How Many Turns Are Needed for a Solenoid with Given Parameters?

[fisixC]

Homework Statement

It is desired to construct a solenoid that has a
resistance of 9.94 Ω (at 20◦C) and produces a
magnetic field at its center of 0.0224 T when
it carries a current of 5.17 A. The solenoid is
to be constructed from copper wire having a
diameter of 0.517 mm.
Find the number of turns of wire needed if
the radius of the solenoid is to be 2.2 cm. The
resistivity of the copper is 1.7 × 10−8Ω · m .
Answer in units of turns.

Homework Equations

magnetic field = B = (vacuum permeability/2)*((number of turns*current)/(resistance))
Resistance = resistivity*(length/area)

The Attempt at a Solution

To start off I solved for the number of turns using the first relevant equation.

number of turns = ((2*resistance*B)/(vacuum permeability*current))

At this point I thought: Well the wanted resistance must be the resistance and so I went on plugging the numbers in and it was wrong...

Then I noticed that I need to specify how large the radius is and I don't see a radius variable inside of one of the equations. I'm basically stumped, and I feel like I'm missing an equation or something...Also any ideas as to why they are giving so much information?

 

[Zryn]

The radius of a wire is related to the (cross sectional surface) area in your Resistance calculation.

 

[gneill]

First check to see what magnetic field strength is possible for the given current and wire.

B = μ0*I*N/L

For wire diameter d, if the turns are close-wound then each turn adds d to the length. So L = N*d, and the equation for the field strength becomes:

B = μ0*I*N/(N*d) = μ0*I/d

Plugging in the given values for I and d yields B = 0.0126T, which is smaller than the desired field strength (which is 0.0224T).

So it would appear that a single layer coil is not going to achieve the desired field strength. This is going to complicate matters... So, given that there must be more than one layer of wire, is the given solenoid radius the inner radius or the outer radius of the assembly?

Where did you find your equation for the magnetic field strength?