Step 1: The given reaction involves the substitution of a halide in an alkyl halide (\( {CH}_3{Br} \)) by a fluoride ion (\( {F}^- \)) from silver fluoride (\( {AgF} \)), resulting in the formation of methyl fluoride (\( {CH}_3{F} \)) and silver bromide (\( {AgBr} \)).
Step 2: This reaction is known as the Swarts reaction, which is a type of nucleophilic substitution reaction. In the Swarts reaction, a halide (usually fluorine) is substituted for another halide in the presence of a metal fluoride. It is specifically used to prepare alkyl fluorides from alkyl halides using silver fluoride (\( {AgF} \)) as the fluorine source.
Step 3: The other reactions listed do not involve halide exchange in the same manner:
- The Finkelstein reaction involves the exchange of halides using sodium halides (e.g., NaCl or NaI).
- The Wurtz reaction is a coupling reaction of two alkyl halides with sodium metal.
- Sandmeyer's reaction involves the substitution of halides with aryl groups, using copper salts.
- Williamson’s reaction involves the preparation of ethers by reacting an alkoxide ion with an alkyl halide.
Thus, the correct answer is option (E), Swarts reaction.