Step 1: Reaction of phenol with NaOH.
Phenol reacts with NaOH to form sodium phenoxide.
\[
\mathrm{C_6H_5OH + NaOH \rightarrow C_6H_5ONa + H_2O}
\] Step 2: Reaction with methyl iodide.
Sodium phenoxide undergoes Williamson ether synthesis with CH\(_3\)I.
\[
\mathrm{C_6H_5ONa + CH_3I \rightarrow C_6H_5OCH_3 + NaI}
\] Step 3: Identify product B.
The product formed is anisole (methoxybenzene). Step 4: Conclusion.
Product B is anisole.