1. Capacity to store charge
Capacitance
Capacity to store charge
C = Q/V L a b
Q = Ll
E = l/2pe0r
V = -(l/2pe0)ln(r/a)
C = 2pe0L/ln(b/a)
Dimension e0 x L
(F/m) x m

2. Capacity to store charge
Capacitance
Capacity to store charge
C = Q/V
Q = Ars
E = rs/e0
V = Ed
C = e0A/d A d
(F/m) x m
Increasing area increases Q and decreases C
Increasing separation increases V and decreases Q

3. 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

4. Increasing C with a dielectric- +
bartleby.com
e/e0 = er
C  erC
To understand this, we need to see how dipoles operate
They tend to reduce voltage for a given Q

5. Effect on Maxwell equations: Reduction of E
Point charge
in free space
in a medium
.E = rv/e0er
.E = rv/e0

6. Point Dipole
R >> d
p. R
4pe0R2
V = _________
Note 1/R2 !

7. R >> d Point Dipole p. R 4pe0R2 V = _________ Note 1/R2 ! 
E = -V = -RV/R – (q/R)V/q
p(2Rcosq + qsinq)
4pe0R3
_____________

8. How do fields create dipoles?+ - J + - E J
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

9. How do fields create dipoles?- +
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, ....
…. in other words, generating a dipole

10. + - Rotating a Dipole F+ = qE F- = -qE T = (d/2 x F+) - (d/2 x F+)
= qd x E
= p x E

11. That’s how micro- wave ovens work!
p = 6.2 x 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

12. Polarization and Dielectrics- + + - P - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - +
Instead of creating new dipoles, E could
align existing atomic dipoles (say on H20)
creating a net polarization P  E

13. 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).

14. 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

15. Dipoles Screen field + E=(D-P)/e0 D = e0E + P - P = e0cE D = eE
Displacement vector + - -P
(opposing
polarization
Field)
E=(D-P)/e0
D = e0E + P D
(unscreened
Field)
P = e0cE
D = eE
e = e0(1+c)
Relative
Permittivity er
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

16. Free vs Bound Charges + E=(D-P)/e0 - -P (opposing polarization D
Field)
E=(D-P)/e0 D
(unscreened
Field)
e0.E = rtotal
= .D - .P
= rfree + rbound

17. Effect on Capacitance + - + - e/e0 = K Since the same charge on the
which is why capacitors
employ dielectrics
(bartleby.com) +
Free charges on metal + -
Polarizing charges in dielectric -
e/e0 = K
Since the same charge on the
plates is now supported by a
smaller, screened potential,
the capacitance (charge stored by applying unit volt)
has actually increased by placement of a dielectric inside!
Due to screening, only
few of the field lines
originating on free charges
on the metal plates survive
in the Dielectric inside the
capacitor

18. 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
Integral eqns
 D.dS = q
 E.dl = 0
D = eE
Constitutive Relation
(Thus, E gets reduced by er)
er = 1 (vac), 4 (SiO2), 12 (Si), 80 (H20)
~2 (paper), 3 (soil, amber), 6 (mica),

19. Point charge
in free space
in a medium
.E = rv/e0er
.E = rv/e0

20. Electrostatic Boundary values
Maxwell equations for E
Supplement with constitutive relation
D=eE
.D = r
 x E = 0
D1n
 D.dS=q
 E.dl = 0 rs
D2n
Use Gauss’ law for a short cylinder
Only caps matter (edges are short!)
D1n-D2n = rs
Perpendicular D discontinuous

21. Electrostatic Boundary values
Maxwell equations for E
Supplement with constitutive relation
D=eE
.D = r
 x E = 0
 D.dS=q
 E.dl = 0
E1t
E2t
No net circulation on small loop
Only long edges matter (heights are short!)
E1t-E2t = 0
Parallel E continuous

22. 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)

23. Example: Bending Parallel E continuous
Perpendicular D continuous if no free charge rs at interface e1 e2 q1 q2
E1t=E2t
cosq1=cosq2
tanq1/tanq2 = e2/e1
e1E1n=e2E2n
D1n=D2n
e1sinq1=e2sinq2
e2 > e1 means q2 < q1

24. 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.

25. Field lines near a conductor+ -
Equipotentials bunch up here
 Dense field lines
Principle of operation
of a lightning conductor
Plot potential, field lines

26. Images So can model as Charge above Compare with Ground plane
field of a Dipole!
So can model as
Charge +
Image
Charge above
Ground plane
(fields perp. to surface)
Equipotential on metal enforced by the image

27. Images