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As far as I understand electric current is nothing more than the flow of electrons, that is, charge. So, my question is, what exactly happens when charge/electrons flow through our body and why is this dangerous and potentially fatal?

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The main problem is that our nervous system communicates using electrical signals. These signals are carried by current in the order of milliamperes. A powerful shock could easily stop the heart and cause muscle spasms. Another major problem is the heat produced. The body basically acts like a resistor and when there is sufficient potential difference, a lot of heat can be produced. You will essentially be cooked to death from the inside. Hence electrocution is very dangerous for any living organism.

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Whenever electric current passes through our body, it interferes with the neurological activity of our body.

But if the current somehow passes through the heart it will definitely interfere with its electrical activity which might lead to fibrillation. Also, irregular blood supply for a while will affect the brain.

In the heart electrical impulses for controlling the valves and the chambers' rhythm is generated in the sino-atrial node. The current generated (required) for this is very small.

Moreover, all of the organs in our body contain fluids and if current passes through them (for long enough) it will quickly heat up which may lead to organ damage.

Once we receive a perceptible shock, we may lose control of our muscles and hence not be able to pull away. (That's the most scary and dangerous part)

I've got an image from Google which suggests that a current of $\approx70\mathrm{mA}$ is just sufficient to disrupt cardiovascular activities.

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Hope it helps.

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