# Volume charge density ? and Gauss`s law ```Ex. 2: Integration - Disk of charge
Circular filament
the field at P
from it!)
Field is along k
σ=Q/(πa2)
Integrate over set
of rings, each of
thickness dr, with
charge dQ=
σ2πrdr
r
dE =
k (dq ) z ˆ k (σ 2πrdr ) z ˆ
k
k= 2
( z 2 + r 2 )3 / 2
( z + r 2 )3 / 2
r
2rdr
a
E = kσπzkˆ ∫r = 0 2
( z + r 2 )3 / 2
v

z
E = 2kσπkˆ1 −
2
z + a2

=
2kQ ˆ
z
k 1 −
2
a
z 2 + a2

Volume charge density ρ and Gauss’s law
h
• Volume charge density ρ : dQ=ρdV;
• If volume charge density is uniform, ρ=Q/V
• Example: Infinite “slab” of charge (Gauss’s law)
z
z
Gaussian
surface for
outside field
ρ
Gaussian
surface for
inside field
Φ E = EA + EA =
2 EA =
E =
Q in
ρ (2 z ) A
=
ε0
ε0
2 ρ zA
ε0
ρz
ε0
r
w
ρz ˆ
E =
k .... for z &lt;
2
ε0








w
h
Uniform volume charge density ρ
Field above slab same as that of infinite sheet of charge
with σ = ρw r ρw
E=
2ε 0
kˆ
Inside slab, use Gaussian box going from -z to +z with
area A top and bottom. Again, E = E k, so
Motion of charge in Electric Field
• Nothing new - electric force acts like any other force;
charge q of mass m in field E has acceleration:
r
q r
a =
E
m
• Object containing equal amounts of pos. &amp; neg.
charge will not feel force in uniform E, but will feel
force if E varies with position.
Note that the sign automatically changes for z&lt;0
Question: Start at midpoint and move up; how does E change?
1
Electric Dipole
• Common arrangement of charges:
Forces and Torques on Dipole
-Q
+Q
• Dipole
r placed in
r uniform rfield feels no net force
-
+
• Feels net torque :
l
p
• No net charge on dipole, but charges separated
• Example: water molecule
• Dipole moment p = Q l where l is the vector from
the negative charge to the positive charge
Torque example
• What is the magnitude of the torque on a dipole
with p = 0.01 C•m and in field of strength E = 100
N/C if the angle between p and E is 30&deg;?
• Is the torque on a dipole in an E field ever zero?
Ftotal = QE + (−Q ) E = 0
r
r r
τ total = ∑ ri &times; Fi
i
r
r r
r1 = 0....r2 = l
r
r r
r
r
τ total = l &times; QE = Ql &times; E
r r r
τ = p&times;E
Thus torque on dipole is
Question: Direction of torque?
Electric Potential Energy
• Remember gravitational potential energy U = mgh
for object of mass m at height h above zero potential
reference position (valid near Earth surface)
– Describes energy stored when work done against
conservative force (gravity)
– g is strength of Earth gravitational field (in N/kg)
– We must do work on mass m to increase its
gravitational potential energy
• A charged particle in an electric field has electric
potential energy (electric force is conservative)
Potential Energy
Coupling strength Field
Mass
Gravitational
Gravitational
Charge
Electric
Electric
2
```