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(20 pts)

You are building an FM radio ($f \approx 100$ MHz) and have a power supply and circuitry that generates annoying harmonics in the low frequencies (especially 60 Hz, but also AM stations around 1 MHz contribute) that contaminate your high frequency output and causes your signal to ``buzz'. Naturally, you have a parts box that contains resistors and capacitors. The range of resistors available runs from 1 Ohm through 100,000 Ohms (to one significant digit - don't bother with resistances like 3.845 Ohms as their rated value is generally accurate only to 10% or so anyway - call it 4 Ohms instead), and you have capacitors that range from 1 microfarad to 1 picofarad, but only in multiples of ten (e.g. $10^{-6}$ farads, $10^{-7}$ farads, ..., $10^{-12}$ farads).

Design a ``high pass'' filter built from one resistor and one capacitor (where you get to choose suitable values for $R$ and $C$ as well as their arrangement) that will output more than half the input voltage for all frequencies greater than 10 MHz but strongly attenuates the output voltage for frequencies more than a bit less than this, and derive the expression (for your circuit) for $V_{\rm out}/V_{\rm in}$ as a function of $R$, $C$ and $\omega$. Draw the circuit in the space above, of course, clearly indicating where $V_{\rm in}$ and $V_{\rm out}$ go.

(Hints and Notes: This was a homework problem, so you should know what a high pass filter is. If you don't remember exactly, consider a series combination of $R$ and $C$ and think about what happens to the voltage drops across each one as a function of $\omega$. Your ``output voltage'' will come from a parallel connection across one or the other.)