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Current Question (ID: 16515)

Question:

$\text{The electric potential } V \text{ at any point } (x, y, z), \text{ all in meters in space is given by } V = 4x^2 \text{ volt.}$ $\text{The electric field at the point } (1, 0, 2) \text{ is:}$

Options:

1. $8 \, \text{V/m}, \text{ along the negative } x\text{-axis}$
2. $8 \, \text{V/m}, \text{ along the positive } x\text{-axis}$
3. $16 \, \text{V/m}, \text{ along the negative } x\text{-axis}$
4. $16 \, \text{V/m}, \text{ along the positive } x\text{-axis}$

Solution:

$\text{Hint: } \vec{E} = - \left[ \hat{i} \frac{\partial V}{\partial x} + \hat{j} \frac{\partial V}{\partial y} + \hat{k} \frac{\partial V}{\partial z} \right]$ $\text{Step: Find the electric field at the point } (1, 0, 2)$ $\vec{E} = - \left[ \hat{i} \frac{\partial V}{\partial x} + \hat{j} \frac{\partial V}{\partial y} + \hat{k} \frac{\partial V}{\partial z} \right]$ $\vec{E} = - [\hat{i} (8x)]$ $\Rightarrow \vec{E}_{(1,0,2)} = -8 \hat{i} \, \text{V/m},$ $\text{So the electric field is } 8 \, \text{V/m}, \text{ along the negative } x\text{-axis.}$ $\text{Hence, option (1) is the correct answer.}$

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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
}