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Current Question (ID: 20043)

Question:

$\text{Two point charges, } Q \text{ each, are placed at a distance } d \text{ apart.}$ $\text{A third point charge } q \text{ is placed at a distance } x \text{ from the mid-point on the perpendicular bisector.}$ $\text{The value of } x \text{ at which charge } q \text{ will experience the maximum Coulomb's force is:}$

Options:

1. $x = d$
2. $x = \frac{d}{2}$
3. $x = \frac{d}{\sqrt{2}}$
4. $x = \frac{d}{2\sqrt{2}}$

Solution:

$\text{Hint: } F = \frac{k q_1 q_2}{r^2}$ $\text{Step 1: Draw a diagram showing the force acting on the charge } q.$ $\text{Step 2: Find the net force on charge } q.$ $F_{\text{net}} = 2F \cos \theta$ $\Rightarrow F_{\text{net}} = \frac{2kQq}{x^2 + d^2/4} \times \frac{x}{\sqrt{x^2 + d^2/4}}$ $\Rightarrow F_{\text{net}} = \frac{2kQqx}{(x^2 + d^2/4)^{3/2}}$ $\text{Step 3: Find the value of } x \text{ at which charge, experience maximum force.}$ $\text{For maximum force experienced by charge, } \frac{dF_{\text{net}}}{dx} = 0$ $\Rightarrow \frac{2kQq \left[ (x^2 + d^2/4)^{3/2} \times 1 - 3x^2 (x^2 + d^2/4)^{1/2} \right]}{(x^2 + d^2/4)^3} = 0$ $\Rightarrow x^2 + d^2/4 = 3x^2$ $\Rightarrow x = \frac{d}{2\sqrt{2}}$ $\text{Hence, option (4) is the correct answer.}$

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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
}