Diamagnetism — Question 7

Question 7

A diamagnetic core changes stored energy differently under two electrical constraints

A long coil has empty inductance L0 and filled inductance $L=L_0(1+\chi)$, with -1 less than chi less than zero and no hysteresis. Compare empty and filled states.

1. Find the stored magnetic-energy ratio when current is held fixed.

2. Find the ratio when flux linkage Lambda is held fixed. Explain why the two energy changes need not have the same sign.

Question 7 - Solution

A diamagnetic core changes stored energy differently under two electrical constraints

(a) Method and calculation

$$U_I=\tfrac12LI^2\Rightarrow\boxed{U_{I,\mathrm{filled}}/U_{I,\mathrm{empty}}=1+\chi<1}.$$$$\Delta U_I=\tfrac12L_0I^2\chi<0.$$

(b) Interpretation and further calculation

$$U_\Lambda=\frac{\Lambda^2}{2L}\Rightarrow\boxed{U_{\Lambda,\mathrm{filled}}/U_{\Lambda,\mathrm{empty}}=\frac1{1+\chi}>1}.$$$$\boxed{\Delta U_\Lambda=-\frac{\chi}{1+\chi}\frac{\Lambda^2}{2L_0}>0}.$$

Physical check

A fixed-current source exchanges energy during insertion; field-energy change alone is not the mechanical work in that case.