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• Spherical mirrors

A spherical mirror is a spherical cap where regular reflection of light occurs.

## Spherical Mirrors

Types of Mirrors: Illustration of the types of spherical mirrors. The part on the left is reflective. The red cap illustrates a convex mirror, the blue cap a concave mirror. The dashed lines indicate the back of mirrors, the side that does not reflect.
Spherical Concave Mirror
When the reflecting surface is the internal side of the spherical cap.
Convex Spherical Mirror
When the reflecting surface is the outside of the spherical cap.

### Geometric Elements Scheme of a spherical mirror (red cap) and its elements. The main geometric elements of a spherical mirror are:
Curvature Center $$(C)$$
It is the center of the sphere containing the spherical cap.
The Curvature Radius $$(r)$$
It is the sphere radius of curvature containing the spherical cap.
Mirror Vertex $$(V)$$
It is the midpoint of the spherical cap.
Main Axis $$(u)$$
It is the straight line containing the center $$C$$ and the vertex $$V$$ of the mirror.
Secondary Axis $$(u')$$
It is any line that contains the center $$C$$, but does not contain the vertex $$V$$ of the mirror.
Main Focus $$(F)$$
For Gauss' spherical mirrors, the equation of the focus is $$f=\frac{r}{2}$$.
Secondary Focus $$(F_s)$$
It is the focal point that belongs to a secondary axis. When a beam reaches a spherical mirror parallel to the main axis, the beam is reflected as a convergent beam, in the case of the concave mirror, and as a divergent beam in the convex mirror. This beam defines a secondary axis with a secondary focus.

## The Reflection Properties of Light Beams

When the trajectories of light beams make small angles with the optical axis (paraxial approximation) for a spherical mirror, the following rules for their reflection applies:

• An incident beam, parallel to the main axis, is reflected from the main focus.
• An incident beam, directed to the main focus, reflects parallel to the main axis.
• An incident beam, coincident with the main mirror axis. is reflected on itself.
• An incident beam directed to the mirror vertex, and oblique to an axis, is reflected symmetrically in respect to this axis.
• The reflection of every beam of light that is obliquely incident to the main axis passes through the secondary focus.

## Image Produced by a Convex Mirror

Assuming that the trajectories of light beams from an object form small angles with the optical axis (paraxial approximation), in this situation the image of a convex mirror is always: virtual, upright (erected) and smaller. Image in a convex mirror. Considering that the beams of an object form small angles with the main axis, we can use the properties of reflection of light beams in a spherical mirror to find the image produced by a convex mirror.

## Image Produced by a Concave Mirror

Assuming that the trajectories of light beams from an object form small angles with the optical axis (paraxial approximation), you can find the different images that can produce a concave mirror in this situation (see figure). Image in a concave mirror. Depending on the position that a real object is in relation to the mirror (opaque candles), its image will have different properties (transparent candles). Notice that the red candle has no image represented in the figure. This is because its image forms at infinity. Description of possible images in a concave mirror (see figure):
Green Candle
It is an object beyond the center of curvature, whose image is real, inverted and smaller.
Blue Candle
It is an object on the center of curvature, whose image is real, inverted and of same size.
Orange Candle
It is an object between the center of curvature and the focus, whose image is real, inverted and larger.
Red Candle
It is an object in the focal plane, whose image is inappropriate (forms at infinity).
Gray Candle
It is an object between the focus and the vertex, whose image is virtual, upright and bigger.

## Gauss Referential

To perform an algebraic analysis/study of spherical mirrors, the Gaussian reference can be adopted:

The x-axis (abscissa)
It coincides with the main axis of the mirror, originating from the vertex and oriented in the opposite direction to the incident light.
The y-axi (ordinate)
It is perpendicular to the main axis, originating from the mirror vertex. The ordinate axis is oriented so that the ordinate of the object height is positive.

## Equation of Transverse Linear Increase

Amplification $$A$$, or linear increase of an image on a spherical mirror, is given by \begin{equation} A=\frac{i}{h}=\frac{-p'}{p},\ \end{equation} where $$p$$ is the object's distance to the mirror vertex, $$p'$$ is the image's distance to the mirror vertex, $$h$$ is the size of the object, and $$i$$ is the image size. Please note: for an upright image, we have $$A\gt0$$. For an inverted image, we have $$A\lt0$$.

## Gauss Equation (Conjugated Points)

The image of an object, placed at a distance $$p$$ of a focal distance $$f$$ of the mirror, is formed at a distance $$p'$$ of the mirror, such that \begin{equation} \frac{1}{f}=\frac{1}{p}+\frac{1}{p'}, \end{equation} where $$p$$ is positive for real objects and negative to virtual objects, $$f$$ is positive for the concave mirror and negative for the convex, and $$p'$$ is positive for a real image and a negative for a virtual image. This equation is also valid only for paraxial approximation, i.e., for light beams which form small angles with the main axis of the mirror.

Quantities $$f$$ , $$p$$ , $$p'$$ , $$i$$ , $$h$$ and $$A$$ are algebraic, i.e. they must be introduced in equations with their signs (positive or negative) so they can produce correct results.    