Calculating Larmor Frequency Ratio at Magnetic Latitudes

The Larmor precession frequency ($\omega$) is directly proportional to the magnetic field strength ($B$). For Earth's magnetic field, approximated as a dipole, the field strength at a magnetic latitude ($\lambda$) is given by the formula:

$B(\lambda) = B_0 \sqrt{1 + 3 \sin^2\lambda}$

where $B_0$ is the magnetic field strength at the magnetic equator ($\lambda=0^\circ$).

Magnetic Field Strength Calculation

We need to find the ratio of the Larmor frequencies at magnetic latitudes $\lambda_1 = 30^\circ$ and $\lambda_2 = 60^\circ$. This ratio is equivalent to the ratio of the magnetic field strengths at these latitudes.

• For $\lambda_1 = 30^\circ$:
  • $\sin(30^\circ) = \frac{1}{2}$
  • $\sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$
  • $B(30^\circ) = B_0 \sqrt{1 + 3 \left(\frac{1}{4}\right)} = B_0 \sqrt{1 + \frac{3}{4}} = B_0 \sqrt{\frac{7}{4}}$
• For $\lambda_2 = 60^\circ$:
  • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
  • $\sin^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$
  • $B(60^\circ) = B_0 \sqrt{1 + 3 \left(\frac{3}{4}\right)} = B_0 \sqrt{1 + \frac{9}{4}} = B_0 \sqrt{\frac{13}{4}}$

Ratio of Larmor Frequencies

The ratio of the Larmor precession frequencies is the ratio of the magnetic field strengths:

$\frac{\omega_1}{\omega_2} = \frac{B(30^\circ)}{B(60^\circ)} = \frac{B_0 \sqrt{\frac{7}{4}}}{B_0 \sqrt{\frac{13}{4}}}$

Simplifying the expression:

$\frac{\omega_1}{\omega_2} = \frac{\sqrt{\frac{7}{4}}}{\sqrt{\frac{13}{4}}} = \sqrt{\frac{7/4}{13/4}} = \sqrt{\frac{7}{13}}$

Therefore, the ratio is $\sqrt{7} : \sqrt{13}$.