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Current Question (ID: 16749)

Question:

$\text{An element } \Delta l = \Delta x \hat{i} \text{ is placed at the origin and carries a large current of } I = 10 \text{ A (as shown in the figure). What is the magnetic field on the } y\text{-axis at a distance of } 0.5 \text{ m?}$ $\text{(Given } \Delta x = 1 \text{ cm)}$

Options:

1. $6 \times 10^{-8} \text{ T}$
2. $4 \times 10^{-8} \text{ T}$
3. $5 \times 10^{-8} \text{ T}$
4. $5.4 \times 10^{-8} \text{ T}$

Solution:

$\text{Hint: Use Biot-Savart's law.}$ $\text{Step: Find the magnetic field on the } y\text{-axis at a distance of } 0.5 \text{ m.}$ $\text{The magnetic field on the } y\text{-axis at a distance of } 0.5 \text{ m is given by:}$ $dB = \frac{\mu_0}{4\pi} \left( \frac{Idl \sin \theta}{r^2} \right)$ $dl = \Delta x = 10^{-2} \text{ m}, \ I = 10 \text{ A}, \ r = 0.5 \text{ m} = y, \ \frac{\mu_0}{4\pi} = 10^{-7} \text{ Tm/A}$ $\theta = 90^\circ$ $\Rightarrow dB = \frac{10^{-7} \times 10 \times 10^{-2}}{25 \times 10^{-2}}$ $\Rightarrow dB = 4 \times 10^{-8} \text{ T}$ $\text{The direction of the field is in the } +z\text{-direction.}$ $\text{Hence, option (2) is the correct answer.}$

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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
}