In a series circuit, a generator (1400 Hz, 12.0 V) is connected to a 15.0- resistor, a 3.80-ìF capacitor, and a 5.70-mH inductor. Find the voltages across (a) the resistor, (b) the capacitor, and (c) the inductor.

Xl = 2pi*F*L = 6.28*1400*0.0057 = j50 ohms.

Xc = 1/(2pi*F*C) = 1/(6.28*1400*3.8*10^-6) = -j30 ohms.
Z = R + j(Xl-Xc) = 15 + j(50-30) = 15 + j20 = 25 ohms[53o].
I = E/Z = 12/25[53o] = 0.48A[-53o]. The current lags the voltage by 53o.

a. Vr = I*R = 0.48[-53] * 15 = 7.2V.[-53o].
b. Vc = I*Xc = 0.48[-53o] * 30V[-90o] = 14.4V.[-143o]. Lags current by 90o.
c. Vl = I*Xl = 0.48[-53o] * 50[90o] = 24V.[37o]. Leads current by 90o.

E = Vr+Vc+Vl = 7.2[-53o] + 14.4[-143o] + 24[37o]

Check: E = 12 + 0.027i = 12.0000304V[0.128o].

The magnitude of Vc and Vl is greater than the supply voltage, but the
vector sum = the 12V.-supply as it should.

To find the voltages across the different components in a series circuit, we need to use the concepts of impedance and reactance, which depend on the frequency of the generator.

Let's calculate the impedance for each component using the formulas:

1. Impedance of a resistor (R): Z_R = R

2. Impedance of a capacitor (C): Z_C = 1/(2πfC), where f is the frequency and C is the capacitance.

3. Impedance of an inductor (L): Z_L = 2πfL, where f is the frequency and L is the inductance.

Given:
f = 1400 Hz
V_generator = 12.0 V
R = 15.0 Ω
C = 3.80 µF = 3.80 × 10^(-6) F
L = 5.70 mH = 5.70 × 10^(-3) H

(a) Voltage across the resistor:
The voltage across a component in a series circuit is given by Ohm's law: V_R = I × Z_R, where I is the current.

Using Ohm's law, we need to find the current flowing through the circuit first.

Using the formula: V_generator = I × Z_total, we can find the total impedance (Z_total) of the circuit.

Z_total = Z_R + Z_C + Z_L

Calculating the individual impedances:
Z_R = R = 15.0 Ω
Z_C = 1 / (2πfC) = 1 / (2π × 1400 Hz × 3.80 × 10^(-6) F)
Z_L = 2πfL = 2π × 1400 Hz × 5.70 × 10^(-3) H

Now we can calculate Z_total.

Next, we can find the current I using the formula: I = V_generator / Z_total.

Once we have the current, we can find the voltage across the resistor using Ohm's law: V_R = I × Z_R.

(b) Voltage across the capacitor:
Using Ohm's law, V_C = I × Z_C.

(c) Voltage across the inductor:
Using Ohm's law, V_L = I × Z_L.

Let's go ahead and calculate the answers step-by-step.