Transcript 5 Conductors in Electrostatics Equilibrium

AP Physics C
Mrs. Coyle

Gauss’s Law


surface
Permittivity of free space:
ε0 = 8.8542 x 10-12 C2 / (N m2)
k
1
4 0
 9 x10 Nm / C
9
2
2

E dA 
qin
EdA cos  
qin
surface
0
0
Insulators vs Conductors
 In an insulator, excess charge is not free to move.
 In conductors the electrons are free to move.
Electrostatic Equilibrium of Conductors
 Electrostatic Equilibrium for a conductor
– no net motion of charge within a
conductor.
 Most conductors, on their own, are in
electrostatic equilibrium.
 Ex: in a piece of metal sitting by itself, there
is no “current.”
Characteristics of Conductors in
Equilibrium
1.
The E-field is zero at all points inside a conductor
(regardless if it is hollow or solid).
2.
If an isolated conductor carries excess charge, the excess
charge resides on its surface.
3.
The E-field just outside a charged conductor is
perpendicular to the surface and has magnitude σ/ε0,
where σ is the surface charge density at that point.
4.
On an irregularly shaped conductors the surface charge
density is biggest where the conductor is most sharp.
 If the conductor is placed
in an electric field at first
there is a movement of
electrons(current) but
eventually the movement
stops and their is
equilibrium.
 If the E was not zero inside
the conductor the
movement would continue
and there would not be
equilibrium.
 Note: Inside the cylinder
there are no electric field
lines.
Ex 1: Point charge Inside
a Spherical Metal Shell
 A -5.0μC charge is located as shown in Fig a). If the shell
is electrically neutral, what are the induced charges on
its inner and outer surfaces? Are those charges uniformly
distributed? What is the E-field pattern?
Ex 1: Solution Strategy
 Since the shell is electrically neutral, E=0 inside the
shell.
 Take a Gaussian surface inside the shell.
 This Gaussian surface must encompass an enclosed
charge of zero because E=0 inside the conductor.
 The point charge is –5μC so since the net charge is
zero:
–5μC + x =0  x= 5μC . This x is the charge on the inside
surface of the shell. Since the shell is neutral the
outside surface of the shell must have a charge of
–x=-5μC
Ex 1: Solution Strategy cont’d
 Since the point charge is not in the center of the spherical
shell but off-centered, there will be more positive charges
closer to the point charge. The charge distribution in the
inner wall of the shell will be more dense closer to the
point charge.
 The field lines between the point charge and the shell
will be closer together nearest to the point charge.
 However, in the outer surface of the shell the negative
charges will be evenly distributed. This is the case no
matter where inside the shell, the point charge is located.
 The field lines are shown in figure b)
 E-lines are always perpendicular to the conductor surface.
Hollow Conductors
Charge placed INSIDE induces
balancing charge ON INSIDE
+
- - +
+ - +
+q
+ -- - - +
+
+
Hollow Conductor
A charge placed OUTSIDE induces
charge separation ON OUTSIDE surface.
+q
-
-
+
E=0
+
+
Ex 2: Sphere inside a Spherical Shell
A solid insulating sphere of
radius a carries a uniformly
distributed charge, Q.
A conducting shell of inner
radius b and outer radius c is
concentric and carries a net
charge of -2Q.
a) Find the E-field in regions 1-4
using Gauss’s Law.
b) Find the charge distribution
on the shell when it is in
electrostatic equilibrium.
Example #31
Consider a thin spherical shell of radius 14.0 cm
with a total charge of 32.0 μC distributed
uniformly on its surface. Find the electric field
(a)10.0 cm and
(b)20.0 cm from the center of the charge
distribution.
Answer: a) E  0, b) E  7.19 x10 N C
6
Ex. #35
A uniformly charged, straight filament 7.00 m in
length has a total positive charge of 2.00 μC. An
uncharged cardboard cylinder 2.00 cm in length
and 10.0 cm in radius surrounds the filament at its
center, with the filament as the axis of the cylinder.
Using reasonable approximations, find (a) the
electric field at the surface of the cylinder and
(b) the total electric flux through the cylinder.
E  51.4 kN C, radially outward
 E  646 N  m 2 C
Ex. #43
A square plate of copper with 50.0-cm sides has
no net charge and is placed in a region of
uniform electric field of 80.0 kN/C directed
perpendicularly to the plate.
Find
(a) the charge density of each face of the plate and
(b) the total charge on each face.
a) E 

   8.00  104 N / C 8.85 1012 C 2 / Nm2
0


  7.08 107 C m2


b) Q   A  7.08  10 C / m2  0.500m 
7
Q  1.77 107 C
2

Ex. #47
A long, straight wire is surrounded by a hollow metal
cylinder whose axis coincides with that of the wire. The
wire has a charge per unit length of λ, and the cylinder
has a net charge per unit length of 2λ. From this
information, use Gauss’s law to find (a) the charge per
unit length on the inner and outer surfaces of the
cylinder and (b) the electric field outside the cylinder, a
distance r from the axis.
a) Inside Surface: 0    qin 
qin
 
Outside Surface: q= 3
b) E 
2ke  3 
r
6k e 
3


radially outward
r
2 0 r
Faraday’s Ice Pail Experiment
A +charge sphere is brought into a neutral metal ice pail
attached to a neutral electroscope.