What capacitance is needed in series with a 30 Ω resistor at 1.0 kHz for a total impedance of 45 Ω?
A) 34 μF
B) 15 μF
C) 4.7 μF
D) 0.037 μF

Question

Quizplus · Accepted Answer

The total impedance in a series circuit can be calculated as Z = R + jX, where R is the resistance and X is the reactance (either capacitive or inductive). At 1.0 kHz, the reactance of the capacitor needed can be calculated as:

Xc = 1 / (2πfC) = 1 / (2π * 1000 * C)

At resonance (when the total impedance is 45 Ω), the reactance of the capacitor must be equal to the resistance, so we can set up the equation:

30 Ω + jXc = 45 Ω

jXc = 15 Ω

Xc = -15 Ω (because the reactance of a capacitor is negative)

Substituting this value into the reactance equation above, we can solve for the capacitance:

-15 Ω = 1 / (2π * 1000 * C)

C = 1 / (2π * 1000 * -15 Ω)

C = 4.7 μF (rounded to one significant figure)

Therefore, the best choice is C, 4.7 μF.

What Capacitance Is Needed in Series with a 30 Ω