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Current Question (ID: 17543)

Question:

$\text{A liquid of refractive index } \frac{4}{3} \text{ is placed between two identical plano-convex lenses touching each other at their spherical surfaces of radius } R. \text{ If the refractive index of the lens is } 1.50, \text{ then the lens behaves as:}$

Options:

1. $\text{a convergent with power } P = \frac{1}{3R}$
2. $\text{a convergent with power } P = \frac{1}{6R}$
3. $\text{a divergent with power } P = \frac{1}{3R}$
4. $\text{a divergent with power } P = \frac{1}{6R}$

Solution:

$\text{The focal length of the lenses of glass:}$ $\frac{1}{f_g} = \left(1.5 - 1\right) \left[\frac{1}{\infty} - \frac{1}{-R}\right]$ $= \frac{0.5}{R} = \frac{1}{2R}$ $f_g = 2R$ $\text{The focal length of the lens of liquid:}$ $\frac{1}{f_L} = \left[\frac{4}{3} - 1\right] \left[-\frac{1}{R} - \frac{1}{R}\right]$ $= \frac{-2}{3R}$ $f_L = \frac{-3R}{2}$ $\text{Net effective focal length:}$ $\frac{1}{f} = \frac{2}{f_g} + \frac{1}{f_L} = \frac{1}{R} - \frac{2}{3R} = \frac{1}{3R}$ $P_{net} = \frac{1}{f} = \frac{1}{3R} \text{ (converging)}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}