Answer:
B = 38.2μT
Explanation:
By the Ampere's law you have that the magnetic field generated by a current, in a wire, is given by:
(1)
μo: magnetic permeability of vacuum = 4π*10^-7 T/A
r: distance from the center of the cylinder, in which B is calculated
Ir: current for the distance r
In this case, you first calculate the current Ir, by using the following relation:
![I_r=JA_r](https://tex.z-dn.net/?f=I_r%3DJA_r)
J: current density
Ar: cross sectional area for r in the hollow cylinder
Ar is given by ![A_r=\pi(r^2-R_1^2)](https://tex.z-dn.net/?f=A_r%3D%5Cpi%28r%5E2-R_1%5E2%29)
The current density is given by the total area and the total current:
![J=\frac{I_T}{A_T}=\frac{I_T}{\pi(R_2^2-R_1^2)}](https://tex.z-dn.net/?f=J%3D%5Cfrac%7BI_T%7D%7BA_T%7D%3D%5Cfrac%7BI_T%7D%7B%5Cpi%28R_2%5E2-R_1%5E2%29%7D)
R2: outer radius = 26mm = 26*10^-3 m
R1: inner radius = 5 mm = 5*10^-3 m
IT: total current = 4 A
Then, the current in the wire for a distance r is:
(2)
You replace the last result of equation (2) into the equation (1):
![B=\frac{\mu_oI_T}{2\pi r}(\frac{r^2-R_1^2}{R_2^2-R_1^2})](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B%5Cmu_oI_T%7D%7B2%5Cpi%20r%7D%28%5Cfrac%7Br%5E2-R_1%5E2%7D%7BR_2%5E2-R_1%5E2%7D%29)
Finally. you replace the values of all parameters:
![B=\frac{(4\pi*10^{-7}T/A)(4A)}{2\PI (12*10^{-3}m)}(\frac{(12*10^{-3})^2-(5*10^{-3}m)^2}{(26*10^{-3}m)^2-(5*10^{-3}m)^2})\\\\B=3.82*10^{-5}T=38.2\mu T](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B%284%5Cpi%2A10%5E%7B-7%7DT%2FA%29%284A%29%7D%7B2%5CPI%20%2812%2A10%5E%7B-3%7Dm%29%7D%28%5Cfrac%7B%2812%2A10%5E%7B-3%7D%29%5E2-%285%2A10%5E%7B-3%7Dm%29%5E2%7D%7B%2826%2A10%5E%7B-3%7Dm%29%5E2-%285%2A10%5E%7B-3%7Dm%29%5E2%7D%29%5C%5C%5C%5CB%3D3.82%2A10%5E%7B-5%7DT%3D38.2%5Cmu%20T)
hence, the magnitude of the magnetic field at a point 12 mm from the center of the hollow cylinder, is 38.2μT