Magnetism – Proof of Concept,
part 2
The preceding chapter showed that the Velocity Dependant Coulombs Law (VDCL) could be used
to determine the exact result as Maxwell’s magnetic force equations. The example used
however was simplified because it had the same current and crosssectional area in each
wire. Will the equations hold true in other situations? Here we’ll derive a general
equation involving different currents and crosssectional areas.
Consider the two wires in the below diagram:
Wire 1 (at left) has a current of I_{1}
and a crosssectional area A_{1}. Wire 2 has a
current of I_{2} and a crosssectional area A_{2}.
What will be the force experienced by wire 2?
We are dealing with more variables now so the maths becomes more complex. Fortunately most
of the work has already been done in the preceding chapter and it’s mostly a matter
of substituting new values into the old equations.
Let the velocity of the electrons in wires 1 and 2 be v_{1}
and v_{2} respectively. Since the electrons are moving at
different velocities we now also need to determine the force between them. Previously this
was unnecessary because they were motionless relative to each other.
So there are four forces to consider:
F_{ee} = force from the electrons in wire 1 on the electrons in wire 2 (repels)
F_{ep} = force from the electrons in wire 1 on the protons in wire 2 (attracts)
F_{pe} = force from the protons in wire 1 on the electrons in wire 2 (attracts)
F_{pp} = force from the protons in wire 1 on the protons in wire 2 (repels)
We’ll start with F_{ep}. As before, we determine the force on the protons in wire
2 from the electrons at +x
and x (in wire 1). Taking the previous equations for F_{p1}
and F_{p2}; and replacing F_{p1} with F_{ep1},
F_{p2} with F_{ep1}, and v with v_{1} we get:
Next we determine the force from the protons on the electrons, F_{pe}.
This will be same formula as F_{ep} but with v_{1} replaced with
v_{2}:
Next to determine the forces between opposing electrons, F_{ee}. Let the
force from the electrons at +x and x be F_{ee1}
and F_{ee2} respectively. Taking the above equations for F_{ep},
changing the force direction to negative, and replacing v_{1} with (v_{1}
 v_{2}) (the negative difference in velocity) we get:
Finally the force between protons, F_{pp}. This can be determined from
F_{ep1} + F_{ep2} by setting v_{1} to zero and changing force
direction to negative. It is simply:
The total force dF can be obtained by adding these seven
forces together:
Compare this with the corresponding formula in the previous chapter. Notice that the
only differences are that the result has increased by a factor of 2 (12 instead of 6) and v^{2}
has been replaced with v_{1} v_{2}.
The factor of 2 is due to looking at the force on the electrons in wire 2 from the protons wire 1
(F_{pe}), as well as the force on the protons in wire 2 from the electrons in wire
1 (F_{ep}). Whereas before we only looked at the force on the electrons in wire
2 from the protons in wire 1 (F_{e}). In other words we have double
counted. To make this consistent with the previous chapter, we’ll reduce dF by a factor of 2:
Next we need to express the velocities in terms of current, charge density and area. This
yields:
To continue from the calculations in the previous chapter, q_{1}
and q_{2} will become:
Substituting this in gives:
Compare this with the corresponding formula in the previous chapter. Notice the
only difference is that I^{2} has been replaced with I_{1} I_{2}.
Since this expression is a constant, we can skip the integration and substitute this expression
into the final result. This gives:
Substituting the ‘permeability constant’ μ_{0} = 4*pi*10^{7},
the above can be written as:
Which is identical to the conventional magnetic force equation.
Conclusion
The Velocity Dependant Coulombs Law proposed here is able to match the conventional
magnetic force equations in a general situation involving two wires carrying different currents, and
this further indicates that the VDCL could be an accurate description of why magnetism occurs.
An added benefit is that we can use this method to determine the magnetic force in different
situations. For example, we can use it to determine magnetic force between two wires of finite
length. Because the conventional equation requires the ‘source wire’ to be infinitely long (while
the ‘target wire’ can be any length), and this could never represent a real situation.
