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I'm well aware that there are many different factors to be considered, such as ease of installation/maintenance, conversion from AC to DC, environment, distance, speed, frequency of turns, and so forth.

Setting all of those aside for now, suppose we build a 10-mile long demonstration track with some hills and turns along the way and featuring both a third rail and an overhead cable. Suppose further that we have two trains (one equipped with a pantograph, the other with contact shoes). They have the exact same weight and are programmed to accelerate and decelerate at the same rate, and run at the same speed.

Which one will have consumed more electricity at the end of the ten-mile run?

Bonus Question: A third rail is a lot wider and thicker than an overhead wire. Is more electricity wasted that way?

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    $\begingroup$ Do you assume both having the same supply voltage? $\endgroup$ – Ariser Nov 19 '15 at 12:37
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    $\begingroup$ @Ariser: I assume nothing. Rearrange the supply voltage any way you like, use any tricks you can think of, have scads of fun, mock me all you want, but PLEASE put up a meter somewhere akin to those your power company uses, do the appropriate readings at appropriate times, print out the two bills, and tell me which one is larger. $\endgroup$ – Ricky Nov 19 '15 at 12:50
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    $\begingroup$ Interesting. My gut thought is overhead is more efficient...but I'm no electrical engineer. $\endgroup$ – grfrazee Nov 19 '15 at 17:08
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    $\begingroup$ I feel that any electrical differences would be very small compared to all of the other considerations. Do you really care about the electrical characteristics or are you thinking that the electrical differences are the main deciding points? $\endgroup$ – hazzey Nov 20 '15 at 1:39
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    $\begingroup$ Third rail will be more efficient, as the answer says, at the same voltage. However overhead cables are harder to touch or step on ... they are frequently much higher voltage (e.g. 25kv instead of 600v) and that allows higher efficiency. $\endgroup$ – Brian Drummond Jan 19 '17 at 19:45
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Your "Bonus question" holds the answer here. Assuming that the two trains are perfectly identical, and that the power consumption is identical (which, if you have the same weight, motor, and size, it will be), you are down to transmission loss.

The resistance of a conductor is R=L*Rho/A. L=length Rho=resistivity (material property) A=cross sectional area

Assuming that L is the same for both systems, it comes down to cross-sectional area. The rail is larger, so less power is lost in transferring the power to the train.

P=I^2*R

P=power, I=current, R=resistance (from above)

So, in the act of transferring current to the train, power is lost. And more is lost in the lines/rails than in actually transferring to the train, and because the rails have lower resistance, they waste less power.

Note: Typically, when overhead lines are used to power trains, it is due to safety concerns. If people are around, and could potentially walk on the rails, like an above-ground, street level ("trolley") train, it makes a lot of sense to not electrocute them when they step on the rails.

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    $\begingroup$ Either the overhead power lines, or the rails. Both of them can be considered "transmission lines" for the power that goes to the trains. $\endgroup$ – Stack Tracer Nov 20 '15 at 14:36
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    $\begingroup$ Also, see updates to the above answer about safety of overhead vs. rails. $\endgroup$ – Stack Tracer Nov 20 '15 at 14:40
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    $\begingroup$ Yes, he is saying the third rail will deliver the power to the train with less loss. The way I always remember electrical transmission resistance is as being analogous to fluid flow (maybe that's because I'm a MechE.) As pipes get smaller and longer, it's harder to push fluid through them, and you lose more energy (pressure) in doing so. Same thing with electrical current in wires. $\endgroup$ – Trevor Archibald Nov 20 '15 at 19:34
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    $\begingroup$ @Ricky, Yes. That is correct. $\endgroup$ – Stack Tracer Nov 21 '15 at 11:47
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    $\begingroup$ This assumes that both are made from the same material. In the case of overhead supply you usually have a steel contact cable connected to a copper or aluminium supply cable so there is no reason to suppose that transmission losses would be significantly different between the two. $\endgroup$ – Chris Johns Jan 17 '16 at 12:56

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