1. **Problem 23:** Calculate the number of maxima observed with a diffraction grating. Given: - Wavelength $\lambda = 5.8 \times 10^{-7}$ m - Number of lines per meter $N = 400000$ 2. Calculate the grating spacing $d$: $$d = \frac{1}{N} = \frac{1}{400000} = 2.5 \times 10^{-6} \text{ m}$$ 3. The diffraction maxima condition is: $$d \sin \theta = n \lambda$$ where $n$ is the order of the maximum. 4. The maximum order $n_{max}$ occurs when $\sin \theta = 1$: $$n_{max} = \frac{d}{\lambda} = \frac{2.5 \times 10^{-6}}{5.8 \times 10^{-7}}$$ 5. Calculate $n_{max}$: $$n_{max} = \frac{2.5}{0.58} \approx 4.31$$ Since $n$ must be an integer, the maximum observable order is $n = 4$. **Answer for problem 23:** 4 maxima. --- 1. **Problem 24:** Find the gravitational field strength at the surface of planet P. Given: - Diameter of planet P is one-third of Earth: $D_P = \frac{1}{3} D_E$ - Mass of Earth is 18 times that of P: $M_E = 18 M_P$ - Gravitational field strength at Earth surface: $g$ 2. Gravitational field strength formula: $$g = \frac{GM}{R^2}$$ where $G$ is gravitational constant, $M$ is mass, and $R$ is radius. 3. Radius of planet P: $$R_P = \frac{D_P}{2} = \frac{1}{3} \times \frac{D_E}{2} = \frac{1}{3} R_E$$ 4. Gravitational field strength at P: $$g_P = \frac{G M_P}{R_P^2}$$ 5. Substitute $M_P = \frac{M_E}{18}$ and $R_P = \frac{R_E}{3}$: $$g_P = \frac{G \frac{M_E}{18}}{\left(\frac{R_E}{3}\right)^2} = \frac{G M_E / 18}{R_E^2 / 9} = \frac{G M_E}{R_E^2} \times \frac{9}{18} = g \times \frac{1}{2}$$ **Answer for problem 24:** $\frac{g}{2}$.