Diffraction grating: A diffraction grating has 450 lines per mm. What is the highest order m that contains the entire visible spectrum from 400 nm to 700 nm?
A) m = 2
B) m = 3
C) m = 4
D) m = 5
E) m = 6

Question

Quizplus · Accepted Answer

The highest order m that contains the entire visible spectrum can be found using the grating equation for diffraction maxima, $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the diffraction angle, $m$ is the order of the spectrum, and $\lambda$ is the wavelength of light. The grating spacing $d$ can be found from the given 450 lines per mm, which is equivalent to $d = \frac{1}{450} mm = \frac{1}{450,000} m$. To ensure the entire visible spectrum (400 nm to 700 nm) is within the diffraction limit, we use the condition for the maximum possible order, which occurs at $\sin\theta = 1$, leading to $d = m\lambda$. For the longest wavelength (700 nm or 700x10^-9 m) to fit within an order, we solve for $m$ using the maximum $d$ value: $m = \frac{d}{\lambda} = \frac{1}{450,000} / 700x10^{-9} \approx 3.2$. Since $m$ must be an integer, the highest possible order that can contain the entire visible spectrum is $m = 3$.

Diffraction Grating: a Diffraction Grating Has 450 Lines Per Mm