Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A ray of light travelling along the line \(x+\sqrt{3 y}=5\) is incident on the \(x\)-axis and after refraction it enters the other side of the \(x\)-axis by turning \(\frac{\pi}{6}\) away from the \(x\)-axis. The equation of the line along which the refracted ray travels is (A) \(x+\sqrt{3} y-5 \sqrt{3}=0\) (B) \(x-\sqrt{3} y-5 \sqrt{3}=0\) (C) \(\sqrt{3} x+y-5 \sqrt{3}=0\) (D) \(\sqrt{3} x-y-5 \sqrt{3}=0\)

Step-by-Step Solution

Answer

The equation of the refracted ray is (B) \(x - \sqrt{3}y - 5\sqrt{3} = 0\).

Step 1: Identify the Line Equation

The equation for the incident ray is given as \(x + \sqrt{3}y = 5\). This is a line with direction \((1, \sqrt{3})\).

Step 2: Calculate the Incident Angle

The slope of the incident line is \(-\frac{1}{\sqrt{3}}\), so the angle \(\theta\) with the positive \(x\)-axis is \(\text{tan}^{-1}(-\frac{1}{\sqrt{3}})\), which simplifies to \(-\frac{\pi}{6}\).

Step 3: Determine the Refraction Path

After refraction, the ray turns \(\frac{\pi}{6}\) away from the \(x\)-axis. Since the original angle was \(-\frac{\pi}{6}\), the new angle with the \(x\)-axis is \(0\), indicating the line is parallel to the \(y\)-axis passing through the point already determined.

Step 4: Find the Point of Intersection

Set \(x + \sqrt{3}y = 5\) and \(y = 0\) to find the point of intersection on the \(x\)-axis. Solving gives the point \((5,0)\).

Step 5: Determine the Refracted Line

The refracted line should be \(x + \sqrt{3}y = m\) or similar with the new direction \((1, -\sqrt{3})\). Check by substituting \((5,0)\) into the equations given. Only \(x - \sqrt{3}y = 5\sqrt{3}\) holds with the condition, making option (B) valid.

Key Concepts

Incident Ray

The incident ray is the path that light follows before it interacts with a surface. In this exercise, the incident ray is described by the equation \(x + \sqrt{3}y = 5\). An incident ray travels towards a medium, like a mirror or glass, where it meets a point known as the point of incidence.

• The importance of identifying the incident ray lies in predicting how light will behave upon contact with a different medium.
• This path develops an intersection with the \(x\)-axis, creating specific conditions for further calculations.

In this case, the point where the incident ray meets the \(x\)-axis is pivotal for determining the refracted ray's trajectory.

Refracted Ray

After a ray of light meets a surface, it can change its direction due to a change in medium. This new path is called the refracted ray. In the given problem, the incident ray refraction is manipulated by an angle away from the original path, leading to a new path given by equation (B): \(x - \sqrt{3}y = 5\sqrt{3}\).

• The refracted ray signifies how light exits the incident point, differing from its original path.
• Law of Refraction, commonly known as Snell’s Law, helps predict precisely how light bends.

Understanding refraction is crucial in lenses and optics, affecting how images are perceived.

Angle of Incidence

The angle of incidence is the angle between the incident ray and the perpendicular, or normal, to the surface at the point of incidence. In this exercise, while calculating the slope of the incident line, we deduce the angle of incidence to be \(-\frac{\pi}{6}\).

• It determines how the refracted ray will behave, as according to Snell’s Law the angle of incidence affects the angle of refraction.
• Measured from the incident ray to the normal, it plays a vital role in physics to anticipate reflections.

Calculating this angle accurately allows physicists to predict the path light will take when entering different mediums.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal to the surface at the point of incidence. This exercise shows a change upon refraction of \(\frac{\pi}{6}\) from the normal, leading to the new path of the refracted ray.

• It is calculated based on the angle of incidence and is subject to the laws of refraction.
• This angle is crucial to determining how the refracted ray angles away from the original path.

Fully understanding this angle helps explain phenomena like bending of light in water, aiding in numerous physics applications and practical optics gearing.