Quartz Crystal Optical Rotation Calculation

When plane-polarized light passes through certain materials, such as a quartz crystal, its plane of polarization rotates. This fascinating phenomenon is known as optical activity. It occurs because optically active materials interact differently with right-handed and left-handed circularly polarized light, causing them to travel at different speeds and thus have different refractive indices.

Key Concepts of Optical Activity

• Optical Activity: This is the property of a substance to rotate the plane of polarization of monochromatic plane-polarized light that passes through it.
• Circularly Polarized Light: Plane-polarized light can be thought of as a superposition of two circularly polarized components: a right-handed circularly polarized (RHCP) component and a left-handed circularly polarized (LHCP) component.
• Refractive Indices: In an optically active material like quartz crystal, the refractive index for the LHCP component (\(n_L\)) is slightly different from that for the RHCP component (\(n_R\)). This difference (\(n_L - n_R\)) leads to the rotation of the plane of polarization.

Given Parameters for Quartz Crystal

From the question, we are provided with the following information for the quartz crystal:

• Refractive index for right-handed circularly polarized light, \(n_R = 1.5391\)
• Refractive index for left-handed circularly polarized light, \(n_L = 1.5392\)
• Wavelength of the light, \(\lambda = 762.9 \text{ nm}\)
• Thickness of the quartz crystal plate, \(d = 0.5 \text{ mm}\)

Unit Conversion for Calculation

Before proceeding with the calculation, it's important to convert all units to a consistent system, typically meters (m) for length and nanometers (nm) for wavelength if keeping it separate, or convert everything to meters.

• Wavelength, \(\lambda = 762.9 \text{ nm} = 762.9 \times 10^{-9} \text{ m}\)
• Thickness, \(d = 0.5 \text{ mm} = 0.5 \times 10^{-3} \text{ m}\)

Formula for Angle of Rotation

The angle of rotation (\(\theta\)) produced by an optically active crystal plate is directly proportional to the thickness of the plate and the difference in refractive indices for the two circularly polarized components, and inversely proportional to the wavelength of light. The formula for the angle of rotation in radians is:

\[ \theta_{\text{radians}} = \frac{\pi d (n_L - n_R)}{\lambda} \]

To convert this angle from radians to degrees, we multiply by the conversion factor \(\frac{180^\circ}{\pi \text{ radians}}\):

\[ \theta_{\text{degrees}} = \frac{\pi d (n_L - n_R)}{\lambda} \times \frac{180}{\pi} \]

The \(\pi\) terms cancel out, simplifying the formula for the angle of rotation directly in degrees:

\[ \theta_{\text{degrees}} = \frac{180 d (n_L - n_R)}{\lambda} \]

Step-by-Step Angle of Rotation Calculation

Let's substitute the given values into the simplified formula:

1. Calculate the difference in refractive indices:

\[ n_L - n_R = 1.5392 - 1.5391 = 0.0001 \]

2. Substitute all values into the formula for angle of rotation in degrees:

\[ \theta_{\text{degrees}} = \frac{180 \times (0.5 \times 10^{-3} \text{ m}) \times (0.0001)}{762.9 \times 10^{-9} \text{ m}} \]

\[ \theta_{\text{degrees}} = \frac{180 \times (5 \times 10^{-4}) \times (1 \times 10^{-4})}{762.9 \times 10^{-9}} \]

\[ \theta_{\text{degrees}} = \frac{900 \times 10^{-8}}{762.9 \times 10^{-9}} \]

3. Simplify the expression by handling the powers of 10:

We can rewrite \(10^{-8}\) as \(10 \times 10^{-9}\):

\[ \theta_{\text{degrees}} = \frac{900 \times 10 \times 10^{-9}}{762.9 \times 10^{-9}} \]

The \(10^{-9}\) terms cancel out:

\[ \theta_{\text{degrees}} = \frac{9000}{762.9} \]

4. Perform the final division:

\[ \theta_{\text{degrees}} \approx 11.7976 \text{ degrees} \]

Final Angle of Rotation

Rounding the calculated angle of rotation to one decimal place, which is standard for such measurements and options provided:

\[ \theta_{\text{degrees}} \approx 11.8^\circ \]

This calculated angle of rotation produced by the quartz crystal plate is approximately \(11.8^\circ\).