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Electric switch paradox

  1. Aug 5, 2010 #1
    This relates to another thread about trains and manhole covers at relativistic speeds falling (or not) down gaps that would be normally be too small.

    Consider two hollow cylinders aligned with each other along their long axes with a short gap between the hollow cylinders. They are designed so that a solid bar sliding along the inside of one cylinder can smoothly slide from one cylinder to the next and the bar is long enough that when it is at rest with cylinders, it can comfortably span the gap and complete an electrical circuit and turn on a light bulb.

    Now when the bar is moving at relativistic speeds, length contraction makes the bar shorter than the gap and so when making the transition from one cylinder to the next it is never in electrical contact with both cylinders at the same time in the rest frame of the cylinders. The cylinders are alligned vertically so that we do not need to worry about the bar falling out sideways. In this frame the light bulb never comes on.

    In the rest frame of the sliding bar, the gap is length contracted so when the bar passes the gap it easily completes the circuit and the light bulb comes on in this reference frame.

    Of course there is resolution to this paradox and I suspect I know the answer, but I am interested in what others think of it too :wink:
  2. jcsd
  3. Aug 5, 2010 #2


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    Always a good 'paradox' for lengthy discussion :)

    My 'short take' is simple: an observer in the rest frame of the bar draws the same conclusion as the observer in the rest frame of the cylinders - no lamp comes on at the relativistic speeds described.

    The only simultaneity that counts is in the rest frame of the lamp and its switch contacts (the cylinder frame). The simultaneity as per a bar observer's frame is irrelevant, except when he properly converts it to the simultaneity of the cylinders/lamp/contacts frame.
  4. Aug 5, 2010 #3
    Hi Jorrie :smile:

    Agree with this bit...

    .. but not this bit.

    My take is that in the bar frame, the electrons from the first cylinder effectively charge up the bar and the bar holds the charge until the next cylinder arrives and discharges its excess electrons causing a momentary flash of the light bulb, so that the light bulb does light up in both frames. The duration of the flash in the bar frame is shorter than in the cylinder frame which is what we would expect from time dilation.
  5. Aug 5, 2010 #4


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    Hi Kev.

    I think you are moving onto very difficult terrain here...

    IMO, the moment you deviate from the straight simultaneity considerations and into other physical ones, you have to specify a heck of a lot more - the detail setup of the experiment and characteristics of all the components become extremely important.

    From a pure simultaneity POV, I still stand by my take, i.e., when the sliding bar Lorentz contracts to smaller than the gap distance (in the cylinder frame), no current flows through the lamp from either frame's POV.
    Last edited: Aug 5, 2010
  6. Aug 5, 2010 #5
    I think this is a really interesting question, and the answer is not obvious at all. I think I don't know enough about the E-M theory, and to know how the fields behave when the circuit is connected. Certainly, the usual analysis - when there's a connection, there's a circuit - can't be right in the relativistic case - you can't immediately connect a circuit and get a light coming on at a distant point.

    Great question though - I just hope I can see the answer before the experts come in and solve it completely....
  7. Aug 5, 2010 #6


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    I'm no expert on the precise details of how conductors work, but I suspect kev's answer is along the right lines. Another way of putting it is that if you push electrons into one end of a conductor, you won't get electrons simultaneously coming out the other end. It takes time for a "voltage wave" to travel along the conductor (by which time the other end has moved, in this example). The precise details might be more complicated than this simplistic analysis.
  8. Aug 5, 2010 #7


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    I think Kev is right. The light comes on for a brief time in each frame. The rod will carry its excess/deficit of charge, and when it makes contact at the other side, a current will flow until the potential is flat again.
    Last edited: Aug 5, 2010
  9. Aug 6, 2010 #8
    This seems to be quite similar to some capacitance allegation. I think this could be the case here. These field effects can create electrical current as well as direct contact between conductors.

    Best wishes

  10. Aug 7, 2010 #9


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    Kev, despite what's written later, I still do not like your solution, for this reason. I can make the bar just marginally shorter than the gap to start with (when the bar is at rest in the cylinder frame). Following your argument (bar holds the charge until the next cylinder arrives and discharges its excess electrons causing a momentary flash of the light bulb...), the lamp would still flash if I let the bar move, even at non-relativistic relative speeds. IMO, this destroys the 'relativistic value' of your thought experiment.

    If you want a flashing lamp and illustrate SR, rather than some electrodynamic effect, I would rather propose putting a sensor at each end of the gap that signals when the bar makes contact with the specific cylinder. By a suitable arrangement of logic gates, equidistant (in the cylinder frame) from the two detectors, it is easy to flash a lamp only when the two ends of the gap are in simultaneous contact with the bar. Now the lamp will only flash when the bar is not Lorentz contracted to less than the gap length, i.e. below a certain relative speed. And this is what will be observed/calculated in both inertial frames (cylinder and bar). Purely relativistic. :wink:
  11. Aug 7, 2010 #10
    I quote this one. The "linear circuits" methods everybody knows are just an approximation even if we stick in one frame: when we close a circuit, there are phenomena we neglect because the time it takes to close the switch is small compared to the time the switch stays closed (or open), so we assume that the asymptotic behaviour of charge currents establishes itself immediately. This condition is not relativistically invariant, though, so the approximation may not be true in some references. That is the case here.

    Now, to solve the problem we must solve it in one reference, EXACTLY, and then transform it. And one thing is sure: the number of electrons passing through the bulb is a relativistic invariant. We should be able to answer problems like this: if (in one fixed reference) the circuit is closed for, say 10^-20 seconds, does the bulb light, even for that small amount of time?

    In all cases, I don't know if the bulb lights, but if it does in one reference, it does in all references.
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