Question

A series RLC circuit is used to tune a radio set to receive NOUN RADIO broadcasting at 105.9 MHz in the FM band. The resistance and the inductance of the circuit of the radio set are
12

Ω
and
1.4

μ
H
respectively. What capacitance should the circuit have?

Question 3Select one:

1.51

μ
F


1.33

m
F


2.11

p
F


1.64

p
F

Answer 1

To find the required capacitance \( C \) in a series RLC circuit tuned to a specific frequency, you can use the resonant frequency formula:

\[
f_0 = \frac{1}{2\pi\sqrt{LC}}
\]

Where:
- \( f_0 \) is the resonant frequency in Hertz (Hz),
- \( L \) is the inductance in Henrys (H),
- \( C \) is the capacitance in Farads (F).

Given:
- \( f_0 = 105.9 \text{ MHz} = 105.9 \times 10^6 \text{ Hz} \)
- \( L = 1.4 \, \mu\text{H} = 1.4 \times 10^{-6} \text{ H} \)

Now, first, square both sides of the resonant frequency formula and rearrange it to find \( C \):

\[
C = \frac{1}{(2\pi f_0)^2 L}
\]

Now, plug in the values:

1. Calculate \( 2\pi f_0 \):

\[
2\pi f_0 = 2\pi \times 105.9 \times 10^6 \approx 666.232 \times 10^6 \text{ rad/s}
\]

2. Now, calculate \( (2\pi f_0)^2 \):

\[
(2\pi f_0)^2 \approx (666.232 \times 10^6)^2 \approx 4.44 \times 10^{17}
\]

3. Now substitute into the capacitance formula:

\[
C = \frac{1}{4.44 \times 10^{17} \times 1.4 \times 10^{-6}} \approx \frac{1}{6.216 \times 10^{11}} \approx 1.61 \times 10^{-12} \text{ F}
\]

4. Convert to picofarads:

\[
C \approx 1.61 \, \text{pF}
\]

After calculating, we see that \( C \) is approximately \( 1.61 \, \text{pF} \).

Looking at the options provided:

- 1.51 μF
- 1.33 mF
- 2.11 pF
- 1.64 pF

The closest match is **1.64 pF**. Therefore, the correct answer is:

**1.64 pF**.