Download The electric field between the plates is uniform with the direction

Example (25.2) Electric potential in a uniform electric field.
+σ
V = Ed
V = Ex
d
x
-σ
V = 0V
The electric field between
the plates is uniform with
the
direction
being
perpendicular to the plates
and with a magnitude of
E≈ σ.
ε0
a) From the relationship between the electric potential
and the electric field vector we can find potential at any
point. Choosing the reference at the negatively charged
plate (V = 0V) we can find potential at any point integrating
the electric field vector.
Ar
x
x
σx
r
r
V(r ) = − ∫ E ⋅ ds = − ∫ [− E ,0,0] ⋅ [dx, dy, dz] = ∫ Edx =E ⋅ x =
ε0
O
0
0
b) From the relationship between the electric field vector
and the electric potential
  σ   σ   σ 
 ∂ x  ∂  x  ∂ x  
ε
ε
ε
 σ

r
E = −∇ V(r ) = −   0  ,  0  ,  0   =  − ,0,0
 ∂x
∂y
∂z   ε 0





21
Example (25.3) Electric potential due to point charge.
B
E
θ
ds
r
(Recall that the electric field vector of a
point charge is
r r
Q
E(r ') = k 2 rˆ '
r'
r’
A
rA
Q
)
a) From the relationship between the electric potential
and the electric field vector we can find the potential at any
given point. Choosing the reference potential (VA = 0V) at
point A we can find potential at any point integrating the
electric field vector.
Br
B Q
B Q
r
r
r
V(r ) = VA − ∫ E ⋅ ds = − ∫ k 2 rˆ '⋅d s = − ∫ k 2 rˆ ' cos θds =
A
A r'
A r'
r
1 1 
Q
kQ
= − ∫ k 2 dr ' =
= kQ  − 
r ' rA
rA r '
 r rA 
r
It is convenient to choose the reference
of potential to be zero far away from the
point charge (at rA = ∞). With this
choice
r
Q
V(r ) = k
r
22
b) From the relationship between the electric field vector
and the electric potential
E (r ) = −∇V (r ) =
∂
kq
= − 
 ∂x  x 2 + y 2 + z 2
 ∂
kq
, 
 ∂y  x 2 + y 2 + z 2


 ∂
kq
, 
 ∂z  x 2 + y 2 + z 2
 
 1
2x
1
2y
= − kq  − ⋅
⋅
3 ,−
 2 x 2 + y 2 + z 2 2 2 x 2 + y 2 + z 2
kq
[x , y, z ] = kq ⋅ rˆ
=− 2 ⋅
2
r
x 2 + y2 + z2 r
(
)
(
23
)

 =


1
2z
⋅
3 ,−
2
2 x 2 + y2 + z2
(
)

=
3
2 

22. (25.5)
Electric potential due to a body (continuous
charge distribution).
The electric potential at a certain location produced by
a charged body can be found by adding the contributions to
that electric potential by all the charges within the body
r
dq
V(r0 ) = k ∫
body r
proof. When we divide the body into differential pieces,
each piece can be considered as a point charge dq. The
contribution to the electric potential dV from this piece
depends on the distance of the considered location to the
field source r.
r
dV = k dq
r
dV
dq
24
Example.
Find the electric potential at a point located on the axis of a
(uniformly) charged ring of radius a.
dϕ
dq
ϕ
a
x
P
dq
k
kQ
=
dq
=
2
2 ∫
body r
x + a body
x2 + a2
VP = k ∫
Comment.
Note that the component of the electric field vector along
the x-axis is:
Ex = −
∂V
kxQ
=
∂x
x2 + a2
(
25
)
3
2
Example.
Find the electric potential at a point located on the axis of a
(uniformly) charged disk of radius a.
R
dϕ ϕ
dr
dq = σrdϕdr
r
P
x
R 2 π σ rdϕ dr
R
2π
dq
r
VP = k ∫
= k∫ ∫ 2
= kσ ∫ 2
∫ dϕdr =
2
2
body r '
0 0
0 r +x 0
r +x
[
= 2πkσ r 2 + x 2
]
R
0
[
= 2 πkσ R 2 + x 2 − x
]
Comment.
Note that the component of the electric field vector along
the x-axis is:
Ex = −


