Consider an equilateral prism (refractive index sqrt2$\sqrt{2}$). A ray of light is incident on its one surface at a certain angle i. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to \_\_\_\_.

### Related Formula textMagnification m = fracD_textoutD_textin = fracf_2f_1$$\text{Magnification } m = \frac{D_{\text{out}}}{D_{\text{in}}} = \frac{f_2}{f_1}$$ ### Core Logic For a beam expander using two convex lenses, the lenses are arranged such that their focal points coincide. This creates an afocal system where parallel input rays remain parallel upon output.

By similar triangles at the focal point: tan theta = fracD_1 / 2f_1 = fracD_2 / 2f_2$$\tan \theta = \frac{D_1 / 2}{f_1} = \frac{D_2 / 2}{f_2}$$ ### Step 1: Solving for f2 fracD_1f_1 = fracD_2f_2$$\frac{D_1}{f_1} = \frac{D_2}{f_2}$$ frac240 = frac14f_2$$\frac{2}{40} = \frac{14}{f_2}$$ f_2 = 14 times frac402 = 14 times 20 = 280text mm$$f_2 = 14 \times \frac{40}{2} = 14 \times 20 = 280\text{ mm}$$ ### Pattern Recognition A standard beam expander (Keplerian telescope design used in reverse) has f_2/f_1 = D_2/D_1$f_2/f_1 = D_2/D_1$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula m approx fracLf_0 times fracDf_e$$m \approx \frac{L}{f_0} \times \frac{D}{f_e}$$ ### Core Logic For a compound microscope in normal adjustment (image formed at infinity), the magnifying power is given by the standard approximation: m simeq fracl Df_0 f_e$$m \simeq \frac{l D}{f_{0} f_{e}}$$ where l$l$ is the tube length, D = 25text cm$D = 25\text{ cm}$ is the least distance of distinct vision. ### Step 1: Substitute Values Given values: l = 32text cm$l = 32\text{ cm}$ f_0 = 2text cm$f_0 = 2\text{ cm}$ f_e = 4text cm$f_e = 4\text{ cm}$ D = 25text cm$D = 25\text{ cm}$ (standard assumption when not given) m = frac322 times frac254$$m = \frac{32}{2} \times \frac{25}{4}$$ m = 16 times frac254 = 4 times 25 = 100$$m = 16 \times \frac{25}{4} = 4 \times 25 = 100$$

### Pattern Recognition Compound microscope formulas: Normal adjustment to$\to$ image at infty$\infty$, m = (L/f_0)(D/f_e)$m = (L/f_0)(D/f_e)$. Image at near point D$D$, m = (L/f_0)(1 + D/f_e)$m = (L/f_0)(1 + D/f_e)$. Default to standard approximation when given tube length. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula A = r_1 + r_2$A = r_1 + r_2$ mu = fracsin isin r$$\mu = \frac{\sin i}{\sin r}$$ ### Core Logic

Since the incident ray is parallel to the base, looking at the geometry, the angle of incidence on the first face can be derived. The base angle is 45^circ$45^{\circ}$. Thus, the normal to the first surface makes an angle of 45^circ$45^{\circ}$ with the incident ray. Therefore, i_1 = 45^circ$i_1 = 45^{\circ}$. For the emergent ray to graze the second surface, the angle of emergence e = 90^circ$e = 90^{\circ}$. Applying Snell's Law at the second surface: mu sin r_2 = 1 cdot sin 90^circ$$\mu \sin r_2 = 1 \cdot \sin 90^{\circ}$$ sqrt2 sin r_2 = 1 implies sin r_2 = frac1sqrt2$$\sqrt{2} \sin r_2 = 1 \implies \sin r_2 = \frac{1}{\sqrt{2}}$$ r_2 = 45^circ$$r_2 = 45^{\circ}$$ ### Step 1: First Surface Analysis Applying Snell's Law at the incident surface: 1 cdot sin(45^circ) = mu sin(r_1)$$1 \cdot \sin(45^{\circ}) = \mu \sin(r_1)$$ frac1sqrt2 = sqrt2 sin(r_1)$$\frac{1}{\sqrt{2}} = \sqrt{2} \sin(r_1)$$ sin(r_1) = frac12 implies r_1 = 30^circ$$\sin(r_1) = \frac{1}{2} \implies r_1 = 30^{\circ}$$ ### Step 2: Prism Geometry The prism angle A$A$ at the apex is the sum of the internal angles of refraction: A = r_1 + r_2 = 30^circ + 45^circ = 75^circ$$A = r_1 + r_2 = 30^{\circ} + 45^{\circ} = 75^{\circ}$$ ### Step 3: Final Conclusion In the triangle of the prism, the sum of angles is 180^circ$180^{\circ}$. A + 45^circ + theta = 180^circ$$A + 45^{\circ} + \theta = 180^{\circ}$$ 75^circ + 45^circ + theta = 180^circ$$75^{\circ} + 45^{\circ} + \theta = 180^{\circ}$$ 120^circ + theta = 180^circ implies theta = 60^circ$$120^{\circ} + \theta = 180^{\circ} \implies \theta = 60^{\circ}$$ ### Pattern Recognition Grazing emergence means r_2 = theta_c$r_2 = \theta_c$ (critical angle). Horizontal incidence with a known base angle gives i_1$i_1$. Combining these through A = r_1 + r_2$A = r_1 + r_2$ resolves the full geometry. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula frac1f = frac1f_1 + frac1f_2 - fracdf_1 f_2, quad m = fracvu$$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}, \quad m = \frac{v}{u}$$ ### Core Logic This question was officially dropped by the examining authority. Full marks awarded to all candidates. ### Pattern Recognition Sees: Lens combination with separation gap introduced. Shortcut: Note official exam status (Dropped question). Check: Question dropped in final key. ✓ ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments