Problem 63 - College Physics

Question

Show that when the reflected light is completely polarized, the sum of the angle of incidence and the angle of refraction is equal to \(90^{\circ}\).

Step-by-Step Solution

Answer

When light is completely polarized, the sum of the angle of incidence and the angle of refraction is 90 degrees, consistent with Brewster's Law.

Step 1: Understand the Problem

We're asked to prove a property of light polarization upon reflection: when light is completely polarized upon reflection, the sum of the angle of incidence and the angle of refraction is 90 degrees. This relates to Brewster's angle.

Step 2: Recall Brewster's Law

According to Brewster's Law, complete polarization occurs when the angle of incidence, called the Brewster angle (\(\theta_B \)), satisfies the condition \(\tan \theta_B = n \), where \(n\) is the refractive index of the second medium relative to the first.

Step 3: Express Angles in Mathematical Form

Let \(\theta_i\) be the angle of incidence and \(\theta_r\) be the angle of refraction. According to Snell's law, \(n_1 \sin \theta_i = n_2 \sin \theta_r\). According to Brewster's law, when light is completely polarized, \(\theta_i + \theta_r = 90^\circ\).

Step 4: Use Snell's Law

Apply Snell's law: \(\sin \theta_i = n \sin \theta_r\). With \(\theta_i + \theta_r = 90^\circ\), \(\sin \theta_r = \cos \theta_i\) because \(\theta_r = 90^\circ - \theta_i\). Thus, \(\sin \theta_i = n \cos \theta_i\).

Step 5: Solve the Equation

From \(\sin \theta_i = n \cos \theta_i\) divide both sides by \(\cos \theta_i\): \(\tan \theta_i = n\). This is Brewster's Law, verifying that when light is completely polarized, \(\theta_i + \theta_r = 90^\circ\).

Step 6: Conclusion

Since the calculation derived \(\tan \theta_i = n\) and supports both Brewster's condition and the angle sum property, we have shown that when the reflected light is completely polarized, \(\theta_i + \theta_r = 90^\circ\).

Key Concepts

Brewster's Law

Brewster's Law is a fascinating concept in the study of light and polarization. It tells us about the condition under which light will become completely polarized upon reflection. Brewster's Law is mathematically stated as \( \tan \theta_B = n \), where \( \theta_B \) is the angle of incidence known as Brewster's angle. Here, \( n \) represents the refractive index of the medium into which the light is entering from another medium. This law highlights that when unpolarized light strikes a medium at Brewster's angle, the reflected light will be entirely polarized parallel to the surface.

• The Brewster angle is the special angle of incidence at which the reflectance for light with one particular polarization is minimized.
• This results in the reflected light being completely polarized.
• Natural examples include glare from water or roads, commonly reduced by polarizing sunglasses.

Understanding Brewster's Law is crucial for applications such as designing optical instruments and reducing unwanted reflections.

Angle of Incidence

The angle of incidence is a fundamental concept when studying wave reflections, including light waves. It is defined as the angle between the incident ray of light and the normal (a line perpendicular) to the surface at the point of incidence. In the context of Brewster's Law, the angle of incidence, especially when it equals Brewster's angle, plays a significant role in determining the polarization of reflected light.

• A larger angle of incidence can increase the amount of polarized light reflected at specific conditions.
• At Brewster’s angle, the reflected and refracted light rays are perpendicular.
• Modifying this angle can change the amount and type of light passing through or reflecting off surfaces.

Understanding the angle of incidence helps in controlling light-energy interactions in various optical devices.

Snell's Law

Snell's Law is a key principle in optics that defines how light bends, or refracts, as it moves from one medium to another. The relationship is given by \( n_1 \sin \theta_i = n_2 \sin \theta_r \), where \( n_1 \) and \( n_2 \) are the refractive indices of the two media, \( \theta_i \) is the angle of incidence, and \( \theta_r \) is the angle of refraction.

• When light enters a denser medium, it slows down and bends towards the normal.
• Conversely, moving to a less dense medium, it speeds up and bends away from the normal.
• Snell's Law helps to determine various optical properties related to lenses, prisms, and other refractive devices.

This law is essential for understanding phenomena such as total internal reflection and is foundational for technologies like fiber optics and corrective lenses.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal to the surface. It is directly related to the angle of incidence and the refractive indices of the two materials according to Snell’s Law. When light passes through different media, the angle of refraction dictates how much the light bends.

• The angle of refraction depends on both the wavelength of the light and the refractive index of the mediums involved.
• If the refractive index of the second medium is greater, the angle of refraction is less than the angle of incidence.
• At Brewster's angle, the refracted ray and the reflected ray are perpendicular to each other.

The concept of the angle of refraction is vital for designing systems that rely on precise light manipulation, such as cameras and telescopes.