According to Bohr's theory, one of the postulates specifies that the angular momentum of an electron in orbit around a nucleus is quantized. This quantization can be expressed by the formula:

$$ L = n\frac{h}{2\pi} $$

Where:

• $L$ is the angular momentum of the electron,
• $n$ is the principal quantum number (or the orbit number in simpler terms), which can take positive integer values (1, 2, 3, ...),
• $h$ is Planck's constant ($6.62607015 \times 10^{-34} m^2 kg / s$), and
• $\frac{h}{2\pi}$ is often denoted as $\hbar$ (h-bar), known as the reduced Planck's constant.

For an electron in the 4th orbit ($n = 4$) of a hydrogen atom, we substitute $n = 4$ into the equation:

$$ L = 4\frac{h}{2\pi} $$

Therefore, the moment of momentum (or angular momentum) of an electron in the $4^{\text{th}}$ orbit of a hydrogen atom is:

$$ L = 4\frac{h}{2\pi} = 2\frac{2h}{2\pi} = 2\frac{h}{\pi} $$

Hence, the correct option is:

Option A: $2 \frac{h}{\pi}$