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Current Question (ID: 20367)

Question:

$\text{A coil of radius } R \text{ centred at } O \text{ carries a current } i. \text{ Point } P \text{ is on the axis of coil at a distance } R \text{ from the centre } O \text{ as shown in the figure.}$ $\text{The ratio of the magnetic field at point } O \text{ to the magnetic field at point } P \text{ is equal to:}$

Options:

1. $2$
2. $2\sqrt{2}$
3. $1\sqrt{2}$
4. $\frac{1}{2\sqrt{2}}$

Solution:

$\text{Hint: } B_p = \frac{\mu_0 i R^2}{2 \left( x^2 + R^2 \right)^{3/2}}$ $\text{Step 1: Find the magnetic field at point } O \text{ and point } P.$ $\text{The magnetic field at the centre point } O \text{ is given by:}$ $B_O = \frac{\mu_0 i}{2R} \quad \cdots (1)$ $\text{The magnetic field at the point } P \text{ is given by:}$ $B_P = \frac{\mu_0 i R^2}{2 \left( x^2 + R^2 \right)^{3/2}} = \frac{\mu_0 i R^2}{2 \left( R^2 + R^2 \right)^{3/2}} = \frac{\mu_0 i}{4\sqrt{2} R} \quad \cdots (2)$ $\text{Step 2: Find the required ratio.}$ $\text{From equations (1) and (2) we get:}$ $\frac{B_O}{B_P} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$ $\text{Therefore, the ratio of the magnetic field at point } O \text{ to the magnetic field at point } P \text{ is } 2\sqrt{2}.$ $\text{Hence, option (2) is the correct answer.}$

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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
}