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

Solution: First we need to find the permeability of copper $\mu$ given by the equation
$\mu =\mu _{0}(1+\chi _{m})$

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

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

Failed to parse (unknown function "\time"): {\displaystyle  R_g = \frac {g}{\mu_0 (\sqrt{A} + g)^2} = \frac {.01}{4 \times \pi \times 10^{-7} (\sqrt{.1} + .01)^2} = 74.8 \time 10^{3}  }