Question

What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light?

Short answer

The maximum number of lines is$1.28\times {10}^4$lines per centimeter.

Step-by-step solution

Description of the diffraction grating

The diffraction grating is made of equally spaced parallel lines, which can be considered as a slit. The lightwave when passes through these parallel slits result in interference patterns of different orders.

Evaluating the maximum number of lines

The number of slits is inversely proportional to the slit separation, of the value $\mathbf{d}$. So, for the sake of getting the maximum number of lines per centimeter, we are supposed to minimize the value $\mathbf{d}$. The equation of the constructive interference:

$\mathbf{\theta sin\lambda }$ reveals that the smaller the value $\mathbf{d}$, the smaller will be $\mathbf{\lambda }$.

Then, to produce a complete first-order spectrum for visible light, we are supposed to evaluate the minimum $\mathrm{\lambda }$has to belocalid="1654054017267" $780\mathrm{nm}$.

The value must be localid="1654054020898" $\mathrm{m}=1,\mathrm{and}90.0^{°}\mathrm{in}\mathrm{\theta sin\lambda }$, and we get,

localid="1654245378786" $$\begin{gathered} \mathrm{d}=\frac{\mathrm{m\lambda }}{\mathrm{sin\theta }} \\ =\frac{1\times 780\times {10}^{‐9}\mathrm{m}}{\sin \left(90.0^{°}\right)} \\ =7.8\times {10}^{‐7} \end{gathered}$$

The number of lines per cm is then evaluated as:

$$\begin{gathered} \mathrm{n}=\frac{1\mathrm{cm}}{\mathrm{d}} \\ =\left(\frac{1\mathrm{cm}}{7.8\times {10}^{‐7}\mathrm{m}}\right)\left(\frac{1\mathrm{m}}{100\mathrm{cm}}\right) \\ =1.28\times {10}^4 \end{gathered}$$

Therefore, the maximum number of lines is $1.28\times {10}^4$lines per centimeter.