140
6 Longitudinal Magnetic Field Effect
helical form. The potential difference is given by a curvilinear integral of the electric
field and is independent of the integral path C. Hence, the same potential difference
results either from the integral path C along the flux line or from the straight integral
path C
. Here we note that (B × v) · ds = (ds × B) · v. Since ds is parallel to B on
the integral path C, the curvilinear integral along it is zero. As a result, we have
C
(B × v) · ds = 0.
(6.55)
Hence, the term B × v does not contribute to the potential difference between two
points separated by a sufficiently longer distance than the helical pitch. The loss
component is contained again in the second term in (6.1), similarly to the case of
the non-steady state. Hence, the second term −∇φ can be estimated from the work
done by the force-free torque in a unit time:
P =
dθ 0
dt
.
(6.56)
Although the details of the analysis are omitted, the electric potential at z
(longitudinal position) and ϕ (azimuthal angle) on the surface is given by [19]
V = V (z, ϕ) − V (0, 0)
= Bv 2
z
2
sinθ R − Rcosθ R sin
ϕ −
z
R
tanθ R
,
(6.57)
where θ R is the angle of the magnetic field on the surface measured from the z-axis.
The second term is the contribution from B × v. The observed electric potential at
three positions along the length of a Pb–Tl cylindrical superconductor [5] and the
corresponding prediction of (6.57) are shown in Fig. 6.20a and b, respectively. The
qualitative agreement is good, and it supports the helical flux flow. In particular,
it is found that the electric field is negative around the area where the variation
in the electric potential along the length of the sample is reversed. In this area the
velocity of flux lines is directed outward from the superconductor, and the term B × v
gives a negative longitudinal component for helical flux lines (see Fig. 6.21). It was
shown in another experiment that Poynting’s vector is directed outward from the
superconductor where the electric field is negative, directly indicating that flux lines
flow out of the superconductor there [20]. Thus, the negative electric field comes
from the unimportant term B × v, and it does not mean a nucleation of energy. The
current flows helically along flux lines and it should be noted that i · E > 0.
Another characteristic point of the helical flux flow is the appearance of a radial
electric field from the second term of (6.57). On the medium line of the theoretical
prediction in Fig. 6.20b, it can be found that the maximum transverse electric field,
6 Longitudinal Magnetic Field Effect
helical form. The potential difference is given by a curvilinear integral of the electric
field and is independent of the integral path C. Hence, the same potential difference
results either from the integral path C along the flux line or from the straight integral
path C
. Here we note that (B × v) · ds = (ds × B) · v. Since ds is parallel to B on
the integral path C, the curvilinear integral along it is zero. As a result, we have
C
(B × v) · ds = 0.
(6.55)
Hence, the term B × v does not contribute to the potential difference between two
points separated by a sufficiently longer distance than the helical pitch. The loss
component is contained again in the second term in (6.1), similarly to the case of
the non-steady state. Hence, the second term −∇φ can be estimated from the work
done by the force-free torque in a unit time:
P =
dθ 0
dt
.
(6.56)
Although the details of the analysis are omitted, the electric potential at z
(longitudinal position) and ϕ (azimuthal angle) on the surface is given by [19]
V = V (z, ϕ) − V (0, 0)
= Bv 2
z
2
sinθ R − Rcosθ R sin
ϕ −
z
R
tanθ R
,
(6.57)
where θ R is the angle of the magnetic field on the surface measured from the z-axis.
The second term is the contribution from B × v. The observed electric potential at
three positions along the length of a Pb–Tl cylindrical superconductor [5] and the
corresponding prediction of (6.57) are shown in Fig. 6.20a and b, respectively. The
qualitative agreement is good, and it supports the helical flux flow. In particular,
it is found that the electric field is negative around the area where the variation
in the electric potential along the length of the sample is reversed. In this area the
velocity of flux lines is directed outward from the superconductor, and the term B × v
gives a negative longitudinal component for helical flux lines (see Fig. 6.21). It was
shown in another experiment that Poynting’s vector is directed outward from the
superconductor where the electric field is negative, directly indicating that flux lines
flow out of the superconductor there [20]. Thus, the negative electric field comes
from the unimportant term B × v, and it does not mean a nucleation of energy. The
current flows helically along flux lines and it should be noted that i · E > 0.
Another characteristic point of the helical flux flow is the appearance of a radial
electric field from the second term of (6.57). On the medium line of the theoretical
prediction in Fig. 6.20b, it can be found that the maximum transverse electric field,