## Magnetism – Proof of Concept
## Force between wiresTo test the theory out, we’ll look at the magnetic force between parallel wires. Suppose we have two parallel wires as shown below: The wires carry currents of I
respectively and are separated by a distance _{2}D. What will be the magnetic
force F experienced by each wire per unit length dL be?
Magnetic force equations tell us that:To simplify the calculations, let’s say the current in both wires is And this is the result we are aiming for.
## Using the VDCLNow let’s see what we predict for magnetic force using the VDCL.
First we note that that each wire holds equal numbers of protons and electrons, giving the
wires a neutral charge. Most of those electrons are fixed while others are ‘free
floating’ and able to carry a current. We can eliminate the fixed electrons and a
corresponding number of protons from our consideration because they have no net effect.
What remains then is only the floating (current carrying) electrons and an equal number of
unmovable protons. Let’s look at the force produced in a small section of the right-hand
wire of length dL
segment be q._{2}The electrons in both wires are moving with identical velocity because the current and cross sectional area are the same. This means that the electrons are standing still relative to one another. So the horizontal force F from the
electrons can be calculated using the standard Coulombs law:_{e}Where r is the distance from Now to calculate the force from the protons. To do this we must consider
the velocity of the protons relative to the electrons. The velocity x relative to dL can be determined from the
rate of change of the distance r.Using this velocity function in the VDCL, the (stronger) force from the
protons at - The (weaker) force from the protons at + The total force can be calculated by adding forces from both protons and
electrons at both positions (+ This shows the force in terms of the velocity Where q in similar
terms:_{2}Substituting these values of velocity and charge into the force expression we get: This gives the force from just the two points at Coulombs constant Maxwell’s and Ampere’s laws tell us that: Extracting k we see that:Substituting this value of This is the exact result we were after! Namely, the VDCL has computed the magnetic force equation.
## ConclusionBased on the above calculations, we can see that the VDCL can be used to
determine the exact result as Maxwell’s magnetic force equations.
## Wait... a mistake?The above calculation only determines the force from the protons in the
Left wire upon the electrons in the Right wire. But the Right wire also has protons that
will be attracted to the electrons of the Left. If this is included the total force per
unit length will be twice as large. Are our results too small by a factor of 2? |

Copyright © 2011 Bernard Burchell, all rights reserved.