Electricity and Magnetism: The Surprise Connection

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You have two like charges with the same velocity and a displacement between them perpendicular to the velocity. Moving charges produce a magnetic field. The charges should magnetically attract one another.

Now consider the reference frame of the charges. There is no motion so no magnetic field exist. How is this possible?

The two charges exert a repulsive force on each other due to their electric field. How do these two forces relate to one another?

The answer (if you can figure it out) might surprise you.

 

[Yuqing]

The charges will repel in either frame, just the magnitude of the repulsion will differ. In a moving frame, both a magnetic field and an electric field are present. The two fields are the same thing simply viewed through different frames.

 

[Nabeshin]

So... Lorentz transformations much?

 

[Bob S]

The Coulomb repulsion always exceeds the magnetic attraction, except when the velocities are extremely relativistic and the two forces cancel. In a Lorentz frame where the two particles are stationary, only the repulsive Coulomb force remains. See thumbnail derivation.

Bob S

 

[{~}]

nice.

What I did is looked at the ratio of magnetic force to electric force. Most of the terms canceled each other an what I was left with is

FB/FE = v^2 * permeability of free space * the permeativity of free space

The only variable is time so I calculated the product of our constants, saw that it was a very small value. I knew from the equation that it is an inverse speed squared quantity because booth sides should be unit less ratios. Sure enough

FB/FE = v^2/c^2