The related concept to solve this exercise is given in the expressions that the magnetic field has both as a function of the number of loops, current and length, as well as inductance and permeability. The first expression could be given as,
The magnetic field H is given as,
![H = \frac{nI}{l}](https://tex.z-dn.net/?f=H%20%3D%20%5Cfrac%7BnI%7D%7Bl%7D)
Here,
n = Number of turns of the coil
I = Current that flows in the coil
l = Length of the coil
From the above equation, the number of turns of the coil is,
![n = \frac{Hl}{I}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7BHl%7D%7BI%7D)
The magnetic field is again given by,
![H = \frac{B}{\mu_t}](https://tex.z-dn.net/?f=H%20%3D%20%5Cfrac%7BB%7D%7B%5Cmu_t%7D)
Where the minimum inductance produced by the solenoid coil is B.
We have to obtain n, that
![n = \dfrac{\frac{B}{\mu_t}l}{I}](https://tex.z-dn.net/?f=n%20%3D%20%5Cdfrac%7B%5Cfrac%7BB%7D%7B%5Cmu_t%7Dl%7D%7BI%7D)
Replacing with our values we have that,
![n = \dfrac{\frac{1.1Wb/m^2 }{200000}(2m)}{4mA}](https://tex.z-dn.net/?f=n%20%3D%20%5Cdfrac%7B%5Cfrac%7B1.1Wb%2Fm%5E2%20%7D%7B200000%7D%282m%29%7D%7B4mA%7D)
![n = \dfrac{(\frac{1.1Wb/m^2 }{200000})(\frac{10^4 guass}{1Wb/m^2})(2m)}{4mA(\frac{10^{-3}A}{1mA})}](https://tex.z-dn.net/?f=n%20%3D%20%5Cdfrac%7B%28%5Cfrac%7B1.1Wb%2Fm%5E2%20%7D%7B200000%7D%29%28%5Cfrac%7B10%5E4%20guass%7D%7B1Wb%2Fm%5E2%7D%29%282m%29%7D%7B4mA%28%5Cfrac%7B10%5E%7B-3%7DA%7D%7B1mA%7D%29%7D)
![n = 27.5 \approx 28](https://tex.z-dn.net/?f=n%20%3D%2027.5%20%5Capprox%2028)
Therefore the number of turn required is 28Truns