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Current Question (ID: 8734)

Question:

$\text{Planck's constant (h), speed of light in the vacuum (c), and Newton's gravitational constant (G) are the three fundamental constants. Which of the following combinations of these has the dimension of length?}$

Options:

1. $\frac{\sqrt{hG}}{c^{3/2}}$
2. $\frac{\sqrt{hG}}{c^{5/2}}$
3. $\frac{\sqrt{hG}}{G}$
4. $\frac{\sqrt{Gc}}{h^{3/2}}$

Solution:

$\text{Hint: Recall the dimensions of physical quantities.}$ $\text{Step: Find the dimensions of the length.}$ $\text{In terms of h, c, and G length can be expressed as,}$ $L = (h)^p(c)^q(G)^r$ $\text{Writing dimensions on both sides, we get:}$ $[M^0LT^0] = [ML^2T^{-1}]^p[LT^{-1}]^q[M^{-1}L^3T^{-2}]^r$ $= [M^{p-r}L^{2p+q+3r}T^{-p-q-2r}]$ $\text{On comparing the powers of M, L, and T on both sides, we get;}$ $p - r = 0, \quad 2p + q + 3r = 1 \quad \text{and} \quad -p - q - 2r = 0$ $\text{On solving, we get;}$ $p = r = \frac{1}{2} \quad \text{and} \quad q = -\frac{3}{2}$ $\text{Therefore, the dimensions of length are,}$ $L = (h)^{1/2}(c)^{-3/2}(G)^{1/2} = \frac{\sqrt{hG}}{c^{3/2}}$ $\text{Hence, option (1) is the correct answer.}$

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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
}