1. The problem statement, all variables and given/known data We want to construct a solenoid with a resistance of 4.30 Ω and generate a magnetic field of 3.70 × 10−2 T at its center when applying 4.60 A of electrical current. We want to use copper wire with a diameter of 0.500 mm. If we need the solenoid's radius to be 1.00 cm, how many turns of wire will be need? 2. Relevant equations B = μ IN /L 3. The attempt at a solution I was thinking the height of the solenoid would be the diameter, which is the height of a wire, multiplied by the number of coils. I know that the solenoid is proportional to the number of coils. How can I judge the number of coils of a solenoid given (1) the resistance... The greater the number of coils is the greater resistance? By finding the resistance of one loop of copper with certain radius and diameter? (2) magnetic field The only equation I know with magnetic field of a solenoid is one derived using Ampere's law. so B = μ IN /L. Certainly I can find the magnetic field of one loop where L is equal to the diameter of wire. This seems like a solid approach. (3) radius Why does radius matter? If the radius increases, the length must increase sevenfold. My equation leaves out radius so I believe radius does not matter.
The formula you've written for the strength of the magnetic field in the solenoid for a given current I shows that the field is proportional to the number of turns over the length of the solenoid. With the given information you can calculate what N/L needs to be. You can interpret N/L as being the number of turns per unit length. So, for example, if you take a unit length to be one cm, you can determine how many turns need to fit in each cm of solenoid (no matter what its total length is). As for the wire, you're given its diameter and that will determine the minimum width of a single turn for close-wound turns (wire wrapped with no space between turns). From this you can determine the maximum number of turns per centimeter for a single layer of turns. If it happens that you can't get enough turns per centimeter into a single layer to meet the magnetic field's N/L requirement then you'll have to consider wrapping multiple layers of turns. Once you've determined a wrapping pattern that will give you the required turns per centimeter you'll have to work out the length of wire required to do it. I'd suggest determining the length of wire required per centimeter of solenoid. This will involve using either an exact formula for the arc length of a helix, or making an approximation based upon circle circumferences. Loosely-wound turns may suffer accuracy problems with such an approximation. With the length of wire per centimeter of solenoid you should be able to work out the wire resistance per centimeter of solenoid. The total resistance desired then determines the length of the solenoid (and hence the total number of turns). The problem doesn't state whether the given radius is to be taken as the inside radius of the solenoid or the outside radius. This matters when you have multiple layers, because each layer of turns will have its own radius that determines the length of wire per turn.