Final answer:
The complex impedance of a 10 μF capacitor with a voltage v(t) = 10cos(2000πt) applied across it is calculated using Z = 1 / (jωC), where ω = 2000π rad/s. The result is -j50 Ω.
Step-by-step explanation:
The complex impedance of a capacitor is determined using the formula Z = 1 / (jωC), where j is the imaginary unit, ω is the angular frequency of the voltage, and C is the capacitance. In this case, the voltage v(t) = 10cos(2000πt) has an angular frequency of ω = 2000π rad/s, and the capacitance C is given as 10 μF (microfarads) which is equal to 10 x 10-6 F (farads).
To find the complex impedance of the 10 μF capacitor, we first convert the capacitance into farads:
C = 10 μF = 10 x 10-6 F
Next, we can calculate the impedance Z:
Z = 1 / (j2000π x 10 x 10-6)
= -j / (2000π x 10 x 10-6)
= -j / (0.02π)
= -j50 Ω
Therefore, the complex impedance of the capacitance is -j50 Ω.