Unformatted text preview: 8. Lecture 8
8.1 Capacitors in series and parallel
In the same way that one can analyze resistors in series and parallel one can understand
what happens for capacitors in series and parallel. Notice that whereas Ohm’s law is
∆V = IR we have Q = C ∆V . So the potential diﬀerence or voltage ∆V is proportional
to R but inversely proportional to C . As a consequence, and working with the circuits
in ﬁg.45 one ﬁnds that
Series: 1
C = 1
C1 + 1
C2 + 1
C3 Parallel: C = C1 + C2 + C3
Try to derive these relations as an exercise. As a hint notice that when the capacitors
are in series, the conductors in the middle are neutral so the charge in each capacitor
is the same (see ﬁg.45).
+Q −Q +Q −Q +Q C1 C2
+Q −Q C3
−Q 1 1 C1
+Q 2 −Q 2 C2
+Q 3 −Q 3 C3
Figure 45: Capacitors in series and parallel. In parallel they add and in series their inverse
add. Notice the diﬀerence with the case of resistors. – 55 – 8.2 Magnetic ﬁeld
Magnetism is a wellknown phenomenon, for example the magnetic ﬁeld of the Earth has
provided orientation through the use of the compass for thousands of years. Magnetic
ﬁelds can be generated by permanent magnets, electric currents and time dependent
electric ﬁelds. In the case of the magnet the magnetic ﬁeld looks like the one in ﬁg.46
(see also the demo in ﬁg. 48). Outside the magnet it is very similar to the one of
a dipole electric ﬁeld. But inside it is not. The lines of magnetic ﬁeld close, they
do not start or terminate at any point. Without knowing that, one might, at ﬁrst
sight, think that cutting in half a magnet will separate two opposite magnetic charges.
Traditionally they are called the North and South pole. A pure North pole would
have a magnetic ﬁeld as the one in ﬁg.47 However such object has never been observed
in Nature. In particle physics there are theoretical indications that heavy particles
with the properties of magnetic monopoles might exists. For that reason it is still an
open question if monopoles (as they are called) exist or not. In any case if they do
exist they are not common objects and we are not going to consider them here. The
question naturally arises of what happens if we keep dividing the magnet, do we ﬁnd
an ”elementary magnet”?. The answer is yes, in fact the elementary magnet is our
old friend the electron. Electrons not only have charge but they also behave as a tiny
magnet. The superposition of the magnetic ﬁelds of all electrons creates the ﬁeld of the
magnet. Proton and neutrons are also tiny magnets but their strength is 2000 times
smaller than the one of the electrons.
An important consequence of the fact that there are no magnetic monopoles is that
the ﬂux of the magnetic ﬁeld through a closed surface is always zero. This is because a
theorem similar to Gauss’ theorem applies. The total ﬂux is proportional to the total
magnetic charge enclosed but since there is no magnetic charge, the ﬂux is always zero,
namely for each closed surface the same amount of magnetic ﬂux comes in as it goes
out.
The magnetic ﬁeld is a vector usually denoted as B . It can be detected by the
eﬀect it causes on charges. When a charge moves in a magnetic ﬁeld it experiences a
force equal to
F = q E + q × B
v
(8.1)
where is the velocity, q the charge and B the magnetic ﬁeld. We included also an
v
electric ﬁeld E for completeness. We need to explain what the operation × B is. It
v
is known as a vector product and given two vectors it gives another vector:
× B =  B  sin θn
v
v
ˆ – 56 – (8.2) Figure 46: Magnetic ﬁeld of a permanent magnet. Outside is similar to an electric dipole
but inside there are no charges where the lines end. They lines of magnetic ﬁeld are actually
closed. Its magnitude is the product of the modulus of , B and the sine of the angle between
v
them that we denote as θ. Its orientation we denote by the unit vector n and is
ˆ
. We write that as
perpendicular to both and B
v
n ⊥ ,
ˆv
n⊥B
ˆ (8.3) Its orientation is given by the righthand rule. If we move our ﬁngers in a screw motion
going from to B then our thumb points in the direction of the force. This is illustrated
v
in ﬁg.50. Some important consequences are
• If the particle is at rest then = 0 and there is no magnetic force.
v
• If the velocity is parallel to B then θ = 0 and also there is no force.
• The magnetic force is always perpendicular to the velocity so it never does work!. – 57 – ? N Figure 47: A conjectured magnetic monopole will have all lines of magnetic ﬁeld coming
out. It has not yet been observed in Nature but theoretical ideas suggest it might be an
actual but exotic particle. Figure 48: The lines of magnetic ﬁeld can be made evident using iron ﬁlings which orient
themselves parallel to the magnetic ﬁeld. Also form the form of the Lorentz force we see that the magnetic ﬁeld is measured
Ns
in units of Cm , such unit is known as a Tesla:
1T = 1 Ns
Kg
=1
Cm
Cs (8.4) where we used 1N = 1 Kgm . To have an idea the magnetic ﬁeld of the Earth is around
s2
10−5 T but in the lab ﬁelds of 10T can be produced.
Since the magnetic force is perpendicular to the velocity it will change its direction
but not its magnitude. Therefore, if there are no other forces present, the kinetic energy – 58 – B
v q F
Figure 49: A particle moving in a magnetic ﬁeld B experiences a force perpendicular to the
. Its orientation is given by the righthand rule.
velocity and to B E = 1 mv 2 will remain constant, a consequence of the fact that the magnetic force does
2
no work. Furthermore, if the magnetic ﬁeld is constant the magnitude of the force will
be constant and the trajectory of the particle will be a circle in a plane perpendicular
to the magnetic ﬁeld. Indeed, in ﬁg.49 we see that if the particle moves in circles then
the force points always toward the center keeping the particle in its trajectory in the
same way that we can tie an object at the end of a rope and make it move in circles.
Using Newton’s law
F = m
a
(8.5)
we can compute the the period of motion in a similar manner that we discussed at
the beginning of the course for the Moon orbiting the Earth. the acceleration is in the
radial direction, namely it is centripetal acceleration whose magnitude is
  =
a v2
r (8.6) as we remember from the mechanics course. Since the force point in the same direction
we only need to equate their magnitudes:
F  = qvB = m
giving
v= qBr
m – 59 – v2
r (8.7) (8.8) F B v Figure 50: Example of the righthand rule. The force is the vector product of the velocity
and the magnetic ﬁeld and has the orientation indicated here. The period of motion is how long does it take for a particle to go around the circle. It
is given by
2π r
m
2π m
∆t =
= 2π r
=
(8.9)
v
qBr
qB
amazingly it is independent of the radius. Namely if the radius is large the particle
moves faster and takes the same time to go around. Notice that a peculiar property of
this motion is that the particle can move in circles only in one direction. If one tries
to move the particle around the circle in opposite direction then the force will point
outward, it will not stay in the trajectory. Actually will again describe a circle in the
“correct” direction. On the other hand the motion in the other direction is precisely
what happens for a particle of opposite charge.
Although rather simple this idea has had numerous applications. The most signiﬁcant one perhaps is in particle accelerators where magnetic ﬁelds keep particles in
circular trajectories while at the same time electric ﬁelds accelerate them at higher
and higher energies. When these particles collide with ﬁxed targets (or two opposite
beams are brought into collision) new particles are created revealing the fundamental
constituents of matter and their interactions. – 60 – F
v
B
Figure 51: A free particle moving in a constant magnetic ﬁeld B moves in circles since the
magnetic force is oriented toward the center. – 61 – ...
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 Fall '11
 NA
 Magnetic Field

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