9.2.3. Sign Convention for Reflection by Spherical Mirrors

While dealing with the reflection of light by spherical mirrors, we shall follow a set of sign conventions called the New Cartesian Sign Convention. In this convention, the pole (P) of the mirror is taken as the origin. The principal axis of the mirror is taken as the x-axis (X’X) of the coordinate system.

The New Cartesian Sign Convention is a method used to simplify the calculations involving reflection in spherical mirrors. In this system, the pole of the mirror (the point where the light rays strike the mirror) is set as the reference point or origin. The vertical line that goes through the center of the mirror is defined as the x-axis. This standardizes how we measure distances and helps avoid errors in calculations.

Think of using a map where you have set a specific point as your 'home'—all distances from your home can be measured either positively (to the right) or negatively (to the left). Similarly, the pole of the mirror acts as that reference point for measuring distances in the sign convention.

• The object is always placed to the left of the mirror. This implies that the light from the object falls on the mirror from the left-hand side.
• All distances parallel to the principal axis are measured from the pole of the mirror.

In this convention, we always assume that the object is on the left side of the mirror, meaning rays of light travel from the left towards the mirror. This positioning helps us organize our reflections and provides consistency in the calculations. Additionally, distances that we measure along the principal axis (x-axis) start from the pole of the mirror, allowing us to define positive and negative distances depending on their direction.

Imagine shining a flashlight towards a wall. The source of the light (the flashlight) is your object. In our sign convention, we always keep the flashlight on the left side when considering where the light goes. Any measurements we make about the position or distance to the wall (the mirror in this case) start from the place where the light beam first hits, just like we measure distances from the pole of the mirror.

• All the distances measured to the right of the origin (along + x-axis) are taken as positive while those measured to the left of the origin (along – x-axis) are taken as negative.
• Distances measured perpendicular to and above the principal axis (along + y-axis) are taken as positive.
• Distances measured perpendicular to and below the principal axis (along – y-axis) are taken as negative.

In this section of the convention, we define how to classify different distances based on their direction. Distances moving to the right (away from the mirror) are considered positive, while distances to the left (towards the mirror) are negative. Similarly, vertical measurements above the principal axis are positive, and those below are negative. This system allows us to easily describe the location of objects and images related to the mirror during calculations.

Think of a number line that you learned in math class. When you move to the right, the numbers increase and are considered positive; when you move to the left, the numbers decrease and are considered negative. Similarly, in our mirror convention, anything to the right of the pole is positive, while everything to the left is negative, helping us keep track of where things are located relative to the mirror.

The New Cartesian Sign Convention described above is illustrated in figure 9.9 for your reference. These sign conventions are applied to obtain the mirror formula and solve related numerical problems.

Figure 9.9 visually represents how we set our coordinate system in relation to the mirror. It shows the directions for positive and negative distances clearly, which helps in understanding and applying the sign support the calculations related to mirrors. Understanding this visual representation ensures that students can apply the correct principles when using mathematical equations in mirror problems.

Just as you'd refer to a map that shows you where everything is located—like where the streets are and where your destination is—the visual representation of the sign convention provides a clear guide for understanding how to measure and calculate distances in relation to the mirror, making it easier to navigate through problems.

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