Final answer:
The maximum capacitance (Cmax) required for an AM tuner to adjust the oscillation frequency over the broadcast band of 540 to 1600 kHz, using a 2.5 mH inductor, is approximately 209 pF, calculated using the resonant frequency formula of an LC circuit.
Step-by-step explanation:
The student is asking how to determine the maximum capacitance needed for an AM tuner to adjust the oscillation frequency over the broadcast band of 540 to 1600 kHz using a 2.5 mH inductor. To find the maximum capacitance (Cmax), we use the formula for the resonant frequency of an LC circuit, f = 1 / (2π√(LC)), where f is the frequency in hertz, L is the inductance in henrys, and C is the capacitance in farads. To adjust over the broadcast band's range, the LC circuit needs to achieve a frequency of 0.550 MHz at its lowest and 1.600 MHz at its highest, corresponding to the lower and upper ends of the AM broadcast band, respectively.
For the lowest frequency (0.550 MHz), we rearrange the formula to solve for Cmax:
Cmax = 1 / ²π²L²f²
Plugging in the values (L = 2.5 mH = 2.5 × 10−3 H, f = 0.550 MHz = 0.550 × 10¶ Hz), we get:
Cmax = 1 / (4²π² × 2.5 × 10−3 × 0.550 × 10¶)²
After calculation, Cmax is found to be approximately 0.209 × 10−6 F or 209 pF (picoFarads).