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Step-by-Step Solution
1. Understanding the Problem
We have an electric fan motor with self-inductance $L = 10\,\text{H}$. To obtain maximum power at a frequency of $50\,\text{Hz}$, we need to connect it with a suitable capacitor such that the inductive reactance equals the capacitive reactance.
2. Condition for Maximum Power
For maximum power transfer in an AC circuit containing inductance and capacitance, the reactances must be equal:
$X_L = X_C.$
3. Reactances for Inductor and Capacitor
Inductive reactance: $X_L = 2\pi f L.$
Capacitive reactance: $X_C = \dfrac{1}{2\pi f C}.$
4. Equating the Reactances
Set $X_L = X_C$:
$2\pi f L = \dfrac{1}{2\pi f C}.$
Rearranging for $C$ gives:
$C = \dfrac{1}{(2\pi f)^2 \, L}.$
5. Substituting the Given Values
Here, $L = 10\,\text{H}$ and $f = 50\,\text{Hz}$. Substituting these into the formula:
$C = \dfrac{1}{(2\pi \times 50)^2 \times 10}
= \dfrac{1}{4\pi^2 \times 50^2 \times 10}.$
Performing the numerical calculation:
$C = \dfrac{1}{4 \times \pi^2 \times 2500 \times 10}
= \dfrac{1}{\text{(some constant)} \times 25000}
\approx 1 \times 10^{-6} \,\text{F}
= 1\,\mu \text{F}.$
6. Final Result
Hence, the capacitance required is $1\,\mu \text{F}$, which matches the correct option.
