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Consider the pictogram, the cotton balls, damp paper towel and iron nails are all very soaked/covered with conductive salt water and the battery leads are soldered to the nails. Do metal ions travel from cotton ball to cotton ball thru the damp paper towel?

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How about if the cotton balls were partially immersed in a bowl of salt water, the nails not contacting the bowl water and like previously the cotton balls are also soaked with salt water. Would metal ions migrate?

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In both scenarios you've described, some current will flow in the circuit and, within the cotton balls and the paper towel, it will be carried by both metal and non-metal ions making up the salt.

This type of conductivity, when a current flows through paper, fabric, plastic or other porous materials soaked in an electrolyte, is pretty common. Some examples are electrolytic capacitors, battery separators, moisture sensors, etc.

The resulting conductivity depends on various properties of the material and the electrolyte and, generally, would not be as good as the conductivity of a liquid electrolyte (due to reduced ionic mobility), but good enough for many applications.

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