Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A tuner can use one of two coils. Both have the same inductance. The first has a resistance of and the second . The capacitor, the source voltage and the resonant frequency are the same. Compare the two circuits: the ratios of their quality factors, of their largest currents, and of their bandwidths, first to second. Which coil separates two close stations better?
At resonance , so and . There , and . The quality factor is . The bandwidth is , or .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The quality factors. and and are the same. So and .
Step 2. The largest currents. At resonance . So .
Step 3. The bandwidths. and is the same. So .
Answer: The ratios are , and . The first coil gives a taller and four times narrower resonance curve, so it separates close stations better.
Watch out. The narrow peak is a gain and a risk. A band of sound is made of frequencies on both sides of the station frequency. A circuit that is too selective cuts the high notes.
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