Question

You have a 2.2 m long copper wire. you want to make an n-turn current loop that generates a 1.9 mt magnetic field at the center when the current is 2.3
a. you must use the entire wire.

Answer 1

The trick here is that there are two variables, Diameter of the coil and number of turns in it. So, in order to solve it, we will need two equations in D and N
Since we are making coil, formula for magnetic field will be B = μₒni/D
D = μₒni/B = 4π x 10^(-7) x (N) x (2.2) / (1.6)x10^(-3) = 1.7 x 10^(-3) x N = D
Another equation will be from the wire's length.
Total length = Circumference of one loop x No. of turns = πD x N = 2.2
We will eliminate N because we need D.
N = 2.2 / πD
Substitute in 1st equation.
1.7 x 10^(-3) x 2.2 / π = D^2
D = 0.034 m = 3.4 cm