Download Ch 32 Maxwell`s Equations Magnetism of Matter

Ch 32
Maxwell’s
Equations
Magnetism
of Matter
蟹狀星雲
First: conclude our basic discussion of E and B
fields, to summarize in only four equations, known as
Maxwell’s equations.
Second: examine the science and engineering of
magnetic materials. The careers of many scientists
and engineers are focused on understanding (1) why
some materials are magnetic and others are not, and
(2) why Earth has a magnetic field but you do not.
→ Countless applications for inexpensive magnetic
materials in cars, kitchens, offices, etc.
Iron powder onto a transparent sheet placed above
a bar magnet: the powder grains align themselves with
the magnet’s B-field to form a field pattern.
The magnet with its two poles is an example of a
magnetic dipole.
Magnet = magnetic dipole
S
N
S
N
S
N
Suppose we break a bar magnet
into pieces. We should be able to
isolate a single magnetic pole, called
a magnetic monopole (磁單極=磁
荷).
However, we cannot — not even if
we break the magnet down to its
individual atoms and then to its
electrons and nuclei. Each fragment
has a north pole and a south pole.
Gauss’ law for B-field: saying that magnetic
monopoles do not exist. The law asserts that the net
magnetic flux ΦB through any closed Gaussian surface
is 0
磁荷不存在
Contrast this with Gauss’ law for E-field,
電荷存在
In both equations, the integral is taken over a closed
Gaussian surface (封閉の高斯曲面):
(1) Gauss’ law for E-field: net electric flux ΦE through the
surface ∝ net electric charge q enclosed by the surface.
(2) Gauss’ law for B-field: no net magnetic flux ΦB
through the surface because there can be no net
“magnetic charge” (individual magnetic poles) enclosed by
the surface.
The simplest magnetic structure that can exist and be
enclosed by a Gaussian surface is a magnetic dipole,
which consists of both a source (N) and a sink (S) for the
field lines.
There must always be as much magnetic flux into the
surface as out of it (Φin=Φout), and the net magnetic flux
(ΣΦ=0) must always be zero.
Gaussian surfaces I and II:
Φin=Φout
S
N B Enters, so S pole exists
A changing magnetic flux (dΦB/dt) induces an
electric field (E), satisfy Faraday’s law of induction
C
dΦB/dt ≠ 0
Induced E
a closed loop C
Because “symmetry” is often so powerful in physics, we
should ask whether induction can occur in the opposite sense.
Can a changing electric flux dΦE/dt induces B?
The answer is that it can !
The equation governing the induction of B is almost
symmetric with Faraday’s law of induction. We often call
it Maxwell’s law of induction (麥克斯威爾感應定律).
C
B is induced along a closed loop C by the changing
electric flux dΦE/dt in the region encircled by that loop
C. (時變電場(電通量)產生感應磁場)
Example:
The charging of a parallel plate
capacitor with circular plates:
We assume that qC is charged
on C at steady rate by a
constant current i = dq/dt =
constant. Then E between the
plates must also increase at a
steady rate (that is, dE/dt =
const.).
+qC
-qC
Because E-field through the loop (經過point 1之loop C1)
is changing, the electric flux through the loop must also
be changing. According to Eq. 32-3, this changing
electric flux (dΦE/dt) induces a B-field around the loop.
central
axis
C1
central
axis
Experiment proves that a B-field is indeed induced around
such a loop C1. This B-field has the same magnitude at
every point around the loop (半徑r的圓上) and thus
has circular symmetry about the central axis of the
capacitor plates.
C1
central axis
If we consider a larger loop C2 (through point 2
outside the plates) we find that a B-field is induced
around that loop as well.
Thus, while E-field is changing (dΦE/dt ≠ 0), B are
induced between the plates, both inside (r<R) and
outside (r>R) the gap. When E-field stops changing
(dΦE/dt = 0), these induced B disappear.
C2
Equation 32-3 lacks the “ ̶ ” sign of Eq. 32-2, meaning
that the induced E-field and the induced B-field have
opposite directions when they are produced in otherwise
similar situations. An increasing B, directed into the page,
induces E. The induced E is counterclockwise (左圖) ,
opposite the induced B (右圖).
dΦB/dt
Induced E 與 induced B 方向相反
dΦE/dt
The integral of the dot product in Eq. 32-3 around a
closed loop also appears in Ampere’s law:
where ienc is the current encircled by the closed loop.
Thus, our two equations that specify the B-field
produced by i and by a changing electric flux dΦE/dt give
the B-field in exactly the same form. We can combine
Eqs. (32-3) and (32-4) into the single equation
x Ampere-Maxwell law
1 When there is a current but no change in electric
flux (dΦE/dt=0), the first term on the right side of Eq.
32-5=0, and so Eq. 32-5 reduces to Eq. 32-4,
Ampere’s law.
2 When there is a change in electric flux but no
current (ienc=0), the second term on the right side of
Eq. 32-5=0, and so Eq. 32-5 reduces to Eq. 32-3,
Maxwell’s law of induction.
