A telescope with objective diameter \(R\) is used to observe a distant star emitting light of wavelength \(500\) nm, at a resolution of \(5 \times 10^{-7}\) radian. The value of \(R\) is _____ cm.

The limit of resolution of a telescope is given by the formula:

\(\Delta \theta = \dfrac{1.22 \lambda}{R}\)

where \(\lambda\) is the wavelength of light and \(R\) is the diameter of the objective lens.

Given:
\(\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m} = 5 \times 10^{-7} \text{ m}\)
\(\Delta \theta = 5 \times 10^{-7} \text{ rad}\)

Substituting the values into the formula:

\(5 \times 10^{-7} = \dfrac{1.22 \times 5 \times 10^{-7}}{R}\)

\(R = 1.22 \text{ m}\)

Converting to centimeters:

\(R = 1.22 \times 100 \text{ cm} = 122 \text{ cm}\)

Answer: \(122\)