Ferromagnetism — Question 9

Question 9

Competing wall and stray-field energies favor a finite domain count

A toy model for n domains in a specimen assigns energy $E(n)=an+b/n$, with positive constants a and b. Treat n as continuous for minimization, then compare adjacent integers. The first term represents domain-wall cost and the second stray-field energy.

1. Find the continuous optimum and minimum energy, and verify it is a minimum.

2. For b/a=20, compare n=4 and n=5. Explain why neither a single domain nor infinitely many domains is automatically favored.

Question 9 - Solution

Competing wall and stray-field energies favor a finite domain count

(a) Method and calculation

$$\frac{dE}{dn}=a-b/n^2=0\Rightarrow\boxed{n_*=\sqrt{b/a}}.$$$$E^{\prime\prime}=2b/n^3>0,\quad\boxed{E_{\min}=2\sqrt{ab}}.$$

(b) Interpretation and further calculation

$$b/a=20:\quad E(4)/a=4+20/4=9,\quad E(5)/a=5+20/5=9.$$$$\boxed{n=4\text{ and }n=5\text{ tie in this discrete model}}.$$

Physical check

Domains reduce stray-field energy but create walls; the supplied model balances these costs without claiming a universal domain geometry.