Download Historical burdens on physics 96 Permeability

Historical burdens on physics
96 Permeability
Subject:
For the magnetic flux density B in a long electromagnet (length l, number of
turns N, relative permeability of the core material μr) the following formula
can be found in text books:
B = µ0 · µr ·
N ·I
!
l
(1)
It is supposed that the electric current intensity is not too high, because
otherwise the core material might approach saturation.
In some text books an equation is found that is equivalent to (1):
B = μr · B0 .!
(2)
Here, B is the flux density within the core material and B0 is that inside the
empty coil, i.e. the coil from which the core has been removed.
Deficiencies:
Equation (1) and (2) are not correct. They are valid only if the whole space
which is occupied by the field is filled with the material of permeability μr.
“In those cases where a homogeneous and isotropic magnetizable substance occupies the whole space of the magnetic field, or a part of it in such
a way that the lines of induction of the magnetizing field do not traverse the
surface of the magnetized material, inside of the material the relation
B = μr · B0
holds, where μr is the relative magnetic permeability of the magnetizable
substance, which…”. [1]
Thus, equations (1) and (2) are valid for instance for a toroidal coil with a
closed core.
In order to understand why the equations are not valid for a normal straight
electromagnet, let us begin by deriving the correct expressions that correspond to equations (1) and (2) for the case of a coil with a toroidal core that
is provided with a slit, Fig. 1. Thereafter we consider the case of a stretched
coil.
Fig. 1. The flux density inside the iron core and in the slit is proportional to μr only if the slit
width d is sufficiently small.
We suppose the slit width to be so small that the field within the slit can be
considered homogenous.
Since we are far from saturation and since the material is isotropic, we have
everywhere
B = µ0 · µr ·H . !
(3)
Now admit that the coil has N turns and the electric current is I. We then
have:
 
∫ Hdr = N ·I
where the integration is over a path that follows the torus.
We now suppose that the radius of our ring (the great radius of the torus) is
great compared with the radius of a cross section of the ring (the small radius of the torus). We now can easily evaluate the Integral:
b ·H m + d ·H s = N ·I !
(4)
The index m refers to the material of the core, s refers to the slit. b is that
part of the integration path that runs inside the core material, whereas d refers to the path section inside the slit.
Since the B field is divergence-free we have
Bm = Bs = B ,
i.e. B is the same in the material and in the slit.
We then get by using equation (2):
µrH m = H s .
Inserting in equation (4) we get
Hm =
N ·I
b + µrd
and by using equation (3)
B=
µ0 ·µr · N ·I
.!
b + µrd
!
!
!
!
!
!
(5)
We see that the dependence of B on μr is not that of equation (1).
For an empty coil we get
B 0 = µ0 ·
N ·I
,
l
where l is the total integration path length. With this equation (5) becomes
B=
µr ·l
B0 .
b + µrd
It is seen that the relation between B and B0 is not that claimed by equation
(2).
Let us now evaluate equation (5) for two special situations:
1. if the solenoid does not have any slit, i.e. if d = 0, or also approximately if
b  µrd we get
B = µ0 · µr ·
N ·I
.! !
b
!
!
!
!
!
(6)
In this approximation the flux density is independent of the (small) slit width
and proportional to the relative permeability of the core material. Since the
path b inside the material is (almost) equal to the total path l, the equation
turns out to be identical with equation (1). Thus, we see that equation (1) is
valid only when the core does not have a slit or if the slit is small compared
with b/μr .
2. If b  µrd , equation (5) becomes approximately
B = µ0 ·
N ·I
.!
d
!
!
!
!
!
!
(7)
Now B is independent of μr, but inversely proportional to the slit width.
By comparing with the empty coil we get
B=
l
B0 . !
d
!
!
!
!
!
!
!
(8)
Also in this case B is independent of μr.
Now, which of these two approximations correspond to the situation
that we meet at school?
Suppose we have a ring magnet of a total length of 50 cm, that would
result if constructed from typical school material. Further suppose
that we provide the magnet with a slit not smaller than 0.2 cm (otherwise it would not be possible to introduce the Hall probe). If we admit
that μr = 1000 we find that b in the denominator at the right side of
equation (5) is only one forth of μr d. We are thus in the scope of validity of equations (7) and (8), and not that of equation (6).
Actually at school the measurement of μr is not made with a ring
magnet but with a straight solenoid. In this case the distance between the two poles of the iron core is even much greater. So one is
definitely in the range of validity of equation (7). Indeed, these experiments give a value of μr that is too small by more than a factor of
10.
Equations (1) and (2) suggest that the magnetic flux density inside
the core of an electromagnet increases in proportion to the specific
permeability of the material. This would mean that the flux density
increases by the same factor outside at the surface of the core. Then
an electromagnet with a core with μr = 100 000 would have a flux
density that is a hundred times that of a core with μr = 1000. However, equation (7), and also common sense tells us that this cannot
be true. An electromagnet with μr = 500 can hardly be improved by
choosing another core material. (This does not mean that in some
situations a material with a very great μr is not indispensable.)
Origin:
We have found the incorrect equations in all of the five high school text
books that we have consulted, but not in any university text book or encyclopedia. This gives a hint on how the error originated. School physics has
to get along with as few physical quantities as possible. So one tries to introduce the specific permeability without using the magnetic field strength
H. The wrong conclusion might have been, that this can be done in a way
that is analogous to introducing the dielectric constant εr . This is done by
inserting a dielectric into the space between the plates of a capacitor
and measuring the decrease of the potential difference between the
plates. The fields strengths with and without the dielectric material
are related by
E 0 = εr ·E .
This relation is applicable, in contrast to its magnetic counterpart, since in
the case of the capacitor the whole space that is occupied by the field is
filled with the dielectric. And in addition for common dielectric materials εr is
much smaller than μr for typical iron core materials.
Disposal:
The description of magnetostatic phenomena gets clearer when employing
H instead of B. Doing so we can formulate a simple rule:
A softmagnetic material displaces the magnetic field (measured by H) from
its inside in the same way as an electric conductor displaces the electric
field.
For most applications it doesn’t make any difference whether the field is,
according to the value of μr, displaced to 99,9 % or 99,99999 %.
[1] Jaworski, B. M. and Detlaf, A. A.: Physik griffbereit, Vieweg, Brauschweig 1972,
p. 410
Friedrich Herrmann, Karlsruhe Institute of Technology