Question

A radio can be tuned into a particular station frequency by adjusting the capacitance in an L-C circuit. Suppose that the minimum capacitance of a variable capacitor in a radio is 4.18pF.

What is the inductance L of a coil connected to this capacitor if the oscillation frequency of the L-C circuit is 1.69MHz , corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance?

Answer 1

To determine the inductance of a coil in an L-C circuit with a given capacitance and resonant frequency, the formula for the resonant frequency of an L-C circuit is rearranged to solve for L. With a minimum capacitance of 4.18 pF and a frequency of 1.69 MHz, the calculated inductance L is approximately 5.02 μH.

Step-by-step explanation:

To find the inductance L of a coil connected to a capacitor with a minimum capacitance of 4.18 pF in an L-C circuit that oscillates at a frequency of 1.69 MHz, we use the formula for the resonant frequency of an L-C circuit:

f = 1 / (2π√(LC))

Let's rearrange the formula to solve for L:

L = 1 / (4π2f2C)

Now we can substitute the values for f and C into the equation:

L = 1 / (4π2(1.69 x 106 Hz)2(4.18 x 10-12F))

Doing the calculation:

L = 1 / (4π2(2.8561 x 1012)(4.18 x 10-12F))

L = 1 / (47.619 x 1012)(4.18 x 10-12F)

L = 1 / (199.02842 x 1012 x 10-12)

L ≈ 5.02 x 10-6 H or 5.02 μH

The inductance L required for a coil in this L-C circuit is approximately 5.02 μH.