Chain Gang on a Fence
“The person furthest from the fence is the only one feeling the electric shock!”… Debunked!
1. The Discussion
Dining last night with some neighbours, neighbour X came with the following story. “When we were kids”, he said, “we used to give hands to each other, left to right, in a long chain and then the first one used to touch an electric fence. None of us would feel any electric current or shock, but the last one felt it all!”
I reacted that the last person suffering the most could not be the case, but my neighbour replied: “It is surely the case, since we did it and we all experienced it!” I then asked him how it could be the case ‘scientifically’ because surely, the current the last person felt should come through the arms of the previous people and he replied that this is/must be the case because “any but the last person is ‘just a conductor’.
2 Refutation
Depending on the amount of science one understands, one may use different techniques to convince people that the conclusion that the last person suffers the most as well as that the other persons would not feel anything is just wrong.
2.1 Refutation by Showing an Error in the Story Telling or Reasoning
First, surely, the last statement that all but the last person in the row are ‘just conductors’ is wrong or at least inaccurate. The confusion is partly due to natural languages allowing for the all too casual use of the word ‘conductor’, while the word does have a particular scientific meaning. In secondary school we all learn about Ohm’s law and that objects (except for some rare superconducting ones that show that behaviour only at very low temperatures close to 0K) all materials have a certain resistance to electric current (whether is be AC or DC current). (There are also semi-conductors.)
Humans do resist electric current so are ‘resistors’. Wikipedia for example states that a human has a resistance between 100000 Ohm and 700 Ohm, depending on how wet/broken the skin is. Wetter and more broken skin has lower resistance, but never 0 Ohm resistance.
Second, even if humans were ‘just conductors’, meaning they would have 0 Ohm resistance, the last man standing would also have 0 resistance and nothing would be fundamentally different for him, since even then, it would still be true that all the current that goes through him has to pass from the fence through all the intermediate people in the chain and then back through the ground to the battery.
This second reason should be sufficient proof for anyone who understands or even has a vague feeling of how electricity chooses its way from a fence to the ground, through a network of resistors, which is what a bunch of connected humans are, as far as the electrons and so electricity are concerned.
At this point, I considered the story sufficiently debunked.
However, since I experienced this was still unconvincing to some (and I admit this somewhat irritated me), I decided I needed some bigger guns.
2.2 Refutation by Googling
Nowadays, outside science, the quickest path to convincing people of something is to find an internet link (millenials need a youtube video) that supports your ‘opinion’ and somehow seems more reputable than you are to your listening audience. Essentially, you try to find someone with either more authority than yourself that supports your opinion, or better, also explains why your opinion is in fact a widely acceptable fact or/and can be derived from widely accepted facts.
However, while searching you may find opposite opinions and then you should just give in and openly admit you were wrong of course.
2.2.1 Listening to Glen
But so we found this exact question on Quora. And the first answer, from Glen Helgeland who signs with “Studied Robotics at Swinbourne University of Technology” says what is right, yet without starting from basic physics but directly stating results.
“If all the people were insulated from the ground, no one would feel a shock. just as a bird can perch on power-lines.
If one of the people grounds themselves, every person between them and the fence will experience the same shock, as the current flowing through them would all be the same. All the others beyond this circuit will feel nothing.
If multiple people ground themselves (or all of them, as the most probable scenario), then each grounding point will lessen the pain for all those to the next node and so on. As the human body is quite resistive, the power will flow most through the first “circuits”. The pain/shock will become progressively weaker after each grounding point. This will continue till the last ground contact, again every one after this point (not grounded) will not be affected.”
What is helpful and hopefully convincing here is that Glen states experiments from from simple to more complex and as such shows also some consistency in the results.
It may be unclear to some what being grounded means in the above. One is grounded if on conducts to the ground with a zero or negligible resistance (in Ohms, the unit of resistance). Being bare feet and in very wet grass would be the most conductive situation while wearing rubber soles on very dry ground would be the least conductive — so highest resistance — situation.
However, who believes Glen? Is he really an engineer? (I have no reason to not believe him.) But he does not build up an argument starting from scientific facts, so some people may not yet buy his claims.
2.2.2 Listening to Pascal
“Who receives the biggest shock?” was the question.
Pascal Heitz, signing as an engineer, answers this.
Probably the guy who touches the fence, since electricity will use the shortest path, which is from the fence to the ground through his body.
The second person of the chain might feel the shock too, but lowered, since he is only the second choice path for electricity.
The third person will feel it too, but again a little less.
Etc.
So contrary to Glen, Pascal uses reasoning from facts, saying the people closer to the fence should experience more current through their bodies since ‘electricity will use the shortest path’.
This conclusion is correct, but the wording ‘electricity will choose the shortest path’ is somewhat sloppy.
