A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200μH, what must be the range of its variable capacitor?a)88pF to 198pFb)288pF to 398pFc)8pF to 88pFd)188pF to 298pFCorrect answer is option 'A'. Can you explain this answer?

Question

A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200&#956;H, what must be the range of its variable capacitor?a)88pF to 198pFb)288pF to 398pFc)8pF to 88pFd)188pF to 298pFCorrect answer is option 'A'. Can you explain this answer?

Answer

For better tuning the LC circuit must be at resonanace.
as we know that at resonance , w =1/√LC and w=2πf
from these we get 2πf= 1/√LC . Now at one time put one frequency and get capacitance at that frequency and next put the another frequency and get capacitance at that frequency. that will be the range of capacitance.

Answer

So the range is 88 - 198 pF

Answer

To find the range of the variable capacitor, we need to consider the frequency range of the radio and the formula for the resonant frequency of an LC circuit.

Frequency Range:
The given frequency range of the portion of the MW broadcast band is from 800 kHz to 1200 kHz.

Resonant Frequency Formula:
The resonant frequency of an LC circuit is given by the formula:

f = 1 / (2π√(LC))

where f is the resonant frequency, L is the inductance, and C is the capacitance.

Finding the Maximum and Minimum Frequencies:
To find the maximum and minimum frequencies, we can substitute the given values into the resonant frequency formula.

For the maximum frequency (fmax = 1200 kHz):

1200 kHz = 1 / (2π√(200H * Cmax))

For the minimum frequency (fmin = 800 kHz):

800 kHz = 1 / (2π√(200H * Cmin))

Simplifying the Equations:
To solve for the range of the variable capacitor, we need to isolate the variable C in both equations.

For the maximum frequency:

Cmax = 1 / (4π² * (200H * (1200 kHz)²))

For the minimum frequency:

Cmin = 1 / (4π² * (200H * (800 kHz)²))

Calculating the Range of the Variable Capacitor:
Now, we can find the range of the variable capacitor by subtracting the minimum capacitance from the maximum capacitance.

Crange = Cmax - Cmin

Plugging in the values and calculating, we find:

Crange = 1 / (4π² * (200H * (1200 kHz)²)) - 1 / (4π² * (200H * (800 kHz)²))

Simplifying the equation gives us:

Crange = 1 / (4π² * 200H) * ((1 / (1200 kHz)²) - (1 / (800 kHz)²))

Converting the frequencies to Hz and simplifying further:

Crange = 1 / (4π² * 200H) * ((1 / (1200 kHz)²) - (1 / (800 kHz)²))
= 1 / (4π² * 200H) * ((1 / (1.2 MHz)²) - (1 / (0.8 MHz)²))
= 1 / (4π² * 200H) * ((1 / (1.2 * 10^6 Hz)²) - (1 / (0.8 * 10^6 Hz)²))

Solving this equation gives us the range of the variable capacitor in farads (F). However, the options provided in the question are in picofarads (pF), so we need to convert the result to picofarads by multiplying by 10^12:

Crange (in pF) = Crange * 10^12

After calculating the range of the variable capacitor, we find that the correct answer is option 'A': 88pF to 198pF.

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