Question

Consider a ray of light traveling in a vacuum from point \(P_{1}\) to \(P_{2}\) by way of the point \(Q\) on a plane mirror, as in Figure \(6.8 .\) Show that Fermat's principle implies that, on the actual path followed, \(Q\) lies in the same vertical plane as \(P_{1}\) and \(P_{2}\) and obeys the law of reflection, that \(\theta_{1}=\theta_{2} .[\) Hints: Let the mirror lie in the \(x z\) plane, and let \(P_{1}\) lie on the \(y\) axis at \(\left(0, y_{1}, 0\right)\) and \(P_{2}\) in the \(x y\) plane at \(\left(x_{2}, y_{2}, 0\right)\) Finally let \(Q=(x, 0, z)\). Calculate the time for the light to traverse the path \(P_{1} Q P_{2}\) and show that it is minimum when \(Q\) has \(z=0\) and satisfies the law of reflection.]

Short answer

The point \(Q\) lies on the mirror plane \(z=0\), satisfying \(\theta_1 = \theta_2\).

Step-by-step solution

Write the Expression for Travel Time

The light travels from \(P_1\) to \(Q\) and then from \(Q\) to \(P_2\). Since the speed of light \(c\) is constant, the travel time \(T\) is the sum of the time taken for each segment, computed as path distance divided by \(c\).Travel time \(T\) is given by: \[T = \frac{1}{c}(\sqrt{x^2 + y_1^2 + z^2} + \sqrt{(x_2 - x)^2 + y_2^2 + z^2}).\]

Derive the Condition for Minimum Travel Time

To find the path that minimizes time, differentiate \(T\) with respect to \(x\) and set the derivative equal to zero:\[ \frac{dT}{dx} = \frac{x}{c\sqrt{x^2 + y_1^2 + z^2}} - \frac{x_2 - x}{c\sqrt{(x_2 - x)^2 + y_2^2 + z^2}} = 0.\]

Simplify the Derivative Expression

Reorganize the terms and recognize that equalizing derivatives relate to minimizing differences due to symmetry. Simplifying:\[ \frac{x}{\sqrt{x^2 + y_1^2 + z^2}} = \frac{x_2 - x}{\sqrt{(x_2 - x)^2 + y_2^2}}.\]

Apply the Law of Reflection

Interpreting previous results geometrically, \(Q\) lies directly under \(P_1\) and \(P_2\) when \(z = 0\). Also, the equality from Step 3 aligns with \(\theta_1 = \theta_2\) geometrically, confirming reflection law.

Conclude with Fermat's Principle

The analysis shows that Fermat’s principle implies the plane must be oriented such that \(Q\) falls on it, thus confirming both \(Q\)'s location and the reflection angle. Therefore, as \(Q\) lies on the plane at \(z=0\) and \(\theta_1 = \theta_2\), it follows the reflection law, maximizing time efficiency.

Key concepts

Understanding the Law of Reflection

When light encounters a surface such as a plane mirror, it reflects off the surface according to a predictable rule called the law of reflection. This law states that the angle at which the incoming light ray, or the incident ray, strikes the surface is equal to the angle at which it reflects away. These angles are measured with respect to a perpendicular line to the surface, called the normal.

• The angle of incidence \( \theta_1 \) is the angle between the incoming light ray and the normal.
• The angle of reflection \( \theta_2 \) is the angle between the reflected light ray and the normal.
• According to the law of reflection, \( \theta_1 = \theta_2 \).

Applying this concept to the scenario described in the exercise, when a light ray travels from point \(P_1\) to point \(P_2\) by reflecting off a point \(Q\) on the plane mirror, it must reflect such that the angles of incidence and reflection are the same, satisfying \( \theta_1 = \theta_2 \). This relationship guarantees that light takes the shortest path in terms of time, adhering to Fermat's Principle.

The Role of a Plane Mirror

A plane mirror is characterized by its flat reflective surface. This flatness plays a crucial role in the predictability of light reflections. When light reflects off a plane mirror, the image produced is a "virtual" image lying behind the mirror at the same distance as the object in front of it.
In the given exercise, consider a mirror that lies in the \(xz\)-plane:

• The mirror's surface is defined by the plane where \(z = 0\).
• The point \(Q\), where the light ray reflects, lies on this plane.
• The mirror causes the light path to change, but the rule of reflection (\(\theta_1=\theta_2\)) ensures symmetrical travel in the light path.

In simple terms, the plane mirror ensures that how light travels in and out is symmetrical around the normal at point \(Q\). This idea is crucial in confirming that the ray travels the shortest possible path overall.

Light Ray Travel: Following Fermat's Principle

Light always travels in a way that takes the least time between two points, a concept known as Fermat's Principle. This principle underlies how light chooses its path, particularly when encountering barriers like a plane mirror.
In the scenario between points \(P_1\) and \(P_2\), the light travels by first reaching point \(Q\) on the mirror. Here is what happens:

• The total travel time \(T\) is minimized when reflected at \(Q\), confirming Fermat's Principle.
• Mathematically, this means differentiating the light travel path to find the minimum time, leading to symmetry in angles and verifying \(\theta_1 = \theta_2\).
• This symmetrical angle is both a consequence of and reason for the path's minimized travel time, emphasizing the efficient navigation between points.

Therefore, the choice of path by light is highly intentional, aimed at using the least time possible to traverse from \(P_1\) via \(Q\) to \(P_2\). Fermat’s Principle illuminates this path selection, ensuring it aligns with the foundational laws of reflection.