However, Pascal wrote something else as well:
There is a belief that says that the last person gets the biggest shock, but from my little electricity knowledge, I don’t see why it would be that way.
So apparently this is a more wide spread urban legend than just with my neighbour and his chain gang.
This means that this belief deserves to be more widely refuted, with proper wording and derived from scientific laws.
Let’s reword to make the argument more accurate.
2.3 Refutation Based on Very Simple Physics and No Mathematics
Electricity consists of electrons flowing from a cathode of the battery to the (connected to the ground) anode of the battery.
Every electron going through the arms and body of person N (furthest from the fence) has to flow through both arms of each of the previous persons closer to the electric fence. This means the last person cannot possibly receive more current than going through the arms of the previous people. This refutes the belief.
Convinced?
Note that we did not need to calculate anything here, to refute the belief.
As for the currents through the bodies, we need a little mathematics.
2.4 Refutation based on Current Flow Physics and Little Mathematics
If one also wants to know by how much every next person receives less electricity, expressed in terms of ‘arm’-resistance r(a) and the arm to feet called ‘body’-resistance r(b), mathematical modelling comes into play.
The laws of Physics tells us in a very accurate way how current flows through resistor networks.
In fact, for any given resistor network, series or parallel or any combination of it, for which each resistor value is known, one can also solve that for all currents and voltages.
As for the physics, only two laws are needed: the Kirchhoff current law and the Kirchhoff voltage law.
The Kirchhoff Current Law states that in each node, all the current that comes in equals the current that goes out.
The Kirchhoff Voltage Law states that in each loop, the summation of all voltage drops along each resistor (and battery) in the loop adds up to zero.
For our human-chain we have the following network.
In the left of the above figure, a single person is modelled, with a head and two arms. Each arms has a resistance r(i,a), with i the index of the person in the chain gang, from 1 on the left to N on the right. The body and legs will leak current from the center of the two arms to the ground along a resistance we call r(i,b).
We then chain N people in a row on the right of the above figure. They all face the same direction and left hands join right hands. The = sign in the network on the left of the right part represents the battery and has an upper voltage vu(0). The nodes vu(1), vu(2) etc… towards the right represent the voltage every person will perceive between both shoulders. From experience, an electrical engineer will know that one has a voltage drop from left to right. So the rightmost person will perceive a voltage vu(N) that is lower than voltage vu(N-1), that is lower than … vu(0). We also have voltages at ground/’down’ level ‘vd’ and theorize that they could increase by distance to the battery ground voltage vd(0)=0V.
The currents going through the arms are i(a,1), i(a,2), etc… towards the right, and are not necessarily equal for each person.
Each person leaks a current to the ground called i(b,1), i(b,2), … etc towards the right and these are not necessarily equal either.
At the feet/ground of the people, we assume currents i(g,1) ,i(g,2), … etc flowing from right to left in the ground. We assume that people have equal distance from each other and so the ground resistances from one person to the previous can be assumed equal. The currents however are not necessarily equal.
2.4.1 Currents through Arm: Refutation by inspection of Current Flows
If we just look at the current flow from person 1 to 2, we see that the current i(1,a) in the closest-to-the fence arm of person 1 gets split into a current i2a into the closest-to-the-fence arm of person 2 and a current i1b in person 1 his body to the ground. So i(a,2) = i(a,1) — i(b,1), so i(a,1) ≥ i(a,2), so currents through the arms from person 1 to person N get lower and lower i.o. higher! So i(a,1) ≥ i(a,2)≥ i(a,3)≥ … ≥ i(a,N). So this refutes the belief that person N will feel a higher current in his arm than any previous person! Note that this is true, whatever the resistances r(a,i) and r(b,i) are!
2.4.2 Current through Bodies: Refutation by Primary School Mathematics
As for the body currents, i(b,i), assuming the same r(b,i)for all i, and assuming the same ground level voltages vd for now, note that then i(b,i) = vu(i) / r(b,i) and since, from current i(i,a) direction to the right, the vu(i) decrease to the right, as in vu(b,0) ≥ vu(b,2)≥ vu(b,3)≥ … ≥ vu(b,N), it also holds that i(1,b) ≥ i(2,b)≥ i(3,b)≥ … ≥ i(N,b). So, assuming equal body resistance for all people, this also proves that the belief that body currents are larger towards the end of the chain is false.
Even if we assume that only the last person is barefoot in wet grass and the rest of them are all wearing insulating rubber soles, then r(b,N) << r(b,i) for all i and the body current can become larger for person N then for the other people, but even in that case, the full current r(b,N) (and some more!) will still have to go through all the arms of all previous people, again refuting the belief that the last person will get more current through the his whole body than other people experience in their arms.
