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Step-by-Step Solution
Step 1: List the given data
• Frequency: $f = 750 \text{ Hz}$
• RMS voltage: $V_{\text{rms}} = 20 \text{ V}$
• Resistance: $R = 100\,\Omega$
• Inductance: $L = 0.1803 \text{ H}$
• Capacitance: $C = 10\,\mu F = 10 \times 10^{-6} \text{ F}$
• Heat capacity of resistor: $S = 2\, \text{J/}^\circ\text{C}$
• Temperature rise needed: $\Delta \theta = 10^\circ\text{C}$
Step 2: Calculate inductive and capacitive reactance
The angular frequency is
$$\omega = 2\pi f.$$
So,
$$X_L = \omega L = 2\pi f L, \quad X_C = \frac{1}{\omega C}.$$
Substitute the values:
$\omega = 2 \times \pi \times 750 \,\text{rad/s}$
$X_L = 2\pi \times 750 \times 0.1803 \,\Omega$
$X_C = \frac{1}{2\pi \times 750 \times 10 \times 10^{-6}} \,\Omega$
Step 3: Determine the total impedance
The net reactance is
$$(X_L - X_C),$$
and the magnitude of the impedance $|Z|$ of the series circuit is given by:
$$
|Z| = \sqrt{R^2 + (X_L - X_C)^2}.
$$
Numerically, it turns out (as shown in the reference solution) to be:
$$
|Z| \approx 834 \,\Omega.
$$
Step 4: Find the current in the circuit
In an AC circuit, the RMS current is
$$
i_{\text{rms}} = \frac{V_{\text{rms}}}{|Z|}.
$$
Here,
$$
i_{\text{rms}} = \frac{20}{834} \text{ A}.
$$
Step 5: Calculate the power dissipated in the resistor
Only the resistor dissipates power as heat. The power dissipated in the resistor can be written in terms of current and resistance or by using the power factor:
1. Using $i_{\text{rms}}$ and $R$:
$$
P = i_{\text{rms}}^2 \, R.
$$
2. Or using $P = (i_{\text{rms}}\, V_{\text{rms}}) \cos \phi$ and recalling $\cos \phi = \frac{R}{|Z|}$, both yield the same result:
$$
P = \left(\frac{V_{\text{rms}}}{|Z|}\right)^2 R.
$$
Substituting the values:
$$
P = \left(\frac{20}{834}\right)^2 \times 100 \,\text{J/s} \approx 0.0575 \,\text{J/s}.
$$
Step 6: Compute the time to raise the resistor's temperature by $10^\circ\text{C}$
The heat required to raise the temperature of the resistor (with heat capacity $S$) by $\Delta \theta$ is:
$$
H = S \,\Delta \theta.
$$
Since $S = 2\,\text{J}/^\circ\text{C}$ and $\Delta \theta = 10^\circ\text{C}$, the total heat needed is:
$$
H = 2 \times 10 = 20 \,\text{J}.
$$
Time taken ($t$) is given by:
$$
t = \frac{H}{P} = \frac{20}{0.0575} \approx 348 \,\text{s}.
$$
Final Answer
The resistor will get heated by $10^\circ \text{C}$ in approximately 348 seconds.
