Spherical Mirrors

A spherical mirror is a mirror whose reflecting surface is a part of a sphere.

Relation between $f$ and $R$

For a spherical mirror of small aperture:

$$f = \frac{R}{2}$$

Mirror Formula

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

where:

• $u$ = object distance
• $v$ = image distance
• $f$ = focal length

Magnification

$$m = \frac{h'}{h} = -\frac{v}{u}$$

where $h'$ is the height of the image and $h$ is the height of the object.

Sign Convention (New Cartesian)

• All distances are measured from the pole ($P$).
• Distances in the direction of the incident ray are positive.
• Distances opposite to the incident ray are negative.
• Heights above the principal axis are positive; below are negative.

Example

A concave mirror has a focal length of $15\,\text{cm}$. An object is placed $30\,\text{cm}$ in front of it. Find the image position and magnification.

Given: $f = -15\,\text{cm}$, $u = -30\,\text{cm}$

$$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15} - \frac{1}{-30} = -\frac{1}{15} + \frac{1}{30} = -\frac{1}{30}$$ $$v = -30\,\text{cm}$$

The image is formed at $30\,\text{cm}$ in front of the mirror (real, inverted).

$$m = -\frac{v}{u} = -\frac{-30}{-30} = -1$$

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