Capacitance
Capacity to store charge
C = Q/V
a
b
L
Q = Ll
E = l/2pe0r
V = -(l/2pe0)ln(r/a)
C = 2pe0L/ln(b/a)
Dimension e0 x L
(F/m) x m
1
Capacitance
Capacity to store charge
C = Q/V
A
d
Q = Ars
E = rs/e0
V = Ed
C = e0A/d
(F/m) x m
Increasing area increases Q and decreases C
Increasing separation increases V and decreases Q
2
Capacitance
Capacitor microphone – sound vibrations move a
diaphragm relative to a fixed plate and change C
Tuning  rotate two cylinders and vary degree of
overlap with dielectric  change C
Changing C changes resonant frequency of RL circuit
Increasing area increases Q and decreases C
Increasing separation increases V and decreases Q
3
Increasing C
with a dielectric
+ +
+
- -
+
-
+
+
-
e/e0 = er
C  erC
+
-
+
+
-
bartleby.com
To understand this, we need to see how dipoles operate
They tend to reduce voltage for a given Q
4
Effect on Maxwell equations:
Reduction of E
Point charge
in free space
.E = rv/e0
Point charge
in a medium
.E = rv/e0er
5
Point Dipole
R >> d
p. R
V = _________
4pe0R2
Note 1/R2 !
6
Point Dipole
p. R
V = _________
R >> d
4pe0R2
Note 1/R2 !


E = -V = -RV/R – (q/R)V/q


p(2Rcosq + qsinq)
_____________
4pe0R3
7
How do fields create dipoles?
-
+
+
J +
+
+
+
+
+
E
Let’s review what happens in a metal
conductivity
A field creates a current density, J=sE,
which moves charges to opposite ends, creating
an inverse field that completely screens out E
8
How do fields create dipoles?
E
+
-
Charges are not free to move in a dielectric!
But electrons can be driven by E a bit away from the
Nucleus without completely leaving it, creating an excess
Charge on one side and a deficit on the other, ....
9
…. in other words, generating a dipole
Rotating a Dipole
+
F+ = qE
-
F- = -qE
T = (d/2 x F+) - (d/2 x F+)
= qd x E
=pxE
10
That’s how microwave ovens work!
p = 6.2 x 10-30 Cm
2.5 GHz Radio wave source
Absorbed by water, sugar, fats
Aligns dipoles built in water molecule and excites atoms
“Friction” during rotation in opp. directions causes heating
11
Polarization and Dielectrics
- +
- +
- +
- +
-- +
-- +
-- +
-
+
- ++
-P
+
+
+
+
+
- +
+
+
- +
Instead of creating new dipoles, E could
align existing atomic dipoles (say on H20)
12
creating a net polarization P  E
Metal conductor vs Dielectric Insulator
Either way, the end result is excess charge on one end
and a deficit on the other, like a metal…
BUT…
There are differences!!
In a metal, E forms a current of freely moving charge,
and the applied E gets cancelled completely
In a dielectric, E creates a polarization of bound but
aligned, distorted but immobile charges, and the applied
E gets reduced partially (by the dielectric constant).
13
Metal conductor vs Dielectric Insulator
A metal is characterized by a conductivity s which determines
its resistance R to current flow, J=sE
A dielectric is characterized by a susceptibility c (and thus a
dielectric constant) which determines its capacitance C
to store charge, P = Np = e0cE
14
Dipoles Screen field
+
+
- -P +
- (opposing +
-polarization+
- Field) +
+
+
Displacement vector
D
(unscreened
Field)
E=(D-P)/e0
D = e0E + P
P = e0cE
Relative
D = eE
Permittivity
er
e = e0(1+c)
Thus the unscreened external field D gets reduced
to a screened E=D/e by the polarizing charges
For every free charge creating the D field from a
distance, a fraction (1-1/er) bound charges screen D to E=D/e
15
Free vs Bound Charges
+
+
- -P +
- (opposing +
-polarization+
- Field) +
+
+
E=(D-P)/e0
D
(unscreened
Field)
e0.E = rtotal
= .D - .P
= rfree + rbound
16
Effect on
Capacitance
Free charges on metal
+ +
-
+
-
+
+
-
+
Polarizing charges in dielectric
+
- -
-
+
-
-
+
-
e/e0 = K
which is why capacitors
Due to screening, only
Since the same charge
the
employ dielectrics
few ofon
the
field lines
(bartleby.com)
plates is now supported
by aon free charges
originating
smaller, screenedonpotential,
the metal plates survive
the capacitance (charge
stored byinside
applying
in the Dielectric
theunit volt)
has actually increased
by placement of a dielectric inside!
capacitor
17
Effect on Maxwell equations:
Reduction of E
.D = rv
x E=0
Differential eqns (Gauss’ law)
Fields diverge, but don’t curl
Defines Scalar potential E = -U
D = eE
 D.dS = q
 E.dl = 0
Integral eqns
Constitutive Relation
(Thus, E gets reduced by er)
er = 1 (vac), 4 (SiO2), 12 (Si), 80 (H20)
~2 (paper), 3 (soil, amber), 6 (mica),
18
Point charge
in free space
.E = rv/e0
Point charge
in a medium
.E = rv/e0er
19
Electrostatic Boundary values
Maxwell equations for E
.D = r
xE=0
Supplement with constitutive relation
D=eE
D1n
 D.dS=q
rs
 E.dl = 0
Use Gauss’ law for a short cylinder
Only caps matter (edges are short!)
D1n-D2n = rs
D2n
Perpendicular D discontinuous
20
Electrostatic Boundary values
Maxwell equations for E
.D = r
xE=0
Supplement with constitutive relation
D=eE
 D.dS=q
E1t
 E.dl = 0
E2t
No net circulation on small loop
Only long edges matter (heights are short!)
E1t-E2t = 0
Parallel E continuous
21
Electrostatic Boundary values
Perpendicular D discontinuous
Parallel E continuous
Can use this to figure out bending of E at an interface
(like light bending in a prism)
22
Example: Bending
Parallel E continuous
Perpendicular D continuous if no free charge rs at interface
e2
e1
q1
E1t=E2t
cosq1=cosq2
e1DE1n1n=D
=e22nE2n
e1sinq1=e2sinq2
q2
tanq1/tanq2 = e2/e1
e2 > e1 means q2 < q1
23
Conductors are equipotentials
• Conductor  Static Field inside zero (perfect screening)
• Since field is zero, potential is constant all over
• Et continuity equation at surface implies no field
component parallel to surface
• Only Dn, given by rs.
24
Field lines near a conductor
+
+
+
--+ -- -- +
+ + +
Equipotentials bunch up here
 Dense field lines
Principle of operation
of a lightning conductor
Plot potential, field lines
25
Images
Charge above
Ground plane
(fields perp. to surface)
Compare with
field of a Dipole!
Equipotential on metal enforced by the image
So can model as
Charge
+
Image
26
Images
27