Step 1: Recall the definition of the Bohr magneton ($\mu_B$).
It is the natural unit of magnetic moment for an electron.
Step 2: The Bohr magneton in S.I. units is given by:
\[
\mu_B = \frac{e\hbar}{2m_e}
\]
Step 3: Its standard numerical value in S.I. units is:
\[
\mu_B = 9.274 \times 10^{-24} \ \text{J T}^{-1}
\]
Step 4: Among the given options, option (B) correctly represents the Bohr magneton in joule per tesla (J T$^{-1}$), which is the S.I. unit.