1. A 26.0-Ω resistor, a 12.0-µF capacitor, and a 17.0-mH inductor are connected in series with a 150-V generator.
(a) At what frequency is the current a maximum?
The subject of resonant LRC circuits is explained at
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/serres.html
To find the frequency at which the current is a maximum, we can calculate the resonant frequency of the circuit. The resonant frequency is the frequency at which the capacitive and inductive reactances cancel each other out, resulting in maximum current.
The reactance of a capacitor (Xc) can be calculated using the formula:
Xc = 1 / (2 * pi * f * C)
Where:
Xc is the capacitive reactance,
f is the frequency,
pi is a mathematical constant (approximately 3.14159), and
C is the capacitance.
The reactance of an inductor (Xl) can be calculated using the formula:
Xl = 2 * pi * f * L
Where:
Xl is the inductive reactance,
f is the frequency, and
L is the inductance.
In a series circuit, the total impedance (Z) is calculated as the sum of the resistive, capacitive, and inductive reactances:
Z = R + Xc + Xl
At the resonant frequency, Xc and Xl cancel each other out, so the total impedance is equal to the resistance:
Z = R
In this case, the resistance (R) is given as 26.0 Ω.
To find the resonant frequency, we can set the total impedance equal to the resistance and solve for frequency:
26 = 1 / (2 * pi * f * 12e-6) + 2 * pi * f * 17e-3
Simplifying this equation and solving for f will give us the resonant frequency.
Now, we can solve the equation numerically using a computational tool or software, such as a graphing calculator or a numerical solver. By inputting the equation and solving for f, we will find the frequency at which the current is a maximum.
So, the answer to part (a) of the question will be the frequency obtained by solving the equation for f.