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Current Question (ID: 16982)

Question:
$A \ 1 \ \text{m} \ \text{long} \ \text{metallic} \ \text{rod} \ \text{is} \ \text{rotating} \ \text{with} \ \text{an} \ \text{angular} \ \text{frequency} \ \text{of} \ 400 \ \text{rad/s} \ \text{about} \ \text{an} \ \text{axis} \ \text{normal} \ \text{to} \ \text{the} \ \text{rod} \ \text{passing} \ \text{through} \ \text{its} \ \text{one} \ \text{end.}$ $\text{The} \ \text{other} \ \text{end} \ \text{of} \ \text{the} \ \text{rod} \ \text{is} \ \text{in} \ \text{contact} \ \text{with} \ \text{a} \ \text{circular} \ \text{metallic} \ \text{ring.}$ $\text{A} \ \text{constant} \ \text{and} \ \text{uniform} \ \text{magnetic} \ \text{field} \ \text{of} \ 0.5 \ \text{T} \ \text{parallel} \ \text{to} \ \text{the} \ \text{axis} \ \text{exists} \ \text{everywhere.}$ $\text{The} \ \text{emf} \ \text{induced} \ \text{between} \ \text{the} \ \text{centre} \ \text{and} \ \text{the} \ \text{ring} \ \text{is:}$
Options:
  • 1. $200 \ \text{V}$
  • 2. $100 \ \text{V}$
  • 3. $50 \ \text{V}$
  • 4. $150 \ \text{V}$
Solution:
$\text{Hint:} \ \text{One} \ \text{end} \ \text{of} \ \text{the} \ \text{rod} \ \text{has} \ \text{zero} \ \text{linear} \ \text{velocity,} \ \text{while} \ \text{the} \ \text{other} \ \text{end} \ \text{has} \ \text{a} \ \text{linear} \ \text{velocity} \ \text{of} \ \omega l$ $\text{Step} \ 1: \ \text{Calculate} \ \text{emf} \ \text{induced} \ \text{in} \ \text{the} \ \text{rod.}$ $\text{Consider} \ \text{the} \ \text{rod} \ \text{is} \ \text{consisting} \ \text{of} \ \text{so} \ \text{many} \ \text{very} \ \text{small} \ \text{rods.}$ $\text{Consider} \ \text{one} \ \text{very} \ \text{small} \ \text{rod} \ \text{of} \ \text{length} \ dx \ \text{after} \ x.$ $\text{velocity} \ \text{of} \ \text{small} \ \text{rod} \ \text{of} \ \text{length} \ dx \ \text{perpendicular}$ $v = x \omega$ $\text{Emf} \ \text{induced} \ \text{in} \ \text{the} \ \text{small} \ \text{rod} \ \text{of} \ dx \ \text{length}$ $d\varepsilon = B(x\omega) \ dx$ $\text{So} \ \text{emf} \ \text{induced} \ \text{in} \ \text{the} \ \text{full} \ \text{rod} \ \text{of} \ \text{length} \ l$ $\int_{0}^{\varepsilon_{AB}} d\varepsilon = B\omega \int_{0}^{1} x \ dx$ $\varepsilon_{AB} = \frac{B\omega l^2}{2}$ $\text{Step} \ 2: \ \text{Put} \ \text{the} \ \text{values} \ \text{and} \ \text{calculate} \ \text{emf} \ \text{induced} \ \text{between} \ \text{centre} \ \text{of} \ \text{the} \ \text{ring.}$ $\varepsilon_{AB} = \frac{0.5 \times 400 \times 1^2}{2} = 100 \ \text{V}.$

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
}