Download Ohm`s Law - Physics of Magnetism and Photonics Research Division

FI 2201 Electromagnetism
Alexander A. Iskandar, Ph.D.
Physics of Magnetism and Photonics Research Group
Electrodynamics
ELECTROMOTIVE FORCE AND
FARADAY’S LAW
1
Ohm’s Law
• To make a current flow, we have to push the charges. How
fast they move depends on the nature of the material.
r
• For most substances, the current
density
is proportional
J
r
to the force per unit charge, f :
r
r
J =σ f
the proportionality “constant” σ is a second rank tensor, as
are the susceptibilities, but many common media are
“linear” in the sense that the conductivity σ can be
considered a scalar
scalar.
• A perfect conductor has σ → ∞, and by contrast, a resistor
has small conductivity.
• The reciprocal of conductivity is called resistivity, ρ = 1 σ ,
which is a characteristics of the material.
Alexander A. Iskandar
Electromagnetism
3
Ohm’s Law
• In principle, the force could be anything (chemical etc),
but we concentrate on the electromagnetic force
(
r
r r r
F = q E+v×B
)
→
r r r r
f = E+v×B
• In real substance, the velocity is very small, thus the
second term above is neglected. Hence, we have
r
r
J =σ E
• This relation is called Ohm’s Law.
• There is no contradiction with the fact that inside a
conductor
d t th
the electric
l t i fi
field
ld iis zero, since
i
iin an electrical
l ti l
circuit, the wires are made of good conductors, thus
r
r J
E = →0
σ
• Example 7.2, Problem 7.3
Alexander A. Iskandar
Electromagnetism
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2
Ohm’s Law
• Ohm’s law should strike you as strange, at first, because
you also know that
r
r
r
J =ρv
r
r
• Suppose
r J is constant. Then v and E should be constant.
r
r
But if E is constant, charges should accelerate at a = qE m ,
r
rendering it impossible to have a constant v . So which
relation is wrong?
• What happens is that collisions between free charges in a
current (like electrons) with fixed or slower moving charges
(like the ions the electrons leave behind), and other free
current carriers, keep the acceleration from going on for
very long.
Alexander A. Iskandar
Electromagnetism
5
Ohm’s Law
• In collisions with the ions, the kinetic energy gained by the
free carriers from the field is largely transferred to the ions,
and the electron starts over.
• We therefore reconcile Ohm’s Law with the definition of
current density by supposing that the collision
process
r
results in a well-defined average velocity, v , also called
the drift velocity which is a constant, and write
r
r
J =ρv
• In fact
fact, we don’t
don t need to suppose; we can show that that’s
that s
the way it is, in a crude classical model of what is mostly a
quantum phenomenon.
Alexander A. Iskandar
Electromagnetism
6
3
Electromotive Force (emf)
• Note that in a typical electric circuit (for example, a light
bulb) connected to a battery, the current is the same all
the way around the loop.
• Why is it constant around the loop ?
r
• Recall that the only driving force on the charges, f s , is
confined on the source (battery).
• Suppose that there is an accumulation of charges on
some part of the wire, such that the current is not
constant This accumulation of charges will create an
constant.
electrostatic field that will smooth out the flow of charges.
• Thus, we can write
r r r
f = f s + Eelectrostatic
Alexander A. Iskandar
Electromagnetism
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Electromotive Force (emf)
• To calculate the work done by this force in taking a charge
around the loop, we line integrate around the closed loop
to yield
r r
r r r
r r
∫ f ⋅ dl = ∫ ( f
C
C
s
)
+ E ⋅ dl = ∫ fs ⋅ dl ≡ E
C
• This non-zero result is called the Electromotive force (emf)
of the circuit.
Alexander A. Iskandar
Electromagnetism
8
4
Motional emf
• Moving a conducting wire in a magnetic field can also
resulted in motion of charges in the wire. This is called
motional emf.
• Consider the experiment of moving a loop in a magnetic
field. In this case, the force that push the charges in
r
motion is the magnetic force
v
⊗
r
B
r r r
fs = v × B
h
R
x
Alexander A. Iskandar
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Electromagnetism
Motional emf
• The motional emf is calculated as before
r r
dx
da
dΦ B
E = ∫ f s ⋅ d l = vBh = − Bh = − B
=−
dt
dt
dt
C
the minus sign accounts for the fact the rate of change of
the area a is negative and the magnetic flux is dΦ B = B da .
