Import Question JSON

Current Question (ID: 20109)

Question:

$\text{A solid conducting sphere, having a charge } Q, \text{ is surrounded by an uncharged conducting hollow spherical shell.}$ $\text{Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be } V.$ $\text{If the shell is now given a charge of } -4Q, \text{ the new potential difference between the same two surfaces is:}$

Options:

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

Solution:

$\text{Hint: Shell will produce constant potential. Thus new potential difference } = V$ $\text{Step: Find the new potential difference between the same two surfaces.}$ $\text{Let a solid sphere of radius } R_1 \text{ have a charge } Q \text{ and it is surrounded by an uncharged conducting hollow spherical shell of radius } R_2.$ $\text{The potential difference between a sphere and a spherical shell is given by:}$ $\Delta V = kQ \text{in} \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$ $\text{The inner surface of the hollow shell will still have a charge of } -Q, \text{ induced by the charge } Q \text{ on the solid sphere.}$ $\text{The outer surface of the hollow shell will now have a charge of } -4Q + Q = -3Q.$ $\text{The potential difference remains proportional to the original charge (i.e., charge on the conducting sphere) because the geometry and charges balance symmetrically.}$ $\text{Therefore, the new potential difference between the same two surfaces is } V.$ $\text{Hence, option (3) is the correct answer.}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}