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Current Question (ID: 20421)

Question:

$\text{Two long parallel conductors } S_1 \text{ and } S_2 \text{ are separated by a distance of } 10 \text{ cm and carrying currents of } 4 \text{ A and } 2 \text{ A respectively.}$ $\text{The conductors are placed along the x-axis in the X-Y plane.}$ $\text{There is a point P located between the conductors (as shown in the figure).}$ $\text{A charged particle of } 3\pi \text{ coulomb is passing through the point P with velocity}$ $\vec{v} = (2\hat{i} + 3\hat{j}) \text{ m/s; where } \hat{i} \& \hat{j} \text{ represents unit vectors along the x \& y-axis respectively.}$ $\text{The force acting on the charged particle is } 4\pi \times 10^{-5} (-x\hat{i} + 2\hat{j}) \text{ N.}$ $\text{The value of } x \text{ is:}$

Options:

1. $2$
2. $1$
3. $3$
4. $-3$

Solution:

$\text{The force on a charged particle moving in a magnetic field is given by } \vec{F} = q(\vec{v} \times \vec{B}).$ $\text{Given } \vec{F} = 4\pi \times 10^{-5} (-x\hat{i} + 2\hat{j}) \text{ N and } q = 3\pi \text{ C.}$ $\text{The velocity } \vec{v} = (2\hat{i} + 3\hat{j}) \text{ m/s.}$ $\text{Using the cross product, equate the components to solve for } x.$ $\text{The correct value of } x \text{ is } 3.$

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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
}