In an instructable, the formula for calculating antenna length was given as "length in meters =300 / frequency in MHz". The example given was for a Bluetooth antenna with a frequency of 2450MHz. The antenna length was calculated as 300/2450 = 0.1225m = 12cm. The author then divided it by 4 to make a "quarter wave" antenna length of 30mm. Why divide it by 4 and what is the benefit of a "quarter wave" antenna. What is the effect of the conductor leading to the antenna? If the 30mm antenna is soldered to a 4mm lead, does that result in a 34mm antenna? For a fractal antenna, should the 30mm length of wire be folded into a fractal shape or should the total fractal antenna be 30mm in width? I have read somewhere that one of the benefits of a fractal antenna is that they are able to receive signals of different wave lengths.

A quarter wave works better than a random wave antenna because of harmonics, I think that's the right word. You get less SWR (google it) using a quarter wave than you would with a 3/16 or a 5/16 wave antenna.

THe feed line does not enter into the calculation if it is shielded wire or a balance line. If you just feed the antenna with a hunk of wire then all of the wire not becomes an antenna and all of your design and math goes out the window, but less of your signal goes out of your antenna.

An antenna is able to receive a signal of any frequency. But frequencies that are multibles of dividibles of the antenna length are recieved much more stronger than those that are at odds with the math.

Think of a "sine wave" 12 cm long. Draw it on graph paper if you like. The first half goes positive, the second half goes negative

When catching an RF signal, "negative is just a phase change - you can divide the signal in half (6cm) and say it starts and ends at zero.

Dividing AGAIN for quarter wave (3cm): looking at your graph, the sine wave goes from zero to maximum. At the pointy end of your dipole, no current flows (open circuit) so, applying the sine wave, the other end with a connector has maximum signal current.

You could create a simple dipole for your FM radio by taking a piece of two-conductor speaker wire, compute your size by taking the middle of the FM band, 98mhz would be close. 300/98 = 3.06 meters. Split the two wires apart at one end until each side is 1.5 meters long. Tape it up at the junction where you reached 1.5 meters. Spread the two separated sides at a 90 degree angle (in a T shape). The 90 degree turn is what separates(de-couples) the radiator from the feedline. Spread your wires and use thumbtacs to keep it in place and you have a very effective FM antenna. Better than the vertical 1/4 wave on your car.