Why is ${B_2}$ paramagnetic while ${C_2}$ is diamagnetic?

Hint: The electron configuration of a substance can be used to identify its magnetic properties: The substance is paramagnetic if its electrons are unpaired, and diamagnetic if all of its electrons are paired.

Complete answer:
Paramagnetic molecules are those molecules in which unpaired electrons are present while diamagnetic molecules are those in which unpaired electrons are absent.
The molecular orbital electronic configuration of both ${B_2}$ and ${C_2}$ are:
\[{B_2}:{\text{ }}{\left[ {\sigma 1s} \right]^2}{\left[ {\sigma *1s} \right]^2}{\left[ {\sigma 2s} \right]^2}{\left[ {\sigma *2s} \right]^2}{\left[ {\pi 2{p_x}} \right]^1}{\left[ {\pi 2{p_y}} \right]^1}\]
\[{C_2}:{\text{ }}{\left[ {\sigma 1s} \right]^2}{\left[ {\sigma *1s} \right]^2}{\left[ {\sigma 2s} \right]^2}{\left[ {\sigma *2s} \right]^2}{\left[ {\pi 2{p_x}} \right]^2}{\left[ {\pi 2{p_y}} \right]^2}\]
As ${B_2}$ has two unpaired electrons, it is paramagnetic, while ${C_2}$ has no unpaired electrons, hence it is diamagnetic.
Hence, we can conclude that ${B_2}$ is paramagnetic while ${C_2}$ is diamagnetic.

Additional Information: Examples of Paramagnetic materials include magnesium, molybdenum, lithium, and tantalum. Examples of diamagnetic substances are bismuth, carbon, mercury, lead. The paramagnetic materials are attracted to a magnet only weakly. It indicates that under the influence of any applied magnetic field, paramagnetic materials are weakly attracted. These materials are unable to generate a magnetic field due to the paired electrons between the atoms. Any applied magnetic field repels or opposes diamagnetic materials.

Note:
Every substance in our environment has certain magnetic characteristics. In the presence of a magnetic field, different materials exhibit distinct properties. The electrons contained in the atoms or molecules give a substance its magnetic characteristics. Every electron in an atom acts as a little magnet. Electrons are also known as tiny current loops that retain their magnetic properties.