Solution

Statement

$9.2.21.$ Estimate the magnetic field induction at the center of a flat iron ring of thickness $1$ cm with an internal radius of $10$ cm and an external radius of $20$ cm. All iron atoms are oriented along the axis of the ring, the magnetic moment of an iron atom is $2\mu_e = 1.85\cdot10^{-23}$ J/T.

Solution

We will use the solution to the previous problem (see 9.2.20). The field at short distances can be expressed through the result of the previous problem as:

$$B = B_{R_1} - B_{R_2} = \frac{\mu_0 M h}{2} \left( \frac{1}{R_1}-\frac{1}{R_2} \right)$$

Let's estimate the magnetic moment per unit volume (magnetization):

$$M = \frac{2\rho N_A \mu_e}{M_{Fe}}$$

Substituting and calculating:

$$B = \frac{\mu_0\rho N_A \mu_eh}{M_{Fe}} \left( \frac{1}{R_1}-\frac{1}{R_2} \right) \approx 4.9 \cdot 10^{-2} \text{ T}$$