This calculates the possible dimensions: lengths, diameters, etc.,
given the desired self resonant frequency (f_r) of the coil as input.

Assuming that the coil will be a quarter wave oscillator ( λ = 4 ℓ_w ),  the total length of the wire (ℓ_w) can be found by:

  c  =  λ f_r    

  c  =  ( 4 ℓ_w ) f_r    

here,

    •  c   is the speed of light in air, ( 29,970,254,700 cm/s ) 
    •  λ   is the wavelength,  
    •  f_r   is the self resonant frequency and,
    •  ℓ_w   is the total length of the wire.

The coil used here for this calculation is solenoid shaped, and it is assumed to be made of copper.

The input for this calculation requires these properties:

    • resonant frequency desired, f_r,
    • number turns per cm, (ρ _turns), a density of turns,
    • the wire's gauge,
    • and then either the coils diameter or length in centimeters.

( units are in:   Hz, turns / cm, AWG, cm   respectively )

The output calculated properties of the coil will be:

    • the total length of wire required for the coil, ℓ_w,
    • diameter of the coil, d,
    • length of the coil, l,
    • inductance. L,
    • approximate self capacitance, C,
    • mass of the coil, M,
    • and the number of turns, N.

( units are in:   cm, cm, cm, Henries, Farads, kg, and turns,  respectively )

The density of turns per centimeter will have a minimum value for each thickness of wire. For example, for a 24 gauge wire the minimum number of turns per centimeter is 18. But here we can allow that value to be less turns per centimeter if desired.

This tool allows us to try different resonant frequencies, diameters, and lengths, and then it solves for a variety of properties of the resulting coil.

density of copper (ρ)  =  8940 kg / m³

where,

    •  M  = the mass of the coil (kilograms)
    •  ρ   = the density of copper (kg / m³)
    •  ℓ_w   = total length of the wire for the coil 
    •  r    = radius of the copper wire (meters)

where,

    •  N   = the number of turns of the coil 
    •  M   = the mass of the coil (kilograms)
    •  ρ   = the density of copper (kg / m³)
    •  d   = diameter of the coil (meters)
    •  r   = radius of the copper wire (meters)

      total length of the wire for the coil :    ℓ_w = N C    

       ( length of the wire = turns, times, circumference of the coil )

      total length of the wire for the coil :   ℓ_w = M / ( ρ A )

       ( length of the wire = mass of the wire, divided by:
                               density of copper times
                                       the cross-sectional area of the wire )

** The ARRL Handbook for Radio Communications 2011
section 2.8 Inductance and Inductors