Final answer:
To find the circuit's capacitive reactance with the given resistance, inductive reactance, and power factor, you can use the power factor equation in terms of resistance and total impedance, and then calculate the impedance to solve for the capacitive reactance.
Step-by-step explanation:
To calculate the circuit's capacitive reactance XC given a resistance R=180Ω and inductive reactance XL=393Ω, and a power factor cos(φ)=0.707, we use the relationship between the power factor, resistance, and reactances in an RLC circuit. The power factor is equal to the resistance divided by the impedance (Z) of the circuit, cos(φ) = R/Z. Since cos(φ) is given as 0.707, we can rearrange this to find Z: Z = R / cos(φ).
With the values provided, Z = 180Ω / 0.707, which approximates to 254.6Ω. The impedance is also given by the square root of (R2 + (XL - XC)2). Equating the two expressions for Z and solving for XC will give us the capacitive reactance. We find that XL - XC is approximately equal to 254.6Ω, and hence we can solve for the capacitive reactance XC.