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Current Question (ID: 17311)

Question:

$\text{An AC source of variable frequency } f \text{ is connected to an } LCR \text{ series circuit.}$ $\text{Which of the following graphs represents the variation of the current } I \text{ in the circuit with frequency } f?$

Options:

1. $\text{Graph 1}$
2. $\text{Graph 2}$
3. $\text{Graph 3}$
4. $\text{Graph 4}$

Solution:

$\text{At the resonant frequency, } f_r, X_L = X_C \Rightarrow Z = R \text{ (min) and the current is max.}$ $\text{When } 0 < f < f_r; X_L = 2\pi f L \text{ increases from 0 and } X_C = \frac{1}{2\pi f C} \text{ decreases from } \infty \text{ and they become equal to each other at } f = f_r \Rightarrow \text{current will increase to its maximum value.}$ $\text{So, } z = \sqrt{R^2 + (X_L - X_C)^2} \text{ decreases for } 0 < f < f_r.$ $\text{When } f_r < f < \infty; X_L = 2\pi f L \text{ increases further and } X_C = \frac{1}{2\pi f C} \text{ after they become equal to each other.}$ $\text{So, } z = \sqrt{R^2 + (X_L - X_C)^2} \text{ increases.}$ $\Rightarrow \text{current will decrease from its maximum value.}$

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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
}