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Current Question (ID: 16611)

Question:

$\text{A copper wire of length } 10 \text{ m and radius } \left(10^{-2}/\sqrt{\pi}\right) \text{ m has an electrical resistance of } 10 \, \Omega. \text{ The current density in the wire for an electric field strength of } 10 \, (\text{V/m}) \text{ is:}$

Options:

1. $10^5 \, \text{A/m}^2$
2. $10^4 \, \text{A/m}^2$
3. $10^6 \, \text{A/m}^2$
4. $10^{-5} \, \text{A/m}^2$

Solution:

$\text{Hint: } \vec{J} = \sigma \vec{E}$ $\text{Step 1: Find the conductivity of the wire.}$ $\text{The conductivity of the wire is given by:}$ $\sigma = \frac{1}{\rho} = \frac{l}{RA}$ $\Rightarrow \sigma = \frac{l}{RA} = \frac{10}{10 \times \pi \times \left(\frac{10^{-2}}{\sqrt{\pi}}\right)^2}$ $\Rightarrow \sigma = \frac{1}{\pi \times \frac{10^{-4}}{\pi}} = 10^4 \, \Omega^{-1} \text{m}^{-1}$ $\text{Step 2: Find the current density in the wire.}$ $\text{The current density in the wire is given by: } J = \sigma E$ $\Rightarrow J = \sigma E = 10^4 \times 10 = 10^5 \, \text{A/m}^2$ $\text{Hence, option (1) is the correct answer.}$

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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
}