PhysGym

495

OPTICS

In 1845, Faraday studied the influence of electricity and magnetism on polarized light and discovered that heavy glass, originally non-optically active, exhibited optical activity under the influence of a strong magnetic field, causing the plane of polarization of polarized light to rotate. This was the first time humanity recognized the relationship between electromagnetic phenomena and light phenomena. The magneto-optical rotation effect later became known as the Faraday Effect. In the magneto-optical rotation effect, the rotation angle of the vibration plane \(\beta\) is proportional to th …

Solution: The propagation of light waves in a medium under an applied magnetic field involves the motion of electrons: \[m\ddot{\vec{r}} = -k\vec{r} - g\vec{v} - e(\vec{E} + \vec{v} \times \vec{B})\] \(\vec{E}\) is the electric field intensity of the incident light wave; \(\vec{B}\) is the magnetic induction intensity applied to the medium; \(g\) is the damping term, and since the medium does not absorb the incident light, it exhibits weak damping and low loss, thus neglecting the damping term: \[m\ddot{\vec{r}} = -k\vec{r} - e(\vec{E} + \vec{v} \times \vec{B})\] Let \(\vec{B}\) be along the p …

\[v = \frac{NZ e^3}{2c\varepsilon_0 m^2 n} \cdot \frac{\omega^2}{(\omega_0^2 - \omega^2)^2}\]

def env_function(N, Z, e, c, epsilon_0, m, n, omega, omega_0): """ Calculate the Verdet constant for the Faraday Effect. Args: N: Number of atoms per unit volume Z: Number of electrons in the outer shell of each atom e: Elementary charge c: Speed of light in vacuum epsilon_0: Vacuum permittivity m: Electron mass n: Refractive index of the medium omega: Angular frequency of incident light omega_0: Harmonic resonance frequency of outer shell electrons Returns: The Verdet constant …

(N * Z * e**3) / (2 * c * epsilon_0 * m**2 * n) * (omega**2) / ((omega_0**2 - omega**2)**2)

{"A":null,"A_0":null,"B":null,"B_0":null,"D":null,"E":null,"E_0":null,"F":null,"G":null,"H":null,"I":null,"I_0":null}

{"A":null,"Delta_alpha":null,"Delta_y":null,"E":null,"E_1":null,"F":null,"F_max":null,"I":null,"M":null,"N_prime":null,"Omega":null,"Omega_2":null}

{"D":null,"E":null,"E_0":null,"E_m":null,"H":null,"I":null,"L":null,"L_0":null,"M":null,"R":null,"R_0":null,"R_BC_direct":null}

285

MECHANICS

A small bug with a mass of $m$ crawls on a disk with a radius of $2R$. Relative to the disk, its crawling trajectory is a circle of radius $R$ that passes through the center of the disk. The disk rotates with a constant angular velocity $\omega$ about an axis passing through its center and perpendicular to the plane of the disk. The bug's angular crawling velocity relative to the disk is in the same direction as the disk's angular velocity and has the same magnitude. Solve for the maximum force $F_{max}$ between the bug and the disk required to maintain this motion (neglecting gravity).

The problem can use the acceleration transformation formula for rotating reference frames, where the actual acceleration of the small insect $$ \overrightarrow{a}=\overrightarrow{a}+2\overrightarrow{\omega}\times\overrightarrow{v}+\dot{\overrightarrow{\omega}}\times\overrightarrow{r}-\omega^{2}\overrightarrow{r} $$ Here, $\vec{a}^{\prime}$ is the acceleration relative to the rotating system, and since the rotating system is a uniform circular motion, $$ \dot{a_{\mathrm{\scriptsize~=~}}}\omega^{2}R $$ is the speed relative to the rotating system, with a magnitude of $\omega R$. Thus, th …

$$F_{\max} = 5m\omega^2R$$

def env_function(m, R, omega): """ Calculate the maximum force between a bug and a disk required to maintain motion. Args: m: Mass of the bug R: Radius parameter (half of disk radius) omega: Angular velocity Returns: Maximum force between bug and disk Raises: ValueError: If any input parameters are negative """ if m <= 0 or R <= 0 or omega < 0: raise ValueError("Invalid values.") return 5 * m * (omega**2) * R

5 * m * (omega**2) * R

{"A":null,"A_0":null,"B":null,"B_0":null,"D":null,"E":null,"E_0":null,"F":null,"G":null,"H":null,"I":null,"I_0":null}

{"A":null,"Delta_alpha":null,"Delta_y":null,"E":null,"E_1":null,"F":null,"F_max":"Maximum force between the bug and the disk required to maintain the motion (in Newtons, N).","I":null,"M":null,"N_prime":null,"Omega":null,"Omega_2":null}

{"D":null,"E":null,"E_0":null,"E_m":null,"H":null,"I":null,"L":null,"L_0":null,"M":null,"R":null,"R_0":null,"R_BC_direct":null}