a) 0.5 m
b) [tex]4.77\cdot 10^{-3} T[/tex]
Explanation:
a)
At the beginning, the length of the copper wire is:
L = 15.0 m
and the diameter (so, the thickness of each adjacent circle) is
[tex]t=2.20 mm = 0.0022 m[/tex]
While the diameter of one circle of the solenoid is
[tex]d=2.10 cm = 0.021 m[/tex]
So the perimeter of one circle is
[tex]p=\pi d=\pi (0.021)=0.0659 m[/tex]
So the number of complete circles in the solenoid is
[tex]n=\frac{L}{p}=\frac{15.0}{0.0659}=227.6[/tex]
The tickness of one circle is [tex]t[/tex], so the total length of the solenoid will be:
[tex]L'=nt=(227.6)(0.0022)=0.5 m[/tex]
b)
The magnetic field at the center of a solenoid is given by
[tex]B=\mu_0 n I[/tex]
where
[tex]\mu_0[/tex] is the vacuum permeability
n is the number of turns of the solenoid
I is the current in the solenoid
Here we have:
[tex]\mu_0 =4\pi \cdot 10^{-7}H/m[/tex] is the vacuum permeability
[tex]n=227.6[/tex] is the number of turns in the solenoid (calculated in part a)
I = 16.7 A is the current in the solenoid
Substituting, we find:
[tex]B=(4\pi \cdot 10^{-7})(227.6)(16.7)=4.77\cdot 10^{-3} T[/tex]
