Import Question JSON

Current Question (ID: 18927)

Question:

$\text{If } e \text{ is the electronic charge, } c \text{ is the speed of light in free space and } h \text{ is Planck's constant, the quantity } \frac{1}{4\pi\varepsilon_0} \left|\frac{e^2}{hc}\right| \text{ has dimensions of:}$

Options:

1. $[M^0L^0T^0]$
2. $[LT^{-1}]$
3. $[MLT^{-1}]$
4. $[MLT^0]$

Solution:

$\text{Hint: } [hc] = [E\lambda]$ $\text{Step 1: Don't find dimensions of each quantity individually}$ $\text{Step 2: Find dimensions of } \frac{e^2}{4\pi\varepsilon_0}$ $F = \left(\frac{1}{4\pi\varepsilon_0}\right) \frac{q^2}{r^2}$ $\left[\frac{e^2}{4\pi\varepsilon_0}\right] = Fr^2$ $\text{Step 3: Find the dimension of } hc$ $\frac{hc}{\lambda} = E$ $[hc] = [ML^3T^{-2}]$ $\text{Step 4: Find the dimension of the given expression}$ $\left[\frac{1}{4\pi\varepsilon_0} \times \frac{e^2}{hc}\right] = [M^0L^0T^0]$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}