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Current Question (ID: 20020)

Question:

$\text{An electric dipole with dipole moment } \vec{p}, \text{ inclined at an angle } \theta \text{ to the } x\text{-axis, experiences a torque } \vec{\tau}_1 = \tau \hat{k} \text{ in the field } \vec{E}_1 = E \hat{i}, \text{ and a torque } \vec{\tau}_2 = -\vec{\tau}_1 \text{ in the field } \vec{E}_2 = \sqrt{3}E \hat{j}. \text{ The angle } \theta \text{ is:}$

Options:

1. $30^\circ$
2. $45^\circ$
3. $60^\circ$
4. $90^\circ$

Solution:

$\text{Hint: } \vec{\tau} = \vec{p} \times \vec{E}$ $\text{Step 1: Find the torque when electric field is directed along x-axis.}$ $\vec{\tau}_1 = \vec{p} \times \vec{E}_1$ $\Rightarrow \vec{\tau}_1 = pE \sin \theta (-\hat{k})$ $\text{Step 2: Find the torque when electric field is directed along y-axis.}$ $\vec{\tau}_2 = \vec{p} \times \vec{E}_2$ $\Rightarrow \vec{\tau}_2 = p \times \sqrt{3}E \sin(90^\circ - \theta)(\hat{k})$ $\Rightarrow \vec{\tau}_2 = \sqrt{3}pE \cos \theta(\hat{k})$ $\text{Step 3: Find the value of angle } \theta.$ $\vec{\tau}_2 = -\vec{\tau}_1$ $\Rightarrow \sqrt{3}pE \cos \theta(\hat{k}) = -pE \sin \theta(-\hat{k})$ $\Rightarrow \tan \theta = \sqrt{3}$ $\Rightarrow \theta = 60^\circ$ $\text{Hence, option (3) is the correct answer.}$

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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
}