Certainly! Let's solve the problem in the requested **4-step format**. --- ### **Step 1: Introduction/Overview** In this problem, we are given a diffraction grating experiment setup. A 720 nm light source passes through a grating, and its diffraction pattern is projected onto a screen 0.85 m away. The bright spots (maxima) are sketched on a scale from -50.0 cm to +50.0 cm. We are to find the **density of lines per millimeter** on the diffraction grating. --- ### **Step 2: Relevant Formula and Identification of Variables** The position of the bright fringes (maxima) in a diffraction grating is given by: \[ d \sin \theta = m \lambda \] Where: - \( d \) = spacing between adjacent grating lines (grating constant) - \( \theta \) = angle from the central maximum to the m-th order maximum - \( m \) = order of the maximum (0, ±1, ±2, ...) - \( \lambda \) = wavelength of light (720 nm = \(7.20 \times 10^{-7}\) m) The distance \( y \) from the central maximum on the screen to the m-th order maximum is related by: \[ \tan \theta \approx \sin \theta \approx \frac{y}{L} \] (for small angles, which is reasonable here) Given: - Screen distance, \( L = 0.85 \) m - \( y_1 \) (distance from center to first order maximum) can be found from the sketch. From the image, the central maximum is at 0 cm, and first order maxima are at ±24 cm = 0.24 m. --- ### **Step 3: Calculation** #### **A. Find \(\sin \theta\) for the first order (\(m=1\)):** \[ \sin \theta \approx \tan \theta = \frac{y_1}{L} = \frac{0.24}{0.85} = 0.282 \] #### **B. Solve for \(d\):** \[ d \sin \theta = m \lambda \implies d = \frac{m \lambda}{\sin \theta} \] Using \(m = 1\): \[ d = \frac{1 \times 7.20 \times 10^{-7} \text{ m}}{0.282} = 2.553 \times 10^{-6} \text{ m} \] #### **C. Calculate line density (\(\frac{1}{d}\)), convert to lines/mm:** \[ 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \] \[ \text{lines per mm} = \frac{1}{d} \times 10^{-3} \] \[ \text{lines per mm} = \frac{1}{2.553 \times 10^{-6}} \times 10^{-3} \] \[ = 3.92 \times 10^{2} = 392 \text{ lines/mm} \] --- ### **Step 4: Final Answer** \[ \boxed{392} \] **Density of lines for the diffraction grating is \(392\) lines/mm.**