Consider two diffraction gratings. One grating has 3000 lines per cm, and the other one has 6000 lines per cm. Both gratings are illuminated with a beam of the same monochromatic light. Which grating produces the greater dispersion?

Both gratings produce the same dispersion.
The grating with 3000 lines produces the greater dispersion.
The grating with 6000 lines produces the greater dispersion.
The dispersion cannot be determined without additional information.

the 6000 lines/cm has the greatest dispersion

That is what the grating is all about !

but why

The dispersion produced by a diffraction grating depends on the spacing between its lines. The greater the spacing, the greater the dispersion. We are given that one grating has 3000 lines per cm (lp/cm) and the other grating has 6000 lp/cm.

To determine which grating produces greater dispersion, we need to compare their line spacings. The line spacing can be calculated by taking the reciprocal of the number of lines per unit length.

For the grating with 3000 lp/cm, the line spacing can be calculated as 1/(3000 lines/cm) = 0.00033 cm/line.
For the grating with 6000 lp/cm, the line spacing can be calculated as 1/(6000 lines/cm) = 0.00017 cm/line.

Comparing the line spacings, we can see that the grating with 3000 lp/cm has a larger line spacing than the grating with 6000 lp/cm. Therefore, the grating with 3000 lp/cm produces greater dispersion.

So, the correct answer is: The grating with 3000 lines per cm produces the greater dispersion.