**Solution**

Given,

Grating space (d) = ?,

Number of lines (N) per mm = 400 lines/mm = 400,000 lines/m,

\(d=\frac1N=\frac1{400000}=2.5\times10^{-6}\) m

Wavelength, \(\lambda=6000Å=6000\times10^{-10}\) m

We have,

\(d\sin\left(\theta\right)=n\lambda\)

So for the first order,

\(d\sin\left(\theta_1\right)=\lambda=6000\times10^{-10}\)

\(\Rightarrow\theta_1=13.85^\circ\)

For the maximum number of the diffraction maxima, θ = 90°

∴ \(n=\frac d\lambda=\frac{2.5\times10^{-6}}{6000\times10^{-10}}=4.17\approx4\)

Hence the maximum number of maxima grating obtained is 4.