Making a crude radio (Difficult/Long)

[Euclid4]

Homework Statement

Suppose you are stranded on an island and need to make a radio so that you can get weather updates. You know that weather updates for your general area can be found on short-wave radio within the frequency range f-low to f-high . Also, you manage to scavenge some thin, insulated wire of thickness d, of which you can allocate a length l to make a solenoid inductor. You also manage to find a long plastic tube of length a and diameter D, a large roll of foil of width w, a large mylar sheet of thickness 1/100d and dielectric constant κ=3 that will accommodate the area of the foil, a speaker from headphones with impedance R, and a small transistor.
Being extra resourceful, you manage to make a small battery out of a lemon, a galvanized nail, and a nickel that you attach to the transistor and headphone speaker to amplify the radio signal.

Part A: Assuming that you use the entire length l of wire for the inductor, as well as the plastic tube as a form, what is the maximum inductance achievable?

Homework Equations

Part A:
L=NΦ/i Where N is the number of turns, Φ is the magnetic field flux, and i is the instantaneous current provided by the battery and transistor.
Φ = (A)(B)

The Attempt at a Solution

The answers are all variables. I assumed N=1. I broke down A and B into (πr^2)(Nµi/2R) which when plugged into the equation for L cancels out i, which was promising as none of the answer choices had i. The issue I'm coming up with is what to consider r for the area, and R for the magnetic field. Maybe the question's wording is confusing me but I don't know what to do with the insulated wire of thickness d and length l, and the long plastic tube of length a and diameter D. Thanks for anyone who even read this.

 

[gneill]

Hi Euclid4, Welcome to Physics Forums.

Have you selected a suitable circuit for your radio? Maybe browse a few crystal radio related web sites to get some idea of what the challenges are.

I suspect that the long plastic tube is meant to act as a form for an inductor (so you can wind and support a uniform coil). The wire will be used to form the inductor and to wire up your circuit, of course.

You can look up the formula for the inductance of an air-core inductor, or a solenoid based upon its geometry (length, diameter, number of turns,...). The basic solenoid inductance formula is only accurate for fairly long coils as the magnetic field tends to diverge at the coil ends. The electronics industry and amateur radio enthusiasts know of more accurate formulas for shorter coils, if that's what you end up needing.

Your problem statement doesn't appear to include a diode of any sort to act as a detector (demodulator) for the radio. Have you given any thought to how you're going to handle that? How will you handle tuning the radio?

 

[Euclid4]

gneill, thanks for the reply!

My professor handed us this problem with a corresponding diagram and the equation for L with respect to the solenoid inductor. Forgive me if I'm wrong, but in this question, referring to the solenoid inductance, I don't have to worry about the current passing through the diode or transistor as they cancel out in the equation?

After rethinking the question I don't believe assuming N=1 is correct. My question then becomes how is the maximum N (and therefore maximum inductance) found? Is there some sort of ratio for N of the inductor and the length of the wire which makes the inductor?

 

[gneill]

gneill, thanks for the reply!

My professor handed us this problem with a corresponding diagram and the equation for L with respect to the solenoid inductor. Forgive me if I'm wrong, but in this question, referring to the solenoid inductance, I don't have to worry about the current passing through the diode or transistor as they cancel out in the equation?

The inductance of a coil is independent of the current. It's an inherent property of the geometry of the coil, depending on quantities like the number of turns per unit length and the cross sectional area of the turns. Look up "self inductance of a coil" or "self inductance of a solenoid" on the web.

After rethinking the question I don't believe assuming N=1 is correct. My question then becomes how is the maximum N (and therefore maximum inductance) found? Is there some sort of ratio for N of the inductor and the length of the wire which makes the inductor?

Usually a practical inductor consists of a coil of wire of several to many turns. Look up the formula for the inductance of a solenoid. You'll find that the materials you're given will dictate the geometry. Hint: to maximize the inductance of a single-layer coil you want to close-wind the turns so that the turns touch each other (the insulation of the wire that is), maximizing the turns per unit length.

 

[Euclid4]

I have L=NΦ/i as it was given. To find the area for Φ=AB, if the coils are touching, then essentially it becomes a long cylinder of diameter D. And for B the diameter for the coils is D?

 

[rude man]

gneill, thanks for the reply!After rethinking the question I don't believe assuming N=1 is correct. My question then becomes how is the maximum N (and therefore maximum inductance) found? Is there some sort of ratio for N of the inductor and the length of the wire which makes the inductor?

Good start, not letting N=1. Why would you do that?
Assume you are to use the entire length a of the tube for your solenoid, otherwise the problem statement doesn't make sense since you could increase inductance arbitrarily by shortening the effective length of the solenoid.
So, 1st step, do you have a formula for the inductance of a solenoid?

 

[Euclid4]

L=NΦ/i

 

[Euclid4]

L=NΦ/i

That's the equation given.

 

[gneill]

You need another formula for inductance, one that doesn't involve the flux or current directly. The formula should depend only on the geometry of the coil. Google: "Inductance of a coil" .

 

[andrevdh]

The inductor and capacitor is used to build an LC circuit that resonates in the given frequency range. The voltage of this circuit is amplified and rectified by the transistor circuit which then drives the headphone speaker.