Energy Density of a Magnetic Field — Question 7

Question 7

Gap energy predicts magnetic attraction and pressure

Two ideal pole faces of area A bound a uniform vacuum gap g. Fringing and all other field energy are negligible. During a virtual gap change, flux Phi is conserved and no electrical work enters. Positive force is toward increasing g.

1. Derive the force on the movable face from gap energy.

2. Evaluate pressure and force magnitude for B=.800 T and A=.00200 square meters.

Question 7 - Solution

Gap energy predicts magnetic attraction and pressure

(a) Method and calculation

$$U(g)=\frac{\Phi^2g}{2\mu_0A},\quad F_g=-\left(\frac{\partial U}{\partial g}\right)_\Phi.$$$$\boxed{F_g=-\frac{\Phi^2}{2\mu_0A}=-\frac{B^2A}{2\mu_0},\quad p=\frac{B^2}{2\mu_0}}.$$

(b) Interpretation and further calculation

$$p=\frac{(.800)^2}{8\pi\times10^{-7}}=2.54648\times10^5\,Pa.$$$$\boxed{|F_g|=pA=509.296\,N}.$$

Physical check

The negative generalized force closes the gap. Pressure has the same units as energy density.