Does an electromagnet increase in strength more by voltage or wire turns? I'm working on a science project and need to know what makes a strong electromagnet - more wire turns or more voltage? I understand energy can be lost to heat with too many wire turns.
 A: The magnetic field in a solenoid of length $L$ around an iron core with $N$ turns is given by: $$ B = \mu \frac{NI}{L}. $$
Assuming some Ohm's law type of resistance in the wire we can replace $I$ with $V/R$ to get
$$ B = \mu \frac{NV}{LR}. $$
So the magnetic field strength increases linearly with both the number of turns and the voltage. The resistance of a wire is given by
$$ R = \rho \frac{l}{A}, $$
(where $\rho$ is some material property of the wire, $A$ is the cross sectional area and $l$ is the length) so that $LR = \rho Ll/A$ and
$$ B = \mu \frac{NVA}{\rho Ll}. $$
If $r$ is the radius of the wire ($\pi r^2 = A$) then $2 r N \approx L.$ (Think of stacking the rings of wire on top of each other.) These last substitutions yield
$$ B = \frac{\mu \pi r}{2 \rho }\frac{V}{l}. $$
This seems to suggest that increasing the voltage has the same effect as shrinking the length of wire used.
A: When you're increasing the voltage, you're increasing  the current. Recall Ohm's law. Provided a steady current flows through the material at constant temperature, the resistance remains the same. If $I$ is increased, then $V$ has to increase in order for the resistance to remain constant.
The magnetic field along the axis of the solenoid is given by $$B=\mu \frac{NI}{L}$$
The permeability of the medium $\mu$, which can be of the order of some $10^5$ when the coil is wound on a soft iron core, which is really helpful. The current (or voltage) can be increased.

There's a problem with increasing the number of turns. When solving for Ohm's law, the expression when you combine $I=nAev_d$ and $v_d=eE\tau/m$, you get 
$$V=\frac{mL}{nAe^2\tau}I$$
The large expression in the middle is the resistance $R$. It can be seen that $R\ \alpha\ l/A$. It can be seen that the larger the area, the less resistance there is. So if you're using a long coil, make sure that the area occupied by the coil per turn is more. There's another problem if you're using an AC. The number of turns can also increase the inductance of the coil and hence may lead to impedance, which also adds up with the resistance.

Hence, I suggest the use of soft iron core, increase the current and voltage input, use a wire with low specific resistivity $\rho$ like silver or aluminum. And finally - if you require more number of turns, use thick wires and make sure that the coil occupies a larger area.
A: The $B$ field does matter, but the force that is exerted from the field is $$F=qVB\ ,$$
where $q$ is charge, $V$ is voltage, and $B$ is magnetic field. Given the equation for $B\ ,$ 
$$B=\frac{\mu N V }{L R}\ ,$$
we have that $$F= \frac{V^2 N \mu}{L R}\ .$$Thus, in order to optimize the force output you must use voltage more than anything.
