Question
A $15.0 \mu F$ capacitor is connected to a $220 V , 50 Hz$ source. Find the capacitive reactance and the current (rms and peak) in the circuit. If the frequency is doubled, what happens to the capacitive reactance and the current?
✓
Answer
The capacitive reactance is
$
X_C=\frac{1}{2 \pi v C}=\frac{1}{2 \pi(50 Hz )\left(15.0 \times 10^{-6} F \right)}=212 \Omega
$
The rms current is
$
I=\frac{V}{X_C}=\frac{220 V }{212 \Omega}=1.04 A
$
The peak current is
$
i_m=\sqrt{2} I=(1.41)(1.04 A )=1.47 A
$
This current oscillates between $+1.47 A$ and $-1.47 A$, and is ahead of the voltage by $\pi / 2$.
If the frequency is doubled, the capacitive reactance is halved and consequently, the current is doubled.
$
X_C=\frac{1}{2 \pi v C}=\frac{1}{2 \pi(50 Hz )\left(15.0 \times 10^{-6} F \right)}=212 \Omega
$
The rms current is
$
I=\frac{V}{X_C}=\frac{220 V }{212 \Omega}=1.04 A
$
The peak current is
$
i_m=\sqrt{2} I=(1.41)(1.04 A )=1.47 A
$
This current oscillates between $+1.47 A$ and $-1.47 A$, and is ahead of the voltage by $\pi / 2$.
If the frequency is doubled, the capacitive reactance is halved and consequently, the current is doubled.
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