So why should we get magnetic force in absence of
electric force? Note the following points,

1. The concept of increase in charge density is wrong. There is a
fixed relationship between the number of conduction electrons and
atoms.
2. Although conduction elctrons move with a drift velocity, the frames
S' and S overlap. Consider a ring of superconducting material in which
high current is set up by momentory change in extraneous magnetic
field. If the volume of S' contracts then two rings will be formed.
This is physically not possible, particularly because of the fixed
relationship between +ve and -ve charges.

3. These are all pathetic efforts to assign physical meanings to
mathematical symbols. If we treat electric field as real and since the
field is representative of charge, the physical properties of a charge
can be mathematically assigned to field. But there cannot be
duplication.Force is either between the charges or it is the
interaction of fields. Energy is either stored in the field or it is
in the charge assembly, but not in both. Magnetic field is produced
either by the moving charge or by the moving field, but not by the
both. Since in the above case, magnetic field is produced in absence
of electric field, we can safely conclude that the magnetic field is
distinct from electric force and has no relation with the later.

4. Lastly, to ensure that there is no electrostatic field between the
two conductors, cover each with a metallic insulated tube. Ground both
the tubes. Now we can be certain that there is no electrostatic force
between the conductors. This will not however, make an iota of a
difference to the magnetic field.

4. Lastly, to ensure that there is no electrostatic field between the
two conductors, cover each with a metallic insulated tube. Ground both
the tubes. Now we can be certain that there is no electrostatic force
between the conductors. This will not however, make an iota of a
difference to the magnetic field.

First consider a conductor having v as a drift velocity of electrons
in x-direction and a charge Q above it, at a distance y, moving with a
velocity u parallel to the conductor.
Force due to electrons in S' is converted to frame S.
electrostatic forces due to -ve and +ve charges in the conductor
cancel out and only the magnetic force remains. However if the
magnetic force is due to the transformation of the electric force,
then we must find residual electric force along with the magnetic
force.
To get it we follow a different procedure. We have now
three frames. stationary frame S, that of electrons S' and that of
moving charge S". We shall find the force in frame S". In this frame
+ve charges are moving with a velocity -u and so the charge density is
increased to dg1, where d is the linear charge density and g1 is
gamma. Electrons are moving with a velocity (v-u) and we get increased
charge density as dg2. The net surplus density becomes d(g1-g2).
Apply Binomial theorem to g1 and g2, neglect very small terms and find
out electrostatic force in S" and transform it to frame S.
Apart from exactly the same magnetic force as found
by the first method, we get an additional term as Qdv^2/kyc^2. Since v
cannnot be associated with Q, this stands for residual electric
force.(Hence maths used here is not only correct but it is more
correct.)
Any way, this does not lead us to any substantial
conclusion. Real test for the relativistic transformation is the case
when we consider two parallel current carrying conductors.
Relativistic transformation fails in this case.
The conductors are nutral except for the increased
electron density in frame S. If both the conductors carry equlal
currents, then the net surplus charge density is d(g-1) in each
conductor and the electrostatic force is,
F= {d (g-1)}^2/yk

When we expand g^2-2g+1 by using Binomial theorem ,
we get zero for this term, that is if we neglect very small terms as
before.
So why should we get magnetic force in absence of
electric force? Note the following points,

1. The concept of increase in charge density is wrong. There is a
fixed relationship between the number of conduction electrons and
atoms.
2. Although conduction elctrons move with a drift velocity, the frames
S' and S overlap. Consider a ring of superconducting material in which
high current is set up by momentory change in extraneous magnetic
field. If the volume of S' contracts then two rings will be formed.
This is physically not possible, particularly because of the fixed
relationship between +ve and -ve charges.
3. These are all pathetic efforts to assign physical meanings to
mathematical symbols. If we treat electric field as real and since the
field is representative of charge, the physical properties of a charge
can be mathematically assigned to field. But there cannot be
duplication.Force is either between the charges or it is the
interaction of fields. Energy is either stored in the field or it is
in the charge assembly, but not in both. Magnetic field is produced
either by the moving charge or by the moving field, but not by the
both. Since in the above case, magnetic field is produced in absence
of electric field, we can safely conclude that the magnetic field is
distinct from electric force and has no relation with the later.
4. Lastly, to ensure that there is no electrostatic field between the
two conductors, cover each with a metallic insulated tube. Ground both
the tubes. Now we can be certain that there is no electrostatic force
between the conductors. This will not however, make an iota of a
difference to the magnetic field.