### Related Formula frac1f = left(fracmu_gmu_m - 1right)left(frac1R_1 - frac1R_2right)$\frac{1}{f} = \left(\frac{\mu_g}{\mu_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ ### Core Logic In air (mu_m = 1$\mu_m = 1$): frac124 = (1.5 - 1) cdot K = 0.5 K implies K = frac112 quad dots (i)$\frac{1}{24} = (1.5 - 1) \cdot K = 0.5 K \implies K = \frac{1}{12} \quad \dots (i)$ In water (mu_m = 1.33 = frac43$\mu_m = 1.33 = \frac{4}{3}$): frac1f' = left(frac1.54/3 - 1right) cdot K = left(frac4.54 - 1right) cdot K = frac18 K quad dots (ii)$\frac{1}{f'} = \left(\frac{1.5}{4/3} - 1\right) \cdot K = \left(\frac{4.5}{4} - 1\right) \cdot K = \frac{1}{8} K \quad \dots (ii)$ Dividing equation (i)$(i)$ by equation (ii)$(ii)$: fracf'24 = frac0.51/8 = 4$\frac{f'}{24} = \frac{0.5}{1/8} = 4$ f' = 24 times 4 = 96mathrm~cm$f' = 24 \times 4 = 96\mathrm{~cm}$ ### Pattern Recognition Standard relation for standard glass lens (mu=1.5$\mu=1.5$) immersed in water (mu=4/3$\mu=4/3$): the focal length always becomes exactly 4 times its original value in air (f_textwater = 4 f_textair$f_{\text{water}} = 4 f_{\text{air}}$). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula sin mathrmc = frac1mu$\sin \mathrm{c} = \frac{1}{\mu}$ ### Core Logic To see the coin from edge point mathrmE$\mathrm{E}$ at grazing emergency, the light ray travelling from mathrmO$mathrm{O}$ to mathrmE$mathrm{E}$ must strike the flat upper boundary at the critical angle.

Given the hemispherical layout geometry, the ray's angle of incidence at the center of the surface plane satisfies: theta = mathrmc = 45^circ$\theta = \mathrm{c} = 45^{\circ}$ Substituting this value into the critical value expression: mu = frac1sin 45^circ = sqrt2$\mu = \frac{1}{\sin 45^{\circ}} = \sqrt{2}$ ### Step 1: Final Conclusion The minimum refractive index required is sqrt2$\sqrt{2}$, matching option (3). ### Pattern Recognition Grazing boundary ray vision configurations dictate evaluating the specific systemic geometric configuration to compute the critical boundary angle. Here, the radius profile fixes theta = 45^circ$\theta = 45^{\circ}$ deterministically. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula delta = (mu - 1)mathrmA$\delta = (\mu - 1)\mathrm{A}$ ### Core Logic To achieve dispersion without deviation, the net deviation produced by the prism combination must be zero: delta_textnet = 0 implies (mu_1 - 1)mathrmA_1 - (mu_2 - 1)mathrmA_2 = 0$\delta_{\text{net}} = 0 \implies (\mu_1 - 1)\mathrm{A}_1 - (\mu_2 - 1)\mathrm{A}_2 = 0$ Substituting the given parameters into the equation: (1.54 - 1) cdot 4^circ - (1.72 - 1)mathrmA_2 = 0$(1.54 - 1) \cdot 4^{\circ} - (1.72 - 1)\mathrm{A}_2 = 0$ 0.54 cdot 4 = 0.72 cdot mathrmA_2$0.54 \cdot 4 = 0.72 \cdot \mathrm{A}_2$ mathrmA_2 = frac2.160.72 = 3^circ$\mathrm{A}_2 = \frac{2.16}{0.72} = 3^{\circ}$ ### Step 1: Final Angle Value The required angle for the second thin prism is 3^circ$3^{\circ}$, which matches option (2). ### Pattern Recognition For zero deviation conditions using thin components, balance the deviation equations directly: (mu-1)mathrmA = (mu'-1)mathrmA'$(\mu-1)\mathrm{A} = (\mu'-1)\mathrm{A}'$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula Lens Maker's Formula: frac1f = (mu - 1) left(frac1R_1 - frac1R_2right)$\frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ where, f$f$ = focal length of the lens, mu$\mu$ = refractive index of the material, R_1, R_2$R_1, R_2$ = radii of curvature with standard Cartesian