Note that he wasn't just holding onto a single line but was in fact standing on one and holding another.
It's certainly possible for him to conduct sufficient electricity (between the lines) to have electrocuted himself.
An electric fence designed to stop things jumping into and climbing it would have alternate rails electrified with opposite polarity, or AC current with the opposing phase, resulting in exactly this effect.
So, while people often object that it doesn't act like a cattle fence, this is only because cows can't jump or climb, and cow-fences are typically a single rail.
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You are correct that birds aren't shocked because they aren't grounded.
However, on a typical electric fence, the electric conductor is separated from the fence itself and insulated from it.
Not all the fence carries a voltage.
If someone were to touch both the conductor and the metal of the fence then they would be grounded.
The current would be able to flow through them to the fence.
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No he wouldn't have gotten shocked because he's not grounded and there's no way there's three phases running across that fence because all the lines are connected by a mesh.
The whole fence would be shorted out if that was the case.
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Part of the drama of the situation was that the fence was initially off.
We can assume this means the fence was at ground state, i.e. zero volts relative to the ground and relative to little Timmy.
The tension of the scene comes from Dr. Grant and the kids racing against the restart sequence conducted by the other intrepid heroes, who are about to unwittingly electrocute homo sapiens instead of tyrannosaurus rex .
If the fence lines Tim were touching were opposite polarity, that would have been 50,000 volts from his hands to his feet, with a pass through his torso.
Now, the fatality of electrocution is measured in current, not voltage.
Apparently as little as 42 volts is enough to push a fatal charge.
However, as you point out, it all hinges on whether or not the current would actually transmit to him, while he's on the fence.
If the fence lines were the same polarity, there's still the issue of capacitance.
Capacitance is what causes static electricity: a charge builds up on your body while you're ungrounded and discharges when you touch an unsuspecting friend or doorknob.
But in this case, it's the fence that's getting charged up and sending a shock into the ungrounded boy.
Again, while 50,000 volts seems like a lot, a typical, minor shock that get from touching metal after walking on carpet can be as high as 25,000 volts .
Part of what causes the voltage from a static shock to be so high is the collection of a surface charge on the pointy ends of our fingers we tend to touch things with.
A charged fence with a large enough current supply pushing electrons into an ungrounded body may behave differently.
I'm not qualified enough to tackle that one, so I'd suggest asking again on Physics.
SE
if you're interested in more.
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Tim would not have been shocked while hanging on the fence.
As stated by a previous post, the fence is all at the same potential.
Under normal conditions, there is voltage present in the wire, but no current flow.
Current will only flow through the fence wiring when a path to ground is presented.
A practical example is an electrical receptacle.
Voltage is always present at the receptacle, but Current will flow when you plug in a hair dryer or when you stick a fork in the hot side.
Tim is not touching the ground; he is touching four different points of the same potential.
He should have been fine.
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What you are failing to notice is that all the the larger conductors are connected together via the smaller uninsulated wires.
Therefore all of the cables are at the same potential.
The boy should not have received a shock due to current flow through his body as all of the cabling is at the potential.
However in EHV overhead transmission, you will not see birds perched on the cables due to the strong electric field caused by the high potential (between phases and also to ground).
The high strength electric field caused by 50kV would be enough to blow him off the fence, especially when the power was turned on as there would have been a surge of current.
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While it's true he isn't grounded and no CURRENT would run through him, it's definitely the reason he's not instantly fried.
However, static discharges is something everyone has tried when touching a charged surface, like your car or say escalator-handrails.
I doubt this has ever killed anyone, even if (as someone mentioned previously, and this is true)
the voltage experienced can be over 25000.
The reason Tim is "electrocuted", though, and rendered unconscious, is the electric POTENTIAL difference (voltage) between him and a suddenly active 50000 volt fence.
Tim then becomes a capacitor (a device which stores electric charge) whose capacitance is much greater than that of a bird.
The voltage doesn't kill you - the current (if grounded) will.
If you've ever seen live powerline work being performed from a helicopter, you'll notice they use a conductive wand on approach to equalize potential difference between wire and helicopter before "touching" the wires.
This neatly visualizes what happens.
For more on that, refer to this question on Physics.
SE .
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In the Jurassic Park movie, the boy (Tim) is on an electric fence when the power gets turned on. However, birds can sit on power lines. I think this is because they do not touch the ground, only the line. Since Tim is not touching the ground, shouldn't he be safe until he reaches the bottom?
Should the little boy have been shocked?
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<jurassic-park>
scifi.stackexchange.com/questions/93840
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If the fence lines Tim were touching were opposite polarity, that would have been 50,000 volts from his hands to his feet, with a pass through his torso. However, the reason Tim is "electrocuted "and rendered unconscious, is the electric POTENTIAL difference (voltage) between him and a suddenly active 50,000 volt fence. The high strength electric field caused by 50kV would be enough to blow him off the fence, especially when the power was turned on as there would have been a surge of current. Current will only flow through the fence wiring when a path to ground is presented. The voltage won't kill you - the current (if grounded) will. For example, if someone were to touch both the conductor and the metal of the fence then they would be grounded. Theoretically, Tim should not have been electrocuted on the fence because he wasn’t grounded. However, the fact that the 50,000 volt fence was activated so suddenly caused him to be electrocuted while on the fence. 
The fact Tim isn’t grounded should mean he isn’t electrocuted. The sudden voltage difference between Tim and the 50,000 volt fence cause the electrocution.  