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Electric shock occurs upon contact of a (human) body part with any source of electricity that causes a sufficient current through the skin, muscles, or hair.

In metals many electrons are said to be delocalised (i.e, electrons are not bound to any of the atoms). Around one electron per atom is said to be unbound to atom in mettalic lattice. So there will be motion of electrons due to thermal energy, however due to their random motion, there won't be such a significant current produced. When electric field is applied across them there will be motion of electrons in a particular direction, causing significant amount of current. In case of insulators, due to bound electrons we can't expect current through them.

When our body becomes a part of electric circuit, current is said to pass through our skin, muscles, or hair. Does it imply that there are free electrons in skin, hair, or in muscles (like those in metals)?

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conduction rarely involves free electrons usually - charge can move through a chain of capture/ionization events. –  shigeta Dec 23 '13 at 5:04
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Our body contains a lot of salt and which act as electroyte,so when we comes in contact of electric source then the electrolyte helps in the conduction of electric current through out body,

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