sign convention. ### Core Logic For a thin bi-convex lens, using standard coordinate conventions: - R_1 = +10mathrm~cm$R_1 = +10\mathrm{~cm}$ (positive since first surface centers to the right of light trajectory), - R_2 = -15mathrm~cm$R_2 = -15\mathrm{~cm}$ (negative since second surface centers to the left), - Focal length f = +12mathrm~cm$f = +12\mathrm{~cm}$. ### Step 1: Substituting in the Equation Substitute the values into Lens Maker's formula: frac112 = (mu - 1) left(frac110 - frac1-15right)$\frac{1}{12} = (\mu - 1) \left(\frac{1}{10} - \frac{1}{-15}\right)$ frac112 = (mu - 1) left(frac110 + frac115right)$\frac{1}{12} = (\mu - 1) \left(\frac{1}{10} + \frac{1}{15}\right)$ frac112 = (mu - 1) left(frac3 + 230right)$\frac{1}{12} = (\mu - 1) \left(\frac{3 + 2}{30}\right)$ frac112 = (mu - 1) left(frac530right) = (mu - 1) left(frac16right)$\frac{1}{12} = (\mu - 1) \left(\frac{5}{30}\right) = (\mu - 1) \left(\frac{1}{6}\right)$ mu - 1 = frac612 = 0.5 implies mu = 1.5$\mu - 1 = \frac{6}{12} = 0.5 \implies \mu = 1.5$ ### Pattern Recognition Convex lenses always have opposite signs for R_1$R_1$ and R_2$R_2$. The term left(frac1R_1 - frac1R_2right)$\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ is additive: left(frac1|R_1| + frac1|R_2|right)$\left(\frac{1}{|R_1|} + \frac{1}{|R_2|}\right)$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments

### Related Formula For a prism of angle A$A$: - Deviation: delta = i + e - A$\delta = i + e - A$ - Minimum deviation delta_textmin$\delta_{\text{min}}$ occurs when: i = e quad textand quad r_1 = r_2 = fracA2$i = e \quad \text{and} \quad r_1 = r_2 = \frac{A}{2}$ - Under minimum deviation, the ray inside an equilateral/isosceles prism travels symmetrically, making it parallel to the prism base. ### Core Logic Let's check the validity of each statement: - **Statement (A)**: The refracted ray inside the prism becomes parallel to the base at minimum deviation. (True for symmetric prisms) - **Statement (B)**: Larger angle prisms provide smaller minimum deviation. By minimum deviation equation: mu = fracsinleft(fracA+delta_textmin2right)sinleft(fracA2right)$\mu = \frac{\sin\left(\frac{A+\delta_{\text{min}}}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ As A$A$ increases, delta_textmin$\delta_{\text{min}}$ generally increases, not decreases. (False) - **Statement (C)**: Angle of incidence i$i$ and angle of emergence e$e$ become equal (i = e$i = e$) during the minimum deviation state. (True) - **Statement (D)**: The delta-i$\delta-i$ curve is asymmetric and parabolic-like; for any deviation delta > delta_textmin$\delta > \delta_{\text{min}}$, there are always exactly two different incident angles (i$i$ and e$e$) that yield the same deviation, except at the minimum deviation point (which has a single unique value). (True) - **Statement (E)**: Angle of refraction r = A/2$r = A/2$, which is half of the prism angle, not double. (False) ### Step 1: Conclusion Since statements A, C, and D are true, the correct option is (1). ### Pattern Recognition Review the classic parabolic shape of the deviation vs. angle of incidence (delta-i$\delta-i$) graph. Notice that any horizontal line above the minimum point intersects twice (representing i$i$ and e$e$ for that deviation). Minimum deviation is the unique local minimum, where i = e$i = e$ and r_1 = r_2 = A/2$r_1 = r_2 = A/2$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics and Optical Instruments: Refraction through Prism