Energy Density of a Magnetic Field — Question 10

Question 10

A narrow air gap can hold most of a magnetic circuit energy

A closed magnetic circuit has equal cross-sectional area A in its core and gap, core length $\ell_c$, and gap length g. Neglect leakage. The core is linear, lossless, and nonhysteretic with $\mu=\mu_r\mu_0$; use $u=B^2/(2\mu)$ there. The same B threads core and gap.

1. Find the ratio of gap energy to core energy and the gap fraction of total energy.

2. Evaluate the gap fraction for $\mu_r=2000$, $\ell_c=.400$ m, and g=.00100 m.

Question 10 - Solution

A narrow air gap can hold most of a magnetic circuit energy

(a) Method and calculation

$$U_g=\frac{B^2Ag}{2\mu_0},\quad U_c=\frac{B^2A\ell_c}{2\mu_r\mu_0}.$$$$\boxed{U_g/U_c=\mu_rg/\ell_c,\quad f_g=\frac{g}{g+\ell_c/\mu_r}}.$$

(b) Interpretation and further calculation

$$U_g/U_c=(2000)(.00100)/.400=5.$$$$\boxed{f_g=5/6=.833333}.$$

Physical check

High permeability lowers core energy density at a given B, so a short vacuum gap can dominate the stored energy.