Why don't you get electrocuted when you jump and touch an electric fence?

Question

I've read that you won't get electrocuted if you jump and touch an electric fence because you aren't closing the circuit with the ground. Which is also why birds don't get electrocuted when they're standing on power lines.

The way I understand voltage is that it's basically "a pressure of electrons" in the wire where electrons would like to escape from each other because they are the same charge, and they would love to reach the ground where there is plenty of space (i.e "low pressure"). When you touch an electric fence, the electrons see an opportunity to reach the ground (to "fill the space") through you and you get electrocuted because your body resists the current which makes heat.

Alright, but I don't understand why don't electrons want to fill up your body even you're not touching the ground? Especially at high voltages; I mean there is plenty of space in your body. So when you jump and touch a fence, even though you're not touching the ground, why don't electrons see an opportunity and rush to fill up your body and cause you a shock?

To put it another way, let's say a bird lands on the ground and discharges all the excess electrons (becomes neutral with the ground). Now let's say the bird flies off and lands on a power line. Why don't electrons in the wire rush to fill up the bird and electrocute it?

Please help me understand.

Answer 1

Electrons do "fill up your body" when you jump up and hit a high voltage wire - there is a property called the capacitance of the body that determines how much the voltage increases when you add a certain amount of charge - mathematically, $C = \frac{Q}{V}$.

But it's not charge that kills you, it is current: charge flowing per unit time. Since it takes relatively few electrons to bring the body up to 30,000 V or so, there is not much charge flowing and nobody gets killed. But you may have noticed a static "shock" when (especially in winter) you walked across a carpet, then touched a metal door and got a shock. As you walked across the carpet you built up static charge (with an associated potential that could reach several 10's of kV); and all that charge "leaks away" when you touch a grounded (conducting) surface. But while you can "feel" the current it's not enough to kill you.

So how much charge is there on your body when you are charged to 30,000 V? It's a bit hard to estimate the capacitance of a human body, so we'll use the physicist's trick of the "spherical cow": we approximate the human body as a sphere with 1 m diameter. The capacitance of a sphere is given by

$$C_{sphere} = 4\pi\epsilon_0 R = 0.11 nF$$

At 30 kV, that gives a charge of 3.3 µA; if that charge comes out of your body in 1 µs*, it would result in a peak current of 3.3 A which is why it feels like quite a jolt; however, the total amount of energy is only $\frac12 C V^2 = 0.05 J$ - and that is not enough to kill you. It's enough to kill sensitive electronic circuits, which is why you have to be careful how you handle "bare" electronics, especially in winter (low humidity = build up of static electricity as conductivity of air is lower).

EDIT

• if the current flows in 1 $\mu$s, that suggests that the time constant of body capacitance and skin resistance should be on that order. Since time constant is RC, solving for R gives about 10 kOhm. That’s a rather low resistance: skin resistance is higher, so peak current will be lower.