Mirror Formula and Magnification

Understanding the Mirror Formula

The Mirror Formula is a fundamental equation in ray optics that establishes a quantitative link between three essential parameters of a spherical mirror: the distance of the object from the mirror ($u$), the distance of the formed image ($v$), and the focal length of the mirror ($f$).

Cartesian Sign Convention Rules

To ensure calculations are consistent across both concave and convex mirrors, we apply the standard New Cartesian Sign Convention:

• The origin is set at the **Pole (P)** of the mirror. All distances are measured along the principal axis.
• Distances measured in the **direction of incident light** are considered **positive** (typically behind the mirror).
• Distances measured in the **direction opposite to incident light** are considered **negative** (typically in front of the mirror).
• Heights measured **perpendicularly upwards** from the principal axis are **positive**.
• Heights measured **perpendicularly downwards** from the principal axis are **negative**.

Sign Convention Cheat Sheet:

Linear Magnification

The **linear magnification** ($m$) measures the ratio of the height of the image ($h_i$) to the height of the object ($h_o$):

We interpret the sign and magnitude of magnification as follows:

• If **$m$ is positive** ($m > 0$), the image is **upright (erect) and virtual**.
• If **$m$ is negative** ($m < 0$), the image is **inverted and real**.
• If **$|m| > 1$**, the image is **magnified** (larger than the object).
• If **$|m| < 1$**, the image is **diminished** (smaller than the object).
• If **$|m| = 1$**, the image is of the **same size** as the object.

Ray Construction and Image Sharpness

In a physical lab setup, the mirror formula results can be observed directly on a screen. For a concave mirror, light rays diverging from the object reflect and converge at a single spatial plane. Placing a white paper screen at this plane intercepts the rays, projecting a **sharp, focused image**. If the screen is placed too close or too far from the image distance ($v$), the incoming light spreads out, creating a **blurry, out-of-focus image**. For virtual images (where reflected rays diverge), no light crosses behind the mirror, meaning a physical screen will display no image at all.