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Current Question (ID: 16810)

Question:

$\text{Two identical current-carrying coaxial loops carry current } I \text{ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as } C,$ $\begin{array}{ll} (a) & \oint \mathbf{B} \cdot d\mathbf{l} = \pm 2\mu_0 I \\ (b) & \text{the value of } \oint \mathbf{B} \cdot d\mathbf{l} \text{ is independent of the sense of } C. \\ (c) & \text{there may be a point on } C \text{ where } \mathbf{B} \text{ and } d\mathbf{l} \text{ are perpendicular.} \\ (d) & \mathbf{B} \text{ vanishes everywhere on } C. \end{array}$ $\text{Which of the above statements is correct?}$

Options:

1. $(a) \text{ and } (b)$
2. $(a) \text{ and } (c)$
3. $(b) \text{ and } (c)$
4. $(c) \text{ and } (d)$

Solution:

$\text{Hint: The magnetic field depends on the current.}$ $\text{Explanation: Consider a simple amperian loop passing once through both the identical current-carrying coaxial loops.}$ $(i) \text{ According to Ampere circuital law, } \oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0(I - I) = 0. \text{ Hence, option (a) is wrong.}$ $(ii) \text{ As } \oint_C \mathbf{B} \cdot d\mathbf{l} = 0, \text{ therefore } \oint_C \mathbf{B} \cdot d\mathbf{l} \text{ is independent of the sense of } C. \text{ Thus option (b) is correct.}$ $(iii) \text{ There will be a point on loop } C, \text{ lying at the axis of two loops } A \text{ and } B, \text{ where } \mathbf{B} \text{ and } d\mathbf{l} \text{ are perpendicular to each other. Thus option (c) is correct.}$ $(iv) \text{ The value of } \mathbf{B} \text{ does not vanish on various points of } C. \text{ Hence, option (d) is the correct answer.}$

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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
}