To solve the given questions for the RC circuit, let's go through each part:

Given Data

• Resistor $R_1 = 40 \, \Omega$
• Capacitor $C_1 = 50 \, \mu \text{F} = 50 \times 10^{-6} \, \text{F}$
• Voltage $V = 12 \, \text{V}$ at frequency $f = 60 \, \text{Hz}$

Solution Steps

a) The Capacitor $C_1$ Reactance ($X_C$)

The capacitive reactance $X_C$ is given by: $X_C = \frac{1}{2 \pi f C}$ Substitute the values: $X_C = \frac{1}{2 \pi \times 60 \times 50 \times 10^{-6}}$ Calculating this will give us $X_C$.

b) The Phase Shift $\theta$

The phase shift $\theta$ between the voltage and current in an RC circuit is given by: $\theta = \arctan\left(-\frac{X_C}{R}\right)$ Once we have $X_C$ from part (a), we can substitute $R = 40 \, \Omega$ and calculate $\theta$.

c) Total Impedance $Z$

The total impedance $Z$ in an RC series circuit is: $Z = \sqrt{R^2 + X_C^2}$ Again, substituting $R = 40 \, \Omega$ and $X_C$ from part (a) will allow us to find $Z$.

Let’s compute these values now.import math

Given values

R = 40 # resistance in ohms C = 50e-6 # capacitance in farads (50 µF) f = 60 # frequency in Hz

Part a: Capacitive Reactance (X_C)

X_C = 1 / (2 * math.pi * f * C)

Part b: Phase Shift (θ)

theta = math.atan(-X_C / R) # in radians

Part c: Total Impedance (Z)

Z = math.sqrt(R2 + X_C2)

Convert theta from radians to degrees for interpretation

theta_degrees = math.degrees(theta)

X_C, theta_degrees, Z(53.05164769729845, -52.984355426817935, 66.4415331204682)(53.05164769729845, -52.984355426817935, 66.4415331204682)successHere are the results for each part:

1. Capacitor $C_1$ Reactance ($X_C$): $X_C \approx 53.05 \, \Omega$
2. Phase Shift $\theta$: $\theta \approx -52.98^\circ$
3. Total Impedance $Z$: $Z \approx 66.44 \, \Omega$

Would you like further details on any of these calculations, or do you have additional questions?

Related Questions

1. How does the frequency affect the capacitive reactance in an RC circuit?
2. What would be the effect on impedance if we increased the capacitance value?
3. How can the phase shift $\theta$ be interpreted in terms of voltage and current lead or lag?
4. What would be the new impedance if we replaced the capacitor with a different value?
5. How does phase shift $\theta$ change with varying resistance in the circuit?

Tip

In RC circuits, the capacitor’s reactance $X_C$ decreases with increasing frequency, causing changes in impedance and phase shift.