Mirror Sign Conventions and Magnification Principles

Sign Convention for Spherical Mirror Parameters

To ensure consistency and mathematical accuracy when using the mirror equation, a specific sign convention must be followed. All parameters related to the position and size of objects and images are determined based on the following three fundamental rules:

1. All distances must be measured from the pole ($P$) of the spherical mirror.

2. The distances measured in the direction of incident light are to be taken as positive ($+$). Conversely, those distances measured in the direction opposite to the incident light are to be taken as negative ($-$).

3. The height of the object ($h_o$) and the height of the image ($h_i$) are considered positive ($+$) if they are measured upwards from the principal axis. They are considered negative ($-$) if they are measured downwards from the principal axis.

Definition of Magnification ($m$)

Magnification ($m$) describes the relationship between the size of the object and the size of the image produced by a spherical mirror. While images formed by spherical mirrors can vary in multiple dimensions, this discussion focuses specifically on the variation in height.

Magnification is formally defined as the ratio of the height of the image to the height of the object:

$\text{Magnification (m)} = \frac{\text{height of image } (h_i)}{\text{height of object } (h_o)}$

Mathematical Derivation of the Magnification Formula

By observing the geometry of light rays incident on a mirror, specifically those incident at the pole, we can derive a relationship between the heights and the distances of the object and image. According to Fig. 22, a ray originating from the top of the object ($O'$) is incident at the pole ($P$) with an angle of incidence $\theta$. Based on the law of reflection, it is reflected with the same angle $\theta$.

From the geometry of Fig. 22, the triangles $\triangle POO'$ and $\triangle PII'$ are identified as similar triangles. Consequently, the ratios of their corresponding sides are equal:

$\frac{II'}{OO'} = \frac{PI}{PO}$

Applying the established sign convention to the variables in this geometric relationship:

1. The object distance ($PO$) is $-u$.
2. The image distance ($PI$) is $-v$.
3. The height of the object ($OO'$) is $h_o$.
4. The height of the image ($II'$) is $-h_i$ (assuming it is inverted, as shown in the derivation logic).

Substituting these values into the ratio equation:

$\frac{-h_i}{h_o} = \frac{-v}{-u}$

This simplifies to:

$\frac{h_i}{h_o} = -\frac{v}{u}$

Therefore, the final expression for magnification in all cases is:

$m = \frac{h_i}{h_o} = -\frac{v}{u}$

Analysis of Specific Cases and Examples

Students are expected to calculate the magnifications for all five cases previously detailed in Table-2 using the newly derived formula. Understanding the specific value of $m$ allows for the determination of whether an image is enlarged (if $|m| > 1$), diminished (if $|m| < 1$), and whether it is real or virtual based on the sign.

Example Problem Scenario

Consider the following scenario for applying the mirror equation and magnification formula:

An object with a size (height) of $4\,cm$ is placed at a distance of $25\,cm$ in front of a concave mirror. The mirror has a focal length of $15\,cm$. To solve this problem, one must determine:

1. The distance from the mirror at which a screen should be placed to obtain a sharp image (the image distance, $v$).
2. The nature of the image (whether it is real, virtual, erect, or inverted).
3. The size (height) of the image ($h_i$) using the magnification formula.

Summary of Given Data for Example:

• Object height ($h_o$): $4\,cm$
• Object distance ($u$): $-25\,cm$ (due to sign convention)
• Focal length ($f$): $-15\,cm$ (for a concave mirror)