Electrodynamics

Parallel Conductors carrying Current explained by Relativity

While studying electricity and magnetism we would have always been across solving problems involving wires/conductors carrying current. For straight wires carrying current the magnetic field is like revolving around the axis of wire which we can easily show by thumb rule.
Parallel Conductors Carrying Current explained by Relativity
Current carrying parallel conductors
getting attracted : Same direction

(image courtesy : MIT)


But its quite different when we approach the problems of two similar wires,kept parallel, carrying equal currents; in the same direction or opposite direction.

What is so different here? 
As usual current passes, magnetic field arise around them. But apart from this the wires seems to have a small movement. If the wires are carrying the current in same direction the wires are seemed to be attracted,while in the opposite directions;either way the wires get repelled.

Parallel Conductors Carrying Current explained by Relativity
Current carrying parallel conductors 
getting attracted : Opposite direction
(image courtesy : MIT)

The easiest answer was to demonstrate the direction of magnetic field arising due to the motion of electrons in wires. Equations was created even to measure the force between these wires and of course they were perfect matching.

However,even though the scene was promising proper results,the explanations wasn't satisfactory. Also the mutual relationship between electric and magnetic fields and Faraday's law doesn't had any physical explanations. They just exists. 

Parallel Conductors Carrying Current explained by Relativity
Free Diagram Showing the attraction and repulsion shown by parallel conductors carrying current


It was finally on the basis of relativity this got explained. Electrons and Protons are subatomic particles which moves with a speed comparable to that of light. Whatever theory explain their physical phenomena's, it I also have to account relativity.

Consider two conductor of equal length,density and all physical measurements. Both are carrying equal currents. Lets examine the work of relativity here.

Parallel Conductors Carrying Current explained by Relativity
Picture 1 : Conductors at rest,without current


Picture 1 shows two conductors at rest. The electrons and protons are equal in number in both conductors. They are also equally spaced in the conductors. The highlighted length L1 to L6 are mutually equal in magnitude. e represents electron while p represents proton. The direction of motion (if a potential is connected to pass same current in same direction) will be in the directions as shown in the left end for protons and right end for electrons.

Parallel Conductors Carrying Current explained by Relativity
Picture 2 : Conductors,while current is passing through both in same direction


Picture 2 shows the same two conductors when both are connected two equal potentials,that is same current is flowing through them.
Note that the spacing between electrons and protons in both conductors have reduced. The space between them got reduced because of the length contraction phenomena,as the particles move with a speed comparable to light velocity. Here the L1 to L6 will be mutually equally but will be smaller than the L1 to L6 of picture one,that is when they are at rest.

But fortunately,these wires are getting attracted (as we have connected the potential and is now passing equal current in same direction).
What could be the possible reason?
Lets check the next picture.

Parallel Conductors Carrying Current explained by Relativity
Picture 3 : From the perspective of Electron in first conductor


In picture 3,we are considering the electron in first conductor and observing the second conductor. From the first conductor, for an electron, the electrons in the other conductor also moves with same speed in same direction,which makes it to feel almost motionless. But the protons which are moving in opposite direction will seems to be moving with double speed in opposite direction. Or we can assume that the number of protons per unit length will be more (as protons are seemed to be travelling with more velocity,the number of them passing through a point also increases) so for this electron,It feels that there are more protons;to be precise there are more Protonic effect,that is the positive charge. Hence, electron gets attracted to these protons.

Parallel Conductors Carrying Current explained by Relativity
Picture 4 : From the perspective of proton in first conductor


Similarly in the picture 4 , if we consider the proton on first conductor and observes the second conductor,the protons of second conductor will be almost motionless while electrons are super fast,which creates an increment in electronic charge and finally attracts the protons.

The same thing happens if we consider the proton and electron of second conductor and observes the motion of subatomic particles in the first conductor. Every time in these four considered options the result is an attraction. Which totally together makes the conductors to gets attracted.

Okay,then what about,of the currents are in opposite direction?

Imagine an new picture 3 analogous to our picture 3 ,but having opposite directions. 
In our real picture 3, the electron of first and second conductor were in same direction,so that they feel almost motionless while proton seemed to be having double velocity. 
In the new picture 3,its just reverse. The electron first and second conductor has opposite directions,hence feel double velocity,while electron of first conductor feels motionless while comparing the proton of second conductor.
Hence a total electronic charge in second conductor is increased. As like charges repels,this repels the electron in first conductor.

The same gets repeated in picture four. Instead of electron, proton takes the place. Gets repelled by the total Protonic charge developed in the second conductor.

The same results are observed when the electron and proton of the second conductor is kept still and been observed with respect to the motion of electrons and protons if first conductor.

Totally,in the four new cases the particles repels,which totally together repels,to make the conductors repel.

Parallel Conductors Carrying Current explained by Relativity
(image credit : google)

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