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Unformatted text preview: 9. Lecture 9
9.1 Magnetic forces on an electric current
Since an electric current is charges in motion it follows that a wire through which a
current circulates will experience a force in the presence of a magnetic ﬁeld. In fact an
interesting demo (ﬁg.52) makes this eﬀect evident. More precisely, suppose that, inside
the conductor, an electric charge density (charge per unit volume) ρ is moving with
velocity v . Then, during time ∆t all the charge contained in a volume v ∆tA will pass
through a cross section of area A as can be seen in ﬁg.53. The electric current, namely
the amount of charge going through a section of the conductor of area A per unit time
is therefore
ρv ∆tA
∆Q
I=
=
= ρvA
(9.1)
∆t
∆t
On the other hand the magnetic force on a moving charge is
F  = q  B  sin θ = ρAL B  sin θ = ILB  sin θ
v
v (9.2) where we used that the total charge is given by q = ρLA for a conductor of length L.
Therefore the force is proportional to the current and the length, one can deﬁne a force
per unit length as
F 
= I B  sin θ
(9.3)
L
The direction and orientation of the force is given by exactly the same righthand rule
as before. Only that instead of the velocity we use the direction of the current.
A very interesting observation is that the magnetic ﬁeld allows to determine the
sign of the charge density ρ responsible for the current. This is illustrated in ﬁg.54 and
known as the Hall eﬀect. The same current can be produced by positive carriers moving
in one direction along a wire or by negative ones moving in the opposite direction.
However, the magnetic force on the wire is the same in both cases and for that purpose
we do not need to know the sign of the carriers. On the other hand it means that
the carriers would like to accumulate on one side of the wire and therefore if that
side becomes positively charged the carriers are positive and if it becomes negatively
charged they are negative. The experiment shows that the carriers are negative as we
know since they are electrons.
9.2 Magnetic ﬁeld created by a current
9.2.1 Ampere’s law
We mentioned that an electric current creates a magnetic ﬁeld as can be seen with an
electromagnet, a coil through which a current circulates behaves as a magnet. The – 62 – Figure 52: This interesting demo shows that a current feels a force in the presence of a
magnetic ﬁeld. By switching the direction of the current the direction of the force changes
and the aluminum bar rolls one way and then the other. Figure 53: A current is a charge density ρ moving with velocity through a conductor of
v
cross section A and length L. The charge contained in a volume  ∆t A goes through the
v
cross section A during a time ∆t. This gives I = ρ A.
v intensity of the magnetic ﬁeld is given by Ampere’s law. We mentioned that the ﬂux of
the magnetic ﬁeld through a surface is zero. However given a vector ﬁeld there is another
important quantity known as the circulation. Given a closed path one multiplies the
component of the magnetic ﬁeld in the direction of the path times the displacement
and sums over all portions of the path:
Circulation around a path =
B ∆ cos θ
(9.4)
where ∆ is the displacement, θ the angle between the magnetic ﬁeld and the direction
of the path and the sum is over all portions in which we divide the path for convenience – 63 – Figure 54: The same current is produced by positive charges moving to the right as by
negative charges moving to the left. In a magnetic ﬁeld both feel a force as indicated meaning
that diﬀerent charge accumulates on the sides of the conductor depending on the sign of the
carriers. This is the Hall eﬀect and allows to determine the sign of the particles responsible
for the current. of the calculation (see ﬁg.55). It is exactly the same type of formula used to compute
the work done by a force when you move and object from one place to another. Only
that here we use the magnetic ﬁeld and not the force in the computation (the magnetic
force does no work anyway). What Ampere’s law says is that the circulation around
any closed path is proportional to the current that pierces through any surface whose
boundary is the given path.
B ∆ cos θ = µ0 I
(9.5)
where the constant µ0 is µ0 = 4π 10−7 Tm
A (9.6) 9.2.2 Displacement current
Maxwell realized that Ampere’s law contains an ambiguity and should actually be
amended to be true in all cases. Indeed, the circulation of the current around a closed
path is equal to the current piercing a surface whose boundary is the loop. However
there are many such surfaces. In ﬁg. 56 we see two surfaces ending in the circular path.
One (S1 ) is just a disk whereas S2 is a domeshaped surface. By charge conservation,
if the current going through S1 is not the same as the one going through S2 , charge
has to accumulate in the volume between the two surfaces. For example, if we put – 64 – Figure 55: The circulation of the magnetic ﬁeld is deﬁned in a similar way as the work of a
force along a path. In fact the circulation along a path can be deﬁned for any vector ﬁeld. a capacitor as in the ﬁgure, we can charge the capacitor and no current would go
through S2 whereas, at the same time, current I is going through S1 . This makes
Ampere’s law unclear since which surface should we use?. Here Maxwell realized that
a time dependent electric ﬁeld would be piercing surface S2 since the capacitor is being
charged and an increasing electric ﬁeld exists between the two plates of the capacitor.
So he proposed that the circulation of the electric ﬁeld should be equal to the current
plus the variation of the electric ﬂux through the surface. He also realized that this had
profound consequences, a varying electric ﬁeld creates a magnetic ﬁeld. We are going to
see that a varying magnetic ﬁeld creates an electric ﬁeld and so on. This process gives
rises to a wave that propagates in space. In this way Maxwell predicted the existence of
radio waves and later also suggested that light could be such an electromagnetic wave.
It is important to study this reasoning in detail. It illustrates perfectly the way
that theoretical physics works. Starting from Ampere’s law that had been veriﬁed – 65 – experimentally, Maxwell realized an ambiguity, corrected it therefore predicting the
existence of a new phenomenon, electromagnetic waves, which was later veriﬁed by
Hertz. It is very instructive and a great scientiﬁc achievement. Figure 56: The current through two surfaces, S1 and S2 is diﬀerent but through S2 there is
a time varying electric ﬂux. 9.2.3 Magnetic ﬁeld of a wire and a solenoid (coil)
We can use Ampere’s law to compute the magnetic ﬁeld surrounding a cable. For a
straight cable the magnetic ﬁeld goes around as indicated in ﬁg.57. Using Ampere’s
law for a circular path around the cable we get
circulation = 2π rB = µ0 I
We then have (9.7) µ0 I
(9.8)
2π r
We can create a stronger magnetic ﬁeld by putting several cables such that their magnetic ﬁelds add up. In fact we do not need to put diﬀerent wires, we can take one
B  = – 66 – wire and form a coil as in ﬁg.58. In that case the magnetic ﬁeld is nonzero inside and
outside is very small (if the coil is very long). Using a path as in the ﬁgure we get from
Ampere’s law:
BL = µ0 IN
(9.9)
where N is the number of turns of the coil and L is its length. Therefore, the magnetic
ﬁeld inside the coil is
N
B = µ0 I z
ˆ
(9.10)
L Figure 57: Magnetic ﬁeld produced by straight wire. – 67 – Figure 58: Using Ampere’s law to ﬁnd the magnetic ﬁeld inside a coil. – 68 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue.
 Fall '11
 NA
 Charge, Current, Force

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