How radius and length of an electromagnet coil affect its strength
Coil size: radius
The magnetic induction at the center of one extreme of a long solenoid with jointing windings is:
The force of the magnet will be proportional to the energy stored in the magnetic field:
Note that the expressions on the right are only true for a solenoid with l > r.
From the equation above we see that the energy is proportional to the radius squared, indicating that the bigger the radius, the bigger the energy. But we have a given length of magnet wire and the larger the radius the smaller number of turns. Actually:
And replacing N in the energy equation:
As we see, the radius squared in the numerator cancels out the radius squared in the denominator.
Conclusion: Given the limited length of magnet wire, the field energy, and hence the strength of the magnet, is fairly independent of the radius of the coil. A round shape is preferred because it gives the maximum cross section with the minimum perimeter.
Coil size: length
We see in the equation for the energy that the coil length is in the denominator and this can wrongly lead us to think that the shorter the coil the better. But remember the approximation that we made in the energy calculation about the coil being much longer than its diameter.
If we refer to the first part of the energy equation we see that it is proportional to the magnetic induction B. Therefore, we must size our coil to maximize B.
In the expression for B on the last page we see that the length l is in the denominator, but this length defines also the angle . For l very small and increasing, will grow proportionally to l, canceling the effect of the coil length in the denominator. As goes above about 45 degrees, the cosine will increase more slowly, and B will decrease.
It is difficult to analytically evaluate these factors. Here is where magnetic design becomes and art. A finite element model will allow changing the length to radius ratio and observe where the maximum in energy is. As a rule of thumb I would suggest to try to pack the coil in a length about once to twice the radius and see what produces the higher strength. (At l = r the field intensity is still 70% of the maximum.) Packing the coil in more that one layer will increase the effective diameter of the coil, but will increase the turns per unit length.