• It turns out that this last relation is valid much more
generally – independent
of the shape of the loop,
r
homogeneity of B.
r
• Suppose
S
i id the
inside
th wire
i th
the d
drift
ift velocity
l it iis u , th
then th
the
total magnetic force
on
a
charge
can
be
deduced
to
be
r
r
f mag
v
r
u
vB
r
r r
f mag = w × B
r
w
uB
Alexander A. Iskandar
Electromagnetism
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5
Motional emf
• Consider a loop of wire moving, and perhaps even
r
changing shape, through a region with a static B , and
follow the point A.
• In time dt rit moves a distance vdt, and with the line
element d l sweeps out an area
r
r r
da = v dt × d l
A
r
dl
at time t
θ
r
v dt
Alexander A. Iskandar
at time (t + dt)
da
11
Electromagnetism
Motional emf
• The change in magnetic flux through the loop, that’s
admitted by the border ribbon is
∫ B ⋅ da = dt ∫ B ⋅ (v × d l )
r
dΦ B = Φ B (t + dt ) − Φ B (t ) =
r r
r
r
C
ribbon
• Now suppose a current runs in the loop. If the drift velocity
r
of the carriers (relative to the loop) is u , and their total
r
r
r r r
velocity w = v + u , then since u must be parallel to d l then
r r r r
v × dl = w× dl
A
r
dl
θ
r
v dt
Alexander A. Iskandar
at time t
da
Electromagnetism
at time (t + dt)
12
6
Motional emf
• Hence,
(
)
(
)
(
)
r r r
r r r
r r r
r r
B ⋅ w × d l = d l ⋅ B × w = −d l ⋅ w × B = −d l ⋅ f mag
• Thus,
Thus
(
or
)
(
E =−
dΦ B
dt
)
r
r r r
r
r r r
dΦ B
= − ∫ B ⋅ w × d l = − ∫ w × B ⋅ d l = − ∫ f mag ⋅ d l
dt
C
C
C
Alexander A. Iskandar
Electromagnetism
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Electromagnetic Induction
What if the field in the region varies, with the loop stationary?
• Relativity: as long as the relative motion is the same, the
same emf must be obtained as before. (We see this
experimentally too.)
• In this case, though, it’s no longer clear what exerts the
r
force that moves the charges, since v = 0 .
• Faraday gave an ingenious explanation to this. He
postulate an induced, non-electrostatic, electric field can be
obtained from changing of magnetic field :
r r
dΦ B
E = ∫ E ⋅ dl = −
dt
C
• With Stokes theorem
r
r r
r r
r
d r r
dB
∫ E ⋅ d l = S∫ ∇ × E ⋅ da = − dt S∫ B ⋅ da → ∇ × E = − dt
C
(
Alexander A. Iskandar
)
Electromagnetism
14
7
Electromagnetic Induction
r
r
dB
∇× E = −
dt
Faraday’s Law of Induction
• This means that a non-static electric field can be induced
by a nonstatic magnetic field.
• That is, a current canr be induced in a loop of conductor by
changing the flux of B through it, no matterr how the flux
changes: motion of the loop, or change in B .
• The minus sign in Faraday’s law indicates that a changing
magnetic flux will induce an electric field and current such
that the magnetic field induces by the current leads to a flux
change in the opposite direction. This is called Lenz’s Law.
Alexander A. Iskandar
15
Electromagnetism
Induced Electric Field
r
E with Faraday’s Law
• Calculations of induced electric field
r
proceed just like calculations of B from steady currents
using Ampère’s Law.
• Note the following relations:
r
r
dB
∇× E = −
dt
r
r
∇ × B = µ0 J
or in integral form
r r
dΦ
E
∫C ⋅ d l = − dt B
r r
∫ B ⋅ d l = µ0 I enc
C
• Also
r
∇⋅B = 0
r
∇ ⋅ E = 0 ← ρ = 0, only currents changes
• Apparently they’re the same, rwith the interchange of
r
r
E⇔B
Alexander A. Iskandar
r
µ0 J ⇔ −
dB
dt
Electromagnetism
µ0 I enc ⇔ −
dΦ B
dt
16
8
Induced Electric Field
• We can use this similarity to calculate induced electric
fields using the machinery of magnetostatics (namely
Ampere’s Law).
• Example 7.7, 7.8
Alexander A. Iskandar
Electromagnetism
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