Question

The FM radio band is broadcast between \(88 \mathrm{MHz}\) and $108 \mathrm{MHz}$. What range of capacitors must be used to tune in these signals if an inductor of \(3.00 \mu \mathrm{H}\) is used?

Short answer

Answer: The range of capacitors required to tune in the signals between 88 MHz and 108 MHz using an inductor of 3.00 µH is approximately C_min to C_max, where C_min and C_max are calculated using the formula $$C = \frac{1}{(2\pi f)^2 L}$$ and by plugging in the values for L and the lower and upper bound frequencies, 88 MHz and 108 MHz, respectively.

Step-by-step solution

Setup the formula for resonance frequency

The resonance frequency formula for an LC circuit is given by: $$f = \frac{1}{2\pi\sqrt{LC}}$$, where f is the resonance frequency, L is the inductance, and C is the capacitance.

Rearrange the formula to solve for C

We need to find the range of capacitors (C) for the given frequency range, so we'll rearrange the formula to solve for C: $$C = \frac{1}{(2\pi f)^2 L}$$

Calculate the lower bound capacitance

To find the capacitance for the lower bound frequency, plug in the values for L = 3.00 µH and f = 88 MHz into the formula: $$C_{min} = \frac{1}{(2\pi (88\times10^6))^2 (3.00\times10^{-6})}$$ Calculate the value for C_min to get the lower bound capacitance.

Calculate the upper bound capacitance

To find the capacitance for the upper bound frequency, plug in the values for L = 3.00 µH and f = 108 MHz into the formula: $$C_{max} = \frac{1}{(2\pi (108\times10^6))^2 (3.00\times10^{-6})}$$ Calculate the value for C_max to get the upper bound capacitance.

Determine the range of capacitors

Determine the range of capacitors required to tune in the signals between 88 MHz and 108 MHz using the lower and upper bound capacitances (C_min and C_max) calculated in steps 3 and 4. The range will be in the form of C_min to C_max.