P. 19
(a) (r ≤ R) By Ampere –
Maxwell law
ienc=0 between the capacitor plates,
左邊 =
=B
B//ds (在r圓上)
右邊 =
=
ds = B(2πr)
B均勻 (在r圓上)
=
= μ0ε0πr2dE/dt
E//dA 且均勻 (在r圓內)
左邊=右邊
B(2πr)= μ0ε0πr2dE/dt
(r≤R)
(b) 將數值代入
(c) (r ≥ R) By Ampere–Maxwell law (ienc=0)
左邊 =
右邊 =
左邊=右邊
=B
=
=
B(2πr)= μ0ε0πR2dE/dt
ds = B(2πr)
= μ0ε0πR2dE/dt
(r≥R)
(d) Plot B(r)
(r≤R)
(r≥R)
B(r≤R)r=R = B(r≥R)r=R= (μ0ε0R/2)dE/dt
B(r)
(μ0ε0R/2)dE/dt
R
r
We can see that the product ε0(dΦE/dt) must have
the dimension of a current. In fact, that term has been
treated as being a fictitious current (假想電流) called
the displacement current id (位移電流)
“Displacement”
Displacement is poorly chosen
in that nothing is being
displaced (imaging).
id (fictious)
i
i
Nevertheless, we can now rewrite Eq. 32-5 as
in which id,enc is the displacement current that is
encircled by the integration loop.
The real current i that is charging the plates
changes the electric field E between the plates.
The fictitious displacement current id (=ε0dΦE/dt)
between the plates is associated with that changing
field E (changing flux ΦE).
E
Fig. 32-7a
The charge q on the plates at any time is related to
the magnitude E of the field between the plates at that
time by Eq. 25-4 (由電學的高斯定律可知: ∫E•dA = q/ε0)
Assuming that E between the two plates is uniform
(neglect any fringing), we can replace the electric flux
ΦE in that equation with EA. Then
Comparing Eqs. 32-13 and 32-14, we see that the real
current i charging the capacitor and the fictitious
displacement current id between the plates have the
same magnitude:
Analysis
1 We can consider the fictitious
displacement current id to be
simply a continuation (延續) of the
real current i from left plate, across the
capacitor gap, to right plate.
2 Because the E-field is uniformly spread over the
plates, the same is true of this fictitious displacement
current id, as suggested by the spread of current arrows.
¨ id 可假想成均勻通過兩板間截面積A，即電流密度為
id/A。
3 Although no charge actually moves across the gap
between the plates, the idea of the fictitious current id
can help us to quickly find the direction and magnitude
of an induced B-field (next page).
We saw the direction of the B-field produced by a real
current i by using the right-hand rule. We can apply the
same rule to find the direction of an induced B-field
produced by a fictitious displacement current id, as is
shown in the center for a capacitor (下圖).
已知 id → find the magnitude and direction of the Bfield induced by charging capacitor with parallel
circular plates of radius R (by Ampere-Maxwell law).
id
id
By Ampere-Maxwell law
(a) B-field inside capacitor at radius r from center (r<R)
(b) B-field outside capacitor at radius r (r>R)
方向由右手
定則決定
Ampere-Maxwell law is the last of the four
fundamental equations of electromagnetism, called
Maxwell’s equations and displayed in Table 32-1.
They are the basis for the functioning of
electromagnetic devices as electric motors, television
transmitters and receivers (發射與接收器), telephones,
fax machines, radar, and microwave ovens.
Maxwell’s equations are the basis from which many
of the equations can be derived. They are also the
basis of many of the equations you will see in Ch33-36
concerning “Optics”.
(1)
(2)
(3)
(4)
(1) Guass’ law for electricity: relates net electric flux
to net enclosed electric charge
¨ 通過某一封閉曲面之電通量 ∝ 此封閉曲面所包含的淨電荷)
(2) Gauss’ law for magnetism: relates net magnetic
flux to net enclosed magnetic charge
¨ 通過某一封閉曲面之磁通量 = 0 ¨ 磁荷(磁單極)不存在
(3) Faraday’s law: relates induced electric field to
changing magnetic flux
¨ 感應電場於一封閉迴路上的環路積分 ∝ 通過此封閉迴路之
磁通量隨時間變化率 (磁通量隨時間改變會產生感應電場)
(4) Ampere–Maxwell law: relates induced magnetic
field to changing electric flux and to current
¨感應磁場+一般磁場於一封閉迴路上的環路積分 ∝ 通過
此封閉迴路之電通量隨時間變化率+此封閉迴路包含之淨電
流 (前者: 電通量隨時間改變會產生感應磁場)
Homework
Ans: (a)
(b)
Ans: (a)
(b)
(c)
Ans: (a)
(b)
(c)
(d)
Ans: (a)
(b)
(c)
(d)
Ans: (a) 27.9 nT. (b) 15.1 nT.
Ans: (a) -8.8×1015 (b) 5.9×10-7 Wb/m.