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Length contraction with increasing velocity is predicted by special relativity, and helps explain the relationship between electricity and magnetism, as proposed by Einstein. Imagine two lanes of traffic moving in opposite directions at nearly the speed of light. An observer at rest in the reference frame of the road sees cars driving past that can apparently fit neatly into the available parking spaces. However, if the observer begins to move parallel to the road, the parking spaces and cars in one lane will appear to contract while cars in the other lane appear to lengthen.[more]
It is expected that the cars either can or cannot fit into the parking spaces, yet different observers will reach different conclusions. An observer standing in the street will see a single car passing by each parking space at any given moment, while a moving observer might see two cars simultaneously passing each parking space or a single car occupying three spaces at the same time. How is this possible?
Now replace the roads with wires, the parking spaces with stationary positive charges, and the cars with negative charges flowing through the wires. Both wires appear to have roughly equal densities of positive and negative charges, and should therefore be electrically neutral; an observer would measure the Coulomb force between the wires to be zero. However, an observer moving alongside an electron (negative charge) in the lower wire will perceive a different scenario, in which the upper wire has a net negative charge and exerts a repulsive force on the lower-wire electrons. In a moving frame, the electrons in the wires therefore experience a repulsive electric force; how can this be reconciled with the lack of any electric force in the rest frame?[less]
Contributed by: Quentin Sedlacek (March 2011)
Based on a program by: Howard Georgi
Open content licensed under CC BY-NC-SA
The solution to the first traffic paradox lies in the principle of length contraction. Consider what it means for a car to park: the vehicle must come to a halt in the reference frame of the road. This means that every observer, in every frame of reference, will see the relative velocity of the car (and therefore the magnitude of its length contraction) change until it matches that of the parking spaces, revealing whether or not the car can actually park. As it turns out, the cars in this Demonstration are too large to park in the available spaces.
What about the different car-to-parking-space densities observed from a moving reference frame? Their explanation lies in the relativity of simultaneity. In order to determine how many cars are passing each parking space "at the same time", an observer must simultaneously measure the positions of the front and back of each car and each parking space. However, the constancy of the speed of light causes observers in relative motion to disagree about whether or not events separated in space are also separated in time. Therefore, position measurements that a moving car driver makes "at the same time" would appear to happen at different times from the viewpoint of a stationary observer (and vice versa), distorting length measurements and the apparent ratio of cars to parking spaces. In the words of Harvard professor Howard Georgi, "these additional parking spaces are coming in from the past and future".
Looking at moving charges instead of moving cars illuminates the connection between electric and magnetic forces. In a moving frame, observers see an electric force repelling the electrons in each wire; in the rest frame, observers see a magnetic force with the same effect, apparently created by the electric current. Electric and magnetic forces are simply the same force observed in different frames of reference.
Wolfram Demonstrations Project
Published: March 7 2011