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Current Question (ID: 17363)

Question:

$\text{An } LCR \text{ series circuit with } 100 \, \Omega \text{ resistance is connected to an AC source of } 200 \, \text{V} \text{ and an angular frequency of } 300 \, \text{rad/s.}$ $\text{When only the capacitance is removed, the current lags behind the voltage by } 60^\circ.$ $\text{When only the inductance is removed, the current leads the voltage by } 60^\circ.$ $\text{Calculate the power dissipated in the } LCR \text{ circuit.}$

Options:

1. $200 \, \text{W}$
2. $400 \, \text{W}$
3. $300 \, \text{W}$
4. $\text{zero}$

Solution:

$\text{Hint: Find the inductive and capacitive reactance.}$ $\text{Step 1: Find the inductive reactance.}$ $\tan 60^\circ = \frac{V_L}{V_R}$ $\sqrt{3} = \frac{X_L}{R}$ $\Rightarrow X_L = \sqrt{3}R$ $\text{Step 2: Find the capacitive reactance.}$ $\tan 60^\circ = \frac{V_C}{V_R}$ $\Rightarrow X_C = \sqrt{3}R$ $\text{Step 3: Find the power factor of the circuit.}$ $\cos \phi = \frac{R}{Z}$ $Z = \sqrt{R^2 + (X_L - X_C)^2} = R$ $\Rightarrow \cos \phi = 1$ $\text{Step 4: Find the power consumed.}$ $P = \frac{V_{\text{rms}}^2}{Z} \cos \phi$ $P = \frac{(200)^2}{100}$ $\Rightarrow P = 400 \, \text{W}$ $\text{Hence, option (2) is the correct answer.}$

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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
}