$A$ $15.0 \; \mu F$ 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?

(N/A) The capacitive reactance is given by $X_{C} = \frac{1}{2 \pi \nu C}$.
Substituting the values: $X_{C} = \frac{1}{2 \pi (50 \; Hz) (15.0 \times 10^{-6} \; F)} \approx 212 \; \Omega$.
The $rms$ current is $I_{rms} = \frac{V_{rms}}{X_{C}} = \frac{220 \; V}{212 \; \Omega} \approx 1.04 \; A$.
The peak current is $I_{m} = \sqrt{2} I_{rms} = 1.414 \times 1.04 \; A \approx 1.47 \; A$.
If the frequency $\nu$ is doubled,the capacitive reactance $X_{C} = \frac{1}{2 \pi \nu C}$ becomes half of its original value. Since $I = \frac{V}{X_{C}}$,the current in the circuit doubles.