2.5. Refutation by (Repeated) Full Calculation & Secondary School Mathematics & some Engineering
Now, by applying Kirchhoff’s (conservation of) current law in every node where currents can merge or split, we get N equations in N variables I(1,a) to i(N,a), first for the top nodes:
and then N equations in N variables i(1,g) to i(N,g), for the bottom nodes
For the Kirchoff Voltage laws, we have, by going through all lines in the network. For the upper lines, this gives the equations.
For the vertical lines we get:
For the lower lines we get:
This gives a system of 5N linear equations for 3N current variables and 2(N+1) voltage variables, so for 5N+2 variables. However, in addition, we set v(l,0) to 0 and v(u,0) to the battery voltage, so we then have 5N+2 equations as well. That is nicely and easily solvable using some linear algebra.
Even though, there is only one solution here and we do not have an objective, to solve this, we can still use the Integer Linear Programming (ILP) solver Gurobi. There will be only continuous variables and the problem is a feasibility rather than optimisation problem.
For a real implementation of this problem, using the Gurobi solver for solving linear algebra systems, have a look at this gist on github or you can even get the source code from github here and then run it on your own machine, assuming you have a Gurobi license.
The conclusion of the calculation experiments in that notebook is that indeed, in each case the current through the last person’s arm is always lower than the ones through the previous peoples arms. Also the body currents in the last person are always lower than the ones in the previous people, even if the last person is assumed to have (much) lower body resistance than the previous ones, since he is protected by the arm resistance of the chain of people before him.
2.6. Refutation by (Repeated) Experimentation
People may have no belief in the above scientific method, but it is fundamentally founded in the experimental method, where a law only becomes accepted while all experiments support it. A scientific law will only stand until one properly conducted experiment disproves it.
If you are still unconvinced, chances are that you want to experiment and find some willing mates to form a chain-gang and find yourself a ‘victim’ too. You may want to measure currents by holding Amere meters between hands rather than trust your possibly subjective sensation of current though.
I must warn that it is not a safe experiment. So don’t try this at home.
3 Epilogue
3.1 Why do People hold Unfounded Beliefs?
That this belief that the last person suffers most exists, to the degree that they even start believing they feel no current in their own bodies (e.g. ‘because they are just conductors’ or for whatever reason), or at least believe they will feel less so than the last person is altogether surprising!
Maybe people unconsciously think that the electrical laws behave like the ones where on an ice rink, people connect hand to hand in series and the first once swings the other ones around him. The centripetal force acts in a way such that the outer Nth person will pick up the highest speed, as such having an amplifying effect from person 1 to N. Of course, this is not the case for electricity.
But maybe it is just group psychology: “As a group, we will play a prank on a sole individual and make fun of him, bully him, which makes us psychologically strong.” Apparently this can even overrule even physical pain in ones own body!
Another possibility would be that people are more sensitive to the current through their body from neck through legs into the ground compared with current through their arms. In that case the last person could sense a current i(b,N)=C as higher than the previous people experiencing the same or even than a higher current i(a,i)=C+some in their arms. If that would be true, it would be purely subjective. In addition, the opposite is true: the density of the nervous system is larger in the hands than in the back, so one will be more sensitive to hand-to-hand currents than back-to-leg currents, as confirmed to me by a specialist medical doctor.
3.2 A Better way to Play the Prank.
A better way to play the prank would be to — as a pranking group — join all left hands together and join all right hands together, as such shunting each individual as a resistor in parallel with all others. The teasers would then each feel roughly 1/(N-1) of the current that the last person N would feel. This is the setup shown in the following figure.
The left side of the figure shows N-1 people holding hands in parallel and one person (in red) receiving all the current going through the N-1 previous people. The current divides itself equally through the first N-1 people, so the voltages v1 to vN-1 are equal, so can all be joined into one node. Doing that, we get the (grey part in) circuit on the right of the figure, where the N-1 resistances Ra and Rb, combined in parallel, behave as single resistances Ra/(N-1) and Rb/(N-1). So, now, the person N will roughly receive N-1 times as much current as the other N-1 people playing the prank.
This is clearly the ‘somewhat smarter’ way Martin Prince Jr would do it! :)
However, note that person N will still receive somewhat lower currents compared to the case where you tease him into directly touching the fence. So there is no effect of ‘magic current multiplication’ or so happening by insertion of previous people before person N, but just adding extra resistance which lowers the current in person N.
3.3 Conclusion
The myth of person N suffering the most is totally busted. It’s quite the opposite!
One can only conclude science should be taught before pranking age. :)
Peter Sels, June 19th 2020. Written for my neighbour X and his neighbours.
Copyright © 2020 Logically Yours BV.
