Final answer:
To find the voltage across the capacitor given the current, one must integrate the current with respect to time for the time domain and divide the current amplitude by the capacitive reactance in the frequency domain, accounting for a -90° phase shift due to the voltage lagging the current in an ideal capacitor.
Step-by-step explanation:
The question asks to find the voltage across the capacitor in both the time domain and the frequency domain given the current through a 100-uF capacitor is i(t) = 40 sin (500t - 60°) A. To find the voltage V(t) across the capacitor in the time domain, we use the relationship between current and voltage in a capacitor, which is i(t) = C dv(t)/dt, where C is the capacitance. Integrating the current i(t) with respect to time gives us the voltage V(t) across the capacitor.
In the frequency domain, the capacitive reactance Xc is used, which is Xc = 1/(\u03c9C), where \u03c9 is the angular frequency of the current. We then find the voltage by dividing the current amplitude by the capacitive reactance. However, the voltage has a phase shift of -90° relative to the current because in an ideal capacitor, the voltage lags the current by 90°.