∂V
x
= 2 πkσ1 −

2
2
∂x
x +R 

26
23. (25.6) Potential of a conductor
The surface of any conductor in equilibrium is an
equipotential surface. The electric potential has the same
value everywhere inside the conductor and is equal to its
value at the surface.
24. (26.7) Substance in an electric field
When
a substance is placed in an (external) electric
r
field E0, its molecules acquire a dipole moment related to
the external field (either by alignment of the molecules with
a permanent dipole moment or by induction of an electric
dipole moment). This creates an radditional electric field
(internal). The net electric field E is different when the
substance is present than when it is not.
The dielectric constant κ of the substance relates the
two fields:
r 1 r
E = ⋅ E0
κ
Substances for which the dielectric constant is more than
one are called dielectrics.
27
25. (26.6)
Electric dipole in a uniform electric field
r
a) The electric dipole moment p of a dipole is a vector
whose magnitude is
p = qd
and the direction from the negative charge toward the
positive charge.
p
+q
F
θ
d
-F
+
- -q
b) In a uniform electric field the net force exerted on a
v
dipole is a zero vector and the electric torque τ , about any
r
point, exerted on the dipole depends on the electric field E
r
and the dipole moment p
r r r
τ = p×E
This equation represents the general definition of the dipole
moment of an object (system of particles).
28
proof: When the net force exerted on an object is a zero
vector the torque is independent of the reference point.
From the definition (Recall eq. 11.7) the torque (about the
negative charge) is
r
r
r r r r
r r r
τ = d × Fel = d × qE = qd × E = p × E
c) The potential energy of a dipole in a uniform electric
field depends on the orientation of the dipole moment with
respect to the electric field.
r r
U = −p ⋅ E
(The reference zero potential energy is at the orientation of
the dipole perpendicular to the electric field)
proof:
r r
U = − Wel = − ∫ τ cos180° ⋅ dθ' = ∫ pE sin θ' dθ' = − pE cos θ = −p ⋅ E
θ
θ
90 °
90°
29
26. (26.1) Capacitors
A capacitor is an electrical element with two sides,
called plates, for which the potential difference (voltage)
between the plates is proportional to the charge transferred
from one plate to the other
+q
V2
-q
V1
V = V2 - V1
Q = CV
Coefficient C is called the capacitance of the
capacitor. (Capacitance does not say how much charge can
be stored on a capacitor !!!)
The SI unit of capacitance is 1 farad (1F = 1C/1V).
The symbol
circuit diagram.
is used to mark the capacitors on a
30
27. (26.2, 26.5) A parallel plate capacitor
A
d
κ
The capacitance C of a parallel plate capacitor
depends on the geometrical shape of the capacitor and the
dielectric constant κ of a medium between the plates
C = κε0 A
d
where A is the area of the plates, d is the distance between
the plates, and κ is the dielectric constant of the substance
between the plates.
proof:
Assume that charge Q was transferred from one plate
to the other. That transfer causes a potential difference
between the plates:
V = Ed = 1 ⋅ σ ⋅ d = d ⋅ Q
κ ε0
κε0A
Recalling the definition of capacitance we can see that
C = κε0 A
d
31
28. (26.5) Dielectric strength of a substance
The maximum (magnitude of) electric field that will
not cause charge transfer across the dielectric is called its
dielectric strength.
The combination of the capacitance of the capacitor
and the dielectric strength of the dielectric separating the
plates determines what charge can be "stored" on the
capacitor.
29. (26.4) Electrical energy stored in a capacitor
The electric potential energy stored in a capacitor is
Q2 1
U=
= 2 QV = 12 CV 2
2C
The energy stored in a capacitor can be considered as
being stored in the electric field. When the capacitor is
discharged the vanishing electric field "performs" work of
the above amount on the charged particles transferred from
one plate to the other.
32
proof:
The differential work dWel
done by the electric field,
produced by charge q on the
capacitor, on the differential
charge dq, transferred during the
charging from one plate to the
other, is
+q
V2
dq
V1
-q
dWel = − (V+ − V− )dq = − Vdq = −
qdq
C
From the definition of potential energy, with an
assumption that the potential energy of a discharged
capacitor is zero, we find
2 Q
Q qdq
1q
U = 0 − Wel = ∫
=
C 2
0 C
0
Q2
=
2C
(alternative derivation
dWel = − (V+ − V− )dq = − Vdq = − VCdV
From which
V
U = 0 − Wel = ∫ CV ' dV ' = C
0
33
2 V
V'
2
0
1
= CV 2 )
2
30. (26.4)
Energy density in an electric field
a) definition: The energy due to the electric per unit
volume
u = dU
dV
is called the energy density in the electric field.
b) If at a certain location the magnitude of the electric
field vector is E, the energy density at this location is:
1
u = κε0E2
2
proof:
dA
dx
E
1
1 κε0 dA
C(dV )2
(Edx )2 1
u=2
= 2 dx
= κε0 E 2
dxdA
dxdA
2
34
31. Systems of electrical elements
a) A series connection means that the devices (elements)
are connected in such a way that, in any time interval, an
equal charge is transferred through each device.
Q
1
Q
Q
2
n
b) A parallel connection means that the devices are
connected in such a way that the same potentials are
connected to both sides of each device.
1
Va
2
Vb
n
c) There are other possibilities of connections.
35
32. (26.3) System of capacitors
a) The equivalent capacitance of capacitors connected in
parallel is equal to the sum of the capacitances of all the
capacitors.
Cp = C1 + C 2 +... + Cn
proof.
Q = C1V + C2 V +... + Cn V = ( C1 + C2 +... + Cn ) V
Example. What is the capacitance of two 1µC capacitors connected in
parallel?
Cp = 1µC + 1µC = 2µC
b) The inverse of the equivalent capacitance of
capacitors connected in series is equal to the sum of the
inverses of the capacitances.
1
1
1
1
=
+
+...+
C s C1 C2
Cn
proof.
 1
Q Q
Q
1
1 

