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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

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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Free electron is just one type of charge-carrier. Once can also have other types: such as anions and cations to carry charge from one place to another. Therefore causing a current to flow.

Even if skin and hair lack "free electrons" like metal and semiconductor, we definitely have a great amount of anions and cations in our body. One example would be -- as krushna mentioned in his answer -- is $Na^+$ and $Cl^-$ ions from common salt in our body.

Now, not all (type) potential can cause a great deal of current to flow in our body. The 12V lead-acid DC battery definitely does not cause any perceptible current to flow through our body; no matter how many free ions we have on our skin. On the other hand, 440V AC live wire will cause a great harm if we touch it even with dry hands.

If I were to model my skin, I would go for a RC circuit with high R and high C attached in series. Put the appropriate values and simulate this circuit using a circuit simulator after applying test voltage across it. The Human Body Model for capacitance, as defined by the Electrostatic Discharge Association (ESDA) is a 100pF capacitor in series with a 1.5kΩ resistor. Since we have a capacitor in series, it prohibits DC current to flow through it. This is one reason why X Volt of AC is much more harmful than X volt of DC voltage (AC passes through capacitor easily).

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