
u = ε0 E 2
2
1 Q
4πε 0 r 2
Q 2 ∞ dr
Q2
2
=
4πr dr =
Checks!
2
∫
R
8πε0 r
8πε0 R
Outside E =
1  1 Q

U = ∫ udV = ∫ ε0 
2
R 2
 4πε0 r 
outside
∞
U=
+
2
(
)
PHY2049: Chapter 25
8
Dielectrics
ÎDielectric
polarized.
is insulator. In E field, it becomes partly
‹ For
microscopic view, read page 671.
‹ If dielectric fills the gap of charged capacitor, E0 due to charges +q
and –q partly polarizes it, inducing charges –q’ and +q’ near
surfaces.
‹ These in turn produce field that partly cancels E0.
‹ Net field E proportional to, and less than, E0.
ÎWhat’s
the point?
‹ E0
→ E = E0/κ
less than E0
‹ V0
→ V = V0/κ
from definition of V
‹ C0
→ C = κ C0
since C=q/V
Larger than C0, which means capacitor
stores more charge for given potential
difference applied by battery. Beneficial
to fill gap with dielectric.
PHY2049: Chapter 25
9
(continued)
ÎΚis
called dielectric constant. Larger than 1.
ÎInduced
charge q’.
‹ So
far, general to any capacitor. Now restrict
ourselves to parallel-plate capacitor
E0 A =
q
ε0
q + (− q′)
EA =
ε0
E q − q′
=
E0
q
κ
Gauss’ law
 1
q ′ = q 1 −  < q
 κ
Induced charge q’ is less than q.
κ=1 (vacuum, no dielectric) → q’=0
No induced charge.
κ large (strong dielectric) → q’→q
PHY2049: Chapter 25
10
REMINDER
ÎNext
WebAssign due Tomorrow,
Thursday
ÎTest
1 on Monday in Class (Chapters 21–
25) Study Sample Exams
‹ Must
bring Gator1 ID card (Will take away points if you
forget ;< )
‹ Calculator (No formulae allowed on calculator)
‹ One 8.5”x11” formula sheet (May use both sides)
‹ Blank scratch paper (Department has no money to
provide)
‹ Pencil, eraser, and sharpner, as usual
PHY2049: Chapter 25
11