V=
+
+ ... +
= Q ⋅  +
+ ... +
C1 C 2
Cn
C
C
C
 1
2
n 
Example. What is the capacitance of two 1µC capacitors connected in
series?
 1
1 
Cs = 
+

 1µC 1µC
36
−1
= 0.5µC
33. (27.1) Electric (conduction) current
+
+
+
+
+
-
+
+
-
-
+
I
I
a) Transport of charge;
b) The electric current across a surface is defined as the
rate at which charge is transferred through this surface.
I≡
dQ
dt
The SI unit of current is 1A (ampere). (1C=1A⋅1s.)
(According to general agreement its direction is chosen to
coincide with the direction in which positive charge carriers
would move, even if the actual carriers have a negative
charge.)
37
34. (27.1, 27.3) Current density vector (for one type of
charge carriers.)
r r
a) The average velocity v d (r ) of charge carriers over a
r
differential vicinity of a given location r is called the drift
velocity at this location.
r
1 r
vd ≡ ∑ vi
N i
b) The center of charge enclosed in this volume moves
with the drift velocity.
r
r
drcq d  1 N r 
q
dri r
= 
∑ qr  =
∑ = vd
dt
dt  Nq i =1 i  Nq i dt
c) The current rdensity (associated with one type of
charge carriers) J is defined as a product of the drift
velocity, the concentration of charge carriers and the
charge of the carriers:
r
r
J = nqv d
38
35. (27.2) Current density and current
a) Current through a surface is equal to the flux of
current density over that surface.
vd
proof.
The
charge
dq
transferred
through
a
differential surface dA in
time dt is
dQ = ?
dA
θ
v ddt
n
r
r r
r
dq = nq c ⋅ dV = nq c ⋅ (dA ⋅ vd dt ⋅ cos θ) = nq c ⋅ dt ⋅ v d ⋅ dA = dt ⋅ J ⋅ dA
The charge dQ transferred though the entire surface in time
dt is
r r
dQ = ∫ dq = dt ⋅ ∫ J ⋅ dA
surface
surface
Hence, the current through thersurface
r is
I = ∫ J ⋅ dA
surface
b) For uniform current density over a flat surface, its
component JA, in the direction perpendicular to the surface
has a value equal to the current per unit area of that
surface.
I
JA =
A
proof.
For a uniform current density
I = JA cosθ = J A A
39
36. (27.2) Electric current in a conductor
a) When the nonelectric forces in a conductor are due to
scattering only, a steady flow of charge in the conductor is
achieved. (The scattering forces exerted on the charge
carriers, by the medium, balance the electric forces of the
external electric field.) When a steady flow is achieved,
charge is not accumulated in any part of the conductor. In
a conductor, current density is proportional to the electric
field vector (satisfies Ohm's law):
r
r
J = σE
where the constant of proportionality σ is called the
conductivity of the conductor.
b) Under a steady flow of charged particles along a
conductor, the current across any cross section of the
conductor has the same value. We assign this value to the
current in the conductor.
40