Unformatted text preview: 10. Lecture 10
10.1 Force between currents
We saw that a current circulating along a wire creates a magnetic ﬁeld. If another
current is in its vicinity it experiences a force. In ﬁg.59 we see a simple conﬁguration of
two parallel wires carrying currents in the same direction. As discussed in the previous
lecture, the magnetic ﬁeld produced by wire 1 has the direction indicated in the ﬁgure
and its intensity is
µ0
B  =
I1
(10.1)
2π r
According to the righthand rule, the force experienced by the second wire is directed
toward the ﬁrst one (attractive force) and is
µ0 L
 F  = I 2 L B  =
I1 I2
2π r (10.2) where r is the distance between the cables, L their length and I1,2 the respective
currents. Suppose they are separated by 1cm, they are 1m long and I1 = I2 = 10A.
µ0
Using that 2π = 2 × 10−7 mT we ﬁnd
A
F  = 2 × 10−7 mT 1m 2 2
10 A = 2 × 10−3 mT A = 2 × 10−3 N
A 1cm (10.3) A very small but measurable force.
10.2 Magnetic induction
If we move a cable through a magnetic ﬁeld, the charges in the cable will feel a force and
then, if the cable forms a closed loop, a current will circulate. Consider the example of
ﬁg.60. The force on a charge inside the cable is
F  = qvB (10.4)
Since the same force would be produced by an electric ﬁeld of magnitude E  = vB
such force can move charges against a potential diﬀerence ∆V = vBL where L is the
length of the cable. Namely if we have a resistor R in the rest of the circuit a current
I = ∆V will circulate. This is in fact an electric generator!. Not very practical but
R
conceptually simple. In reality, it is better to move the cable in a circular motion so
that it comes back to the original position and can keep moving. Notice that from
the direction of the induced current we see that the moving cable acts as an eﬀective
battery whose positive terminal is in the lower cable. – 69 – Figure 59: Two parallel wires attract if current circulate in the same direction. They repel
is the current circulates in opposite direction. A deeper insight into the problem can be obtained if we compute the magnetic ﬂux
through the loop. Since the magnetic ﬁeld is constant and perpendicular to the loop,
the ﬂux is simply:
Flux = Φ = B Ld
(10.5)
where L is the length of the moving conductor and d is its distance to the resistor. The
distance d changes in time such that
∆d
=v
∆t (10.6) where v is the velocity of the moving wire. Since the loop becomes bigger, namely d is
growing we see that the ﬂux changes as
∆Φ
= B Lv
∆t (10.7) which is nothing else but the emf or voltage between the end points of the moving cable.
Moreover, we see that the current generated in the cable will itself create a magnetic
ﬁeld in the loop which is opposite to the one already present, namely it creates an extra
ﬁeld that opposes the increase in ﬂux. This phenomenon can be exempliﬁed by a demo
(see ﬁg.61 and ﬁg.62) where a loop of wire is moved in the proximity of a magnet and
a current is detected by an Ammeter. The same experiment shows that if we move the – 70 – magnet instead of the loop of wire then the eﬀect is the same. Although this appears
more or less evident it is not clear in the latter case which force is moving the charges
in the loop. Thinking about this problem was one of the reasons Einstein came up with
the theory of relativity that we are going to discuss later in the course. Figure 60: When a conductor moves in a magnetic ﬁeld, the charges inside it feel a force
that induces a current. The resulting emf can be computed from the Lorentz force or from the
Faraday law with identical results. Notice that the induced current also creates a magnetic
ﬁeld, indicated with the green arrows, which opposes the change in ﬂux. For the moment we can summarize these ﬁnding in two important laws, the Faraday
and Lenz law.
Faraday’s law says that if the magnetic ﬂux through a loop changes in time then
an electromotive force is generated proportional to the rate of variation of the ﬂux:
∆ Φ
emf =
(10.8)
∆t
where the Greek letter Φ is used to denote ﬂux. It is completely equivalent as if we put
a battery with ∆V = emf. It is important to know also in which direction the force goes
or equivalently, in such eﬀective battery which is the positive and which the negative
pole. This is given by Lenz law. It says that the current generated in the loop creates – 71 – a magnetic ﬁeld that opposes the change in ﬂux. Namely if the ﬂux increases, the
magnetic ﬁeld created by the current is opposite to the magnetic ﬁeld already present
and vice versa, if the ﬂux decreases, the current tend to reinforce the magnetic ﬁeld.
This phenomenon is known as magnetic induction, a magnetic ﬁeld induces a current.
To exemplify this issues further we can use two interesting demos (see ﬁg.63 and
64). In the ﬁrst the current induced in a ring creates a repulsion that shoots the ring
into the air. In the second the same repulsion is used to stop a pendulum showing how
magnetic braking works. Figure 61: Demo. A current is induced in a loop if we move it in the presence of a magnetic
ﬁeld. See also ﬁg. 62. – 72 – Figure 62: A current is induced in the loop if we either move the magnet or the loop. In
both cases the magnetic ﬂux through the loop changes. – 73 – Figure 63: Demo. Starting up the current through the coil induces a current on a ring
producing a repulsive force and shooting the ring in the air. Cooling the ring decreases the
resistance therefore increasing the induced current and the height reached by the ring. – 74 – Figure 64: Demo. The current induced in the pendulum when it goes through the poles of
the magnet creates a repulsive force that brakes the pendulum halting it. The same principle
can be used in electric car brakes. In that case, the induced current can charge the battery
and the energy reused to propel the car. – 75 – ...
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 Fall '11
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 Current, Force, Flux, Magnetic Field, loop

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