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Difficulty: EasyAlternating Current (AC) Circuits

A capacitor of capacitance 50 μF50\ \mu\text{F} is connected across an alternating current (AC) source operating at a frequency of 100π Hz\frac{100}{\pi}\ \text{Hz}. What is the capacitive reactance of the capacitor?

Answer: 100 \Omega

Answer

The capacitive reactance of the capacitor is 100 Ω100\ \Omega.
Capacitive reactance XCX_C is given by the formula XC=12πfCX_C = \frac{1}{2\pi f C}. Substituting C=50×10−6 FC = 50 \times 10^{-6}\ \text{F} and f=100π Hzf = \frac{100}{\pi}\ \text{Hz} into the formula yields XC=12π(100/π)(50×10−6)=110−2=100 ΩX_C = \frac{1}{2\pi (100/\pi) (50 \times 10^{-6})} = \frac{1}{10^{-2}} = 100\ \Omega.

Step-by-Step Solution

1
Convert capacitance to farads and state all given values
C=50×10−6 FC = 50 \times 10^{-6}\ \text{F} and f=100π Hzf = \frac{100}{\pi}\ \text{Hz}
Calculations require standard SI base units.
2
Apply the formula for capacitive reactance
XC=12πfCX_C = \frac{1}{2\pi f C}
Capacitive reactance measures the opposition offered by a capacitor to alternating current.
3
Substitute the values and calculate the result
XC=12π⋅100π⋅(50×10−6)=110,000×10−6=100 ΩX_C = \frac{1}{2\pi \cdot \frac{100}{\pi} \cdot (50 \times 10^{-6})} = \frac{1}{10,000 \times 10^{-6}} = 100\ \Omega
The factor π\pi cancels out directly, making the arithmetic simple.

Key Concept

Capacitive Reactance in AC Circuits
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Topics

Alternating Current (AC) Circuits

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