Step 1: Understanding the Concept:
Cloning vectors are DNA molecules used to carry foreign genetic material into a host cell for replication or expression.
Different vectors are engineered with distinct cloning capacities depending on their structural stability and host system.
Step 2: Detailed Explanation:
Let us compare the maximum cloning capacities of the listed vectors:
1. Plasmids: These are small, circular, extra-chromosomal DNA molecules found in bacteria.
They are easy to manipulate but display limited stability with larger inserts, carrying a maximum foreign DNA insert size of approximately $10\text{ to }15 \text{ kb}$.
2. Cosmids: These are hybrid vectors containing plasmid sequences and the cohesive (*cos*) sites of bacteriophage lambda.
They are packaged into phage heads, carrying an insert capacity of $30\text{ to }45 \text{ kb}$.
3. Bacterial Artificial Chromosomes (BACs): These are low-copy-number vectors based on the F-plasmid of *E. coli*.
They are highly stable and can carry foreign DNA inserts ranging from $100\text{ to }300 \text{ kb}$.
4. Yeast Artificial Chromosomes (YACs): These are linear vectors engineered with yeast telomeres, a centromere, and an origin of replication, allowing them to replicate as artificial chromosomes inside yeast cells.
They possess the largest cloning capacity of any standard vector system, carrying inserts ranging from $100\text{ to }1000 \text{ kb}$ (and occasionally up to $2000 \text{ kb}$).
Therefore, YACs can carry the longest piece of foreign DNA.
Step 3: Final Answer:
The vector that can carry the longest piece of foreign DNA is YAC, which corresponds to option (B).