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A sphere of radius A has a charge Q uniformly spread
throughout its volume. Find the difference in the electric
potential, in other words, the voltage difference, between
the center and a point 2A from the center.

r2
 
Vr2  Vr1    E  dr  0

r1
since

E  0 inside the conductor.

For any two points r1
and

r2
inside the conductor
Vr1  Vr2
The conductor’s surface is an equipotential.
Equipotential Surfaces
An equipotential surface is a surface on which
the electric potential V is the same at every
point.
Because potential energy does not change as a test charge moves over
an equipotential surface, the electric field can do no work on such a

charge. So, electric field must be perpendicular
to the surface at every
point so that the electric force qE
is always perpendicular to the
displacement of a charge moving on the surface.
Field lines and equipotential surfaces are always
mutually perpendicular.
Method of images: What is a force on the point charge near a conducting plate?
Equipotential surface
-
--
The force acting on the positive charge is exactly the same as it would
be with the negative image charge instead of the plate.
a
The point charge feels a force towards the plate with a magnitude:
1
2
q
F
40 (2a) 2
Method of images: A point charge near a conducting plane.

E ?
Equipotential surface
-
--
P
r
a
E  
1
aq
40 (a 2  r 2 ) 3 2
1
aq
E  
40 (a 2  r 2 ) 3 2
2
aq
E
40 (a 2  r 2 )3 2
Equilibrium in electrostatic field: Earnshaw’s theorem
There are NO points of stable equilibrium in any electrostatic field!
How to prove it? Gauss’s Law will help!
Imaginary surface
surrounding P
P
If the equilibrium is to be a stable one, we require that if we move the
charge away from P in any direction, there should be a restoring force
directed opposite to the displacement. The electric field at all nearby points
must be pointing inward – toward the point P. But that is in violation of
Gauss’ law if there is no charge at P.
Thomson’s atom
1899
If charges cannot be held stably, there cannot be matter made up of
static point charges (electrons and protons) governed only by the laws
of electrostatics. Such a static configuration would collapse!
Capacitors
Consider two large metal plates which are parallel to each other
and separated by a distance small compared with their width.
y
Area A
L








The field between plates is






 
 
V

E
0

 [V (top)  V (bottom)]   E y dy 
dy   L
0
0
0
0
L
L


 A
QL
 [V (top)  V (bottom)]   L  
L
0
0 A
0 A
QL
V 
A 0
The capacitance is:
A 0
Q
Q
C


QL
V
L
A 0
Cylindrical Capacitor
Spherical Capacitor
A 0
Q
Q
C


V QL
L
A 0
Capacitors in series:
1
1
1
1



 ...
Ctot C1 C2 C3
Capacitors in parallel: Ctot  C1  C2  C3  ...
1
1 2
2
W  CV 
Q
2
2C
[C ]  farad
Quiz
1) If a 4-F capacitor and an 8-F capacitor are connected in
parallel, which has the larger potential difference across it?
Which has the larger charge?
2)
A
B
Two capacitors are connected in series as shown. If they were
initially uncharged, what will be the charge inside the dotted box
after connecting points A and B to a battery of voltage V?
3) If the wire connecting the capacitors is bent so that capacitors
look like
A
B
how do you now call the arrangement?
a
C3
C1
C2
b
C5
C4
C1=C5=8.4 F and C2=C3=C4=4.2 F
The applied potential is Vab=220 V.
a)What is the equivalent capacitance of the network
between points a and b?
b) Calculate the charge on each capacitor and the
potential difference across each capacitor.
 
E

d
r

0
in
electrosta
tics


E
 
E  dr  0
B
C

E2

E1
D
A
 
E  dr  0
  D 
 E1dr   E2dr  E1 ( B  A)  E2 ( D  C )  0
B
A
C
E1  E2
Most capacitors have a non-conducting material, or dielectric, between their
conducting plates. When we insert an uncharged sheet of dielectric
between the plates, experiments show that the potential difference
decreases to a smaller value V.
Q
C0 
without dielectric
V0
Q
C
with dielectric
V
C  C0
When the space between plates is completely filled by the dielectric, the ratio
C
K
C0
is called dielectric constant.
Smaller, Denser, Cheaper
Moore’s Law (1965): every 2 years
the number of transistors on a chip is
doubled
Have a great day!