Question:

How does this process take place?

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The thickness of the electroplated layer depends on the magnitude of the current and the duration for which it is applied.
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Solution and Explanation

The process takes place in an electrolytic cell and relies on the principle of electrolysis.
Anode (Positive Electrode): This is made of the metal that will form the coating (in this case, silver). It is connected to the positive terminal of the battery. At the anode, silver atoms are oxidized (lose electrons) to become positive silver ions (Ag$^+$) and dissolve into the electrolyte solution.
Cathode (Negative Electrode): This is the object to be plated (the spoon). It is connected to the negative terminal. The positive silver ions (Ag$^+$) in the solution are attracted to the negatively charged cathode.
Electrolyte: This is a solution containing ions of the plating metal (e.g., silver nitrate, AgNO$_3$).
When the electric current flows, the silver ions (Ag$^+$) migrate to the cathode, where they are reduced (gain electrons) and deposit as a solid layer of silver metal onto the surface of the spoon.
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