Lens and Mirror Formulas

Mirror Formula: For spherical mirrors, the relationship between object distance \( u \), image distance \( v \), and focal length \( f \) is:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

Lens Formula: For thin lenses, the same form is used but with the sign convention for lenses:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

Note: This is often rewritten as:

\[ \frac{1}{v} + \left(-\frac{1}{u}\right) = \frac{1}{f} \]

The radius of curvature \( R \) of a mirror is related to the focal length by:

\[ f = \frac{R}{2} \]

Sign Convention

All distances are measured from the optical center or pole.
Object distance (\( u \)): Always taken as positive in front of mirror/lens.
Image distance (\( v \)):
- Positive if image is on the same side as the outgoing rays (real image)
- Negative if image is on the same side as incoming rays (virtual image)
Focal length (\( f \)):
- Positive for converging (convex lens or concave mirror)
- Negative for diverging (concave lens or convex mirror)

An object is placed 20 cm in front of a concave mirror with a focal length of 10 cm. Find the image distance.

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \Rightarrow \frac{1}{10} = \frac{1}{20} + \frac{1}{v} \]

\[ \frac{1}{v} = \frac{1}{10} - \frac{1}{20} = \frac{1}{20} \Rightarrow v = 20\ \text{cm} \]

The image is real, inverted, and the same size as the object.

An object is placed 10 cm from a concave lens of focal length -20 cm. Find the image distance.

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \Rightarrow \frac{1}{-20} = \frac{1}{v} - \frac{1}{-10} \]

\[ \frac{1}{v} = -\frac{1}{20} + \frac{1}{10} = \frac{1}{20} \Rightarrow v = 20\ \text{cm} \]

The image is virtual, upright, and diminished, located 20 cm on the same side of the lens.

Written by Albert Marin