Example problems of magnetic circuits

Given:

A copper core with susceptibility ${\chi }_m=-9.7\times 10^{-6}$

length of core L = 1 m

Gap length g = .01 m

cross sectional area A = .1 m

current I = 10A

N = 5 turns

Find: $B_g$

Solution: First we need to find the permeability of copper $\mu$ given by the equation
$\mu ={\mu }_0(1+{\chi }_m)$

Which yeilds $\mu =4\times \pi \times 10^{-7}(1+-9.7\times 10^{-6})=1.2566\times 10^{-6}\frac{N}{A^2}$

Now with this, the length and cross sectional area of the core we can solve for reluctance $R_c$ by:

$R_c=\frac{L}{\mu A}=\frac{1}{1.2566\times 10^{-6}\times .1}=7.96\times 10^6$

Similarly to get the reluctance of the gap

$R_g=\frac{g}{{\mu }_0(\sqrt{A}+g{)}^2}=\frac{.01}{4\times \pi \times 10^{-7}(\sqrt{.1}+.01{)}^2}=74.8\times 10^3$

Now Using $B_g=\frac{NI}{(R_g{R}_c)((\sqrt{A}+g{)}^2}$
Yields <math> B_g = \frac{5 \times 10}{74.8 \times 10^{3} \times 7.96 \times 10^{6} \times (\sqrt{.1} + .01)^2} = .789 \times 10^{-9}