When an ordinary beam of light encounters a transparent medium like a rectangular glass plate, its behavior upon reflection and refraction depends significantly on the angle at which it hits the surface. A special angle known as the polarizing angle, or Brewster's angle, is crucial in understanding this phenomenon.

Polarizing Angle and Brewster's Law

The polarizing angle (\(i_p\)) is the specific angle of incidence for which the reflected light is completely plane-polarized. At this angle, the reflected light contains vibrations only perpendicular to the plane of incidence. This principle is governed by Brewster's Law.

Brewster's Law states that the tangent of the polarizing angle is equal to the refractive index (\(n\)) of the transparent medium. Mathematically, it is expressed as:

\(\tan i_p = n\)

Where:

• \(i_p\) is the polarizing angle (angle of incidence).
• \(n\) is the refractive index of the medium (in this case, the glass plate).

Reflected Beams and Refracted Beams at Polarizing Angle

A key consequence of light being incident at the polarizing angle is the relationship between the resulting reflected and refracted beams. According to Brewster's Law:

• When an unpolarized beam of ordinary light is incident on a surface at the polarizing angle, the reflected beam is completely plane-polarized.
• More importantly for this question, the reflected beam and the refracted beam are found to be exactly perpendicular to each other.

This means the angle between the reflected ray and the refracted ray is \(90^\circ\).

Let's consider the angles involved:

• Angle of incidence = \(i_p\) (polarizing angle)
• Angle of reflection = \(i_p\) (from the law of reflection)
• Angle of refraction = \(r\)

From Snell's Law, we have \(\frac{\sin i_p}{\sin r} = n\).

Since \(\tan i_p = n\), we can write \(\frac{\sin i_p}{\cos i_p} = n\).

Comparing these expressions for \(n\), we get \(\frac{\sin i_p}{\sin r} = \frac{\sin i_p}{\cos i_p}\), which implies \(\sin r = \cos i_p\).

We know that \(\cos i_p = \sin (90^\circ - i_p)\).

Therefore, \(\sin r = \sin (90^\circ - i_p)\), which leads to \(r = 90^\circ - i_p\).

Rearranging this equation, we get \(i_p + r = 90^\circ\).

The angle between the reflected ray and the refracted ray is given by \(180^\circ - (\text{angle of reflection} + \text{angle of refraction})\).

Angle between reflected and refracted ray = \(180^\circ - (i_p + r)\)

Since we found \(i_p + r = 90^\circ\), substituting this into the equation gives:

Angle between reflected and refracted ray = \(180^\circ - 90^\circ = 90^\circ\).

Thus, when a beam of ordinary light is incident on a rectangular glass plate at a polarizing angle, the resulting reflected and refracted beams are indeed perpendicular to each other.