Converging & Diverging Lenses

Ray 1 (Parallel Ray): Draw a ray parallel to the principal axis that refracts through the far focal point (converging) or appears to come from the near focal point (diverging).

Ray 2 (Central Ray): Draw a ray passing directly through the optical center of the lens; this ray continues in a straight line without significant deviation.

Ray 3 (Focal Ray): Draw a ray passing through the near focal point (converging) or aimed at the far focal point (diverging), which refracts to become parallel to the principal axis.

The intersection of these refracted rays (or their virtual extensions) determines the location, orientation, and size of the resulting image.

The relationship between object distance ($d_o$), image distance ($d_i$), and focal length ($f$) is given by the Thin Lens Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Magnification ($m$) describes the ratio of image height ($h_i$) to object height ($h_o$), calculated as: $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

A positive magnification indicates an upright image, while a negative magnification indicates an inverted image.

The absolute value of magnification $|m|$ determines if the image is enlarged ($>1$), diminished ($<1$), or the same size ($=1$) as the object.

Real Images are formed on the opposite side of the lens from the object ($d_i > 0$) and can be projected onto a screen.

Virtual Images are formed on the same side of the lens as the object ($d_i < 0$) and cannot be projected; they are seen by looking through the lens.

Sign Convention Check: Always assign a negative value to the focal length ($f$) for diverging lenses and a negative value to the image distance ($d_i$) for virtual images.

Sanity Check: If you calculate a real image ($d_i > 0$) for a converging lens, it MUST be inverted ($m < 0$). If your math says it is upright, re-check your signs.

Diverging Shortcut: Remember that a single diverging lens can NEVER produce a real image or an enlarged image; if a multiple-choice option suggests this, eliminate it immediately.

Object at $2F$: For a converging lens, an object placed at exactly twice the focal length ($2f$) produces a real, inverted image of the exact same size at $2f$ on the other side.