Diffraction Grating Line Density Formula

Steps:

• Diffraction grating equation: $d \sin \theta = m \lambda$, where $d$ is line spacing and $m=2$ for second order.
• Line density $N = 1/d$, so $\sin \theta = m \lambda N$.
• Rearrange: $N = \frac{\sin \theta}{m \lambda}$.
• Substitute $m=2$: $N = \frac{\sin \theta}{2\lambda}$.

Why C is correct:

• Matches the grating equation $d \sin \theta = m \lambda$ with $d=1/N$ and $m=2$, yielding line density $N = \frac{\sin \theta}{2\lambda}$.

Why the others are wrong:

• A: Inverts formula and omits $m=2$, giving spacing instead of density.
• B: Includes $m=2$ but inverts to yield spacing, not density.
• D: Applies to first order ($m=1$), not second order.

Final answer: C