# Spherical Mirrors

Spherical mirrors are surfaces similar to caps or spherical reflective sections , capable of promoting light reflection in a regular way. There are two types of spherical mirrors, concave mirrors and convex mirrors . While concave mirrors converge the reflected light , convex mirrors diverge it.

Both have geometric elements in common — vertex , focal point and center of curvature . These points are used as a reference to define the trajectory of the reflected light rays, as well as the conjugate points equation and the linear transverse increase equation.

## Definition of spherical mirrors

Just like plane mirrors, spherical mirrors obey the laws of reflection, however, the curvature of their surface changes the shape of the images , so they can conjugate them in different positions and in different sizes. To better understand what spherical mirrors are, let’s separate them into: concave and convex.

## concave mirrors

Concave mirrors are shaped like a spherical reflecting cavity. They have a surface that reflects light, and another, opposite and opaque, that is, that does not allow the transmission of light. When a beam of light is incident parallel to the concave mirror’s axis of symmetry, it is reflected towards a point in front of the mirror, known as the focus .

In the figure it is possible to observe the geometric elements of a concave spherical mirror:

F – focus

f – focal length (m)

V – vertex

C – center of curvature

R – radius of curvature (m)

The central point of the concave mirror is called the vertex (V), on the basis of which we draw a horizontal line that separates the hemispheres of the mirror, this line is called the axis of symmetry . Point F is called the focus , and it is at this point that the light rays intersect .

The distance between the apex of the mirror and the focal point is known as the focal length (f). Finally, point C, called the center of curvature , is a distance equal to twice the focal length from the mirror vertex, so we can write the following identity:

Before proceeding, we will remember what real and virtual images are and which of them each type of spherical mirror is capable of producing.

## Real image and virtual image

Real images are those that can be projected, just like a movie projector. They are always inverted, as they are formed by the crossing of rays of light. When we place a magnifying glass against the Sun , concentrating the light rays in a single point, we are actually projecting a real image of the star.

Virtual images , on the other hand, cannot be projected and formed by the crossing of extensions of light rays that occur behind the reflecting surface of the mirror. When we look at our reflection in a flat mirror, the image we see is virtual.

Formation of images in concave mirrors

To understand the formation of images in concave mirrors, we must trace at least two rays of light used as a reference and called remarkable rays . Images are formed at the point where rays of light, reflected by the mirror, intersect. The main notable rays are:

• The ray of light incident parallel to the axis of symmetry and reflected towards the focal point;
• The ray of light that falls towards the focal point and is reflected parallel to the axis of symmetry;
• The ray of light that falls towards the vertex and is reflected symmetrically, that is, with an angle of reflection equal to the angle of incidence;
• The ray of light that falls in the direction of the center of curvature and is reflected in that same direction.

The notable rays described are a device that facilitates the determination of the characteristics of the conjugated image by the concave mirror. Its use is due to the fact that any rays of light that fall on concave reflective surfaces will be reflected as well as remarkable rays are.

RI – incident light ray

RR – ray of reflected light

According to the object’s position in relation to the center of curvature, the concave mirror will produce real images, however, when an object is approached from the mirror at a distance smaller than the focal distance, the conjugated image will be virtual and enlarged.

## convex mirrors

Convex mirrors diverge the light reflected from their surface , that is, the light rays are scattered after reflection, unlike concave mirrors. Whenever we look into a convex mirror, we will see a reduced image of ourselves. This type of mirror is widely used in shops, buses and high-traffic places, where you want to have the largest field of vision possible.

Unlike concave mirrors, convex mirrors are only capable of producing virtual images . The geometric elements of convex mirrors are exactly the same as those of concave mirrors, the difference that still exists, however, is in the position of these elements, which are located behind the reflecting surface.

## Formation of images in convex mirrors

The formation of images in convex mirrors is simple, since there is only one case : the one in which the conjugate image is virtual (direct) and reduced , as we can see:

## Formulas used in spherical mirrors

Let’s get to know the main formulas for calculating spherical mirrors, starting with the Gauss equation , known as the equation of conjugate points, note:

In the equation, p and p’ represent, respectively, the positions of the object and the position of the conjugate image. If the image is conjugated behind the mirror, being, therefore, a virtual image, the sign of p’ must be negative.

Finally, we make use of the linear transverse increase equation , a dimensionless physical quantity used to solve most problems related to spherical mirrors. Transverse linear magnification refers to the size of the image in relation to the size of the object , that is, if the magnification module is greater than 1, the image is enlarged, otherwise, it is reduced.

It is important to understand the meaning of the algebraic sign obtained when calculating the transverse linear increase. The plus sign indicates that the image is direct. already the negative sign indicates that the image is upside down, therefore being real.

## Exercises on spherical mirrors

Question 1) Regarding the formation of images by convex mirrors, mark the correct alternative:

a) Images conjugated by convex mirrors can be projected.

b) When we look at the surface of a convex mirror, what we see is a real image.

c) Convex mirrors can combine both virtual and real images.

d) Images formed by convex mirrors are always virtual, direct and reduced.

e) Images formed by convex mirrors are the result of the crossing of light rays.

Template: letter d

Resolution:

Let’s analyze the alternatives:

a) FALSE. Because they are virtual, these images cannot be projected.

b) FALSE. Real images are always inverted, so what we observe are virtual images.

c) FALSE. Only concave mirrors are capable of producing both types of images.

d) TRUE

e) FALSE. Virtual images produced by convex mirrors are formed behind their surface, by crossing extensions of light rays.

question 2) An object is positioned in front of a spherical mirror, and it is observed that the image produced by the mirror is direct, but also larger than the object. Regarding the situation described, it can be concluded that:

a) The mirror in question is convex, and the object lies between the focal point and the center of curvature of that mirror.

b) The mirror in question is concave, and the object is located between the vertex and the focal point of that mirror.

c) The mirror is convex, and the observed image is real.

d) The mirror is convex, and the observed image is virtual.

e) The mirror is concave, and the object is positioned beyond the center of curvature of that mirror.

Template : letter b

Resolution:

The situation described in the statement states that: when we look at the reflecting surface of the mirror, we see a direct image, therefore, virtual, but also enlarged. In this way, we can exclude the possibility that the mirror in question is convex, since this type of mirror only combines virtual images that are smaller than the object. Therefore, the correct alternative is letter b.

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