Spherical Mirrors

Spherical Mirrors
Category : Optic

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Plane mirror
Spherical Mirrors

Spherical Mirrors

One of the key properties of plane mirrors is that the object size is equal to the corresponding image size. But there are mirrors that images formed by these are not equal to the object size. These mirrors are known as spherical mirrors. Spherical mirrors are a section of a large spherical polished surface. There are two types of spherical mirror: concave and convex. In the concave (convex) case the outer (inner) part of the mirror has polished and is reflecting.

To study the image formation by spherical mirrors, we need a few key concepts. As illustrated in the figure below, the center of curvature of the mirror (the center of the sphere of which the mirror is a part of it) is at $C$ and the vertex of the mirror is at $V$. The line joining the center of curvature to vertex is called optical (principal) axis. The distance between $C$ and $V$ along the optical axis is the radius of curvature of the mirror. When a bundle of rays striking a concave mirror parallel to the optical axis, are reflected such that they come from a same point $f$ on the optical axis. This point $f$ at which the parallel rays to the optical axis converge is called the focal point. The distance from the vertex to the focal point is called focal length and is half of the radius of curvature $R$ i.e. $f=\frac{R}{2}$.

when the rays are too close and parallel to the principal axis they converge at the focal point $f$, in this case we say that these rays are brought to a focus and are called the paraxial rays. but if the rays diverging from the object are too far away from the optical axis they are brought to a focus ever closer to the mirror. This lack of perfect focusing of a spherical mirror is called spherical aberration. If a mirror has spherical aberration the image formed by it is blurred. The approximation in which we neglect the spherical aberration is called the paraxial approximation.

The parabolic mirror is type of a mirror in which the spherical aberration has been canceled. These mirrors are used to collect energy from a distant object such as incoming waves from a satellite or incoming star light and bring them to a focal point. The other commonplace application of these mirrors is in the car headlights. The car headlight consists of a light-bulb that is placed at the focal point of a parabolic reflector. According to the principles of the reflection, when a point source (light-bulb) is located at the focus, the rays emanating from it are reflected parallel to the optical axis.

Image formation by a spherical mirror

As shown in the figure, a bundle of rays from a point object $P$ reflecting from a concave mirror and converging at point $P^{'}$(image point). Since the reflected rays actually pass through this point we say this image is a real image. By placing a photographic film at the image point we can take a photo from the object’s image. This is in contrast to the virtual image formed by a plane mirror since in that case the image point is located behind the mirror and consequently no light-rays actually emanate from it. Thus real images are formed in front of the spherical mirrors.

To find the location of the image in the spherical mirrors by ray tracing, one must use the following special (principal) rays:
    ($i$)The ray passing through the center of curvature is reflected back on itself. From the ordinary geometry we know that this rays are perpendicular to the surface of the mirror. These rays are called Radial rays.
    ($ii$)The ray strikes the mirror parallel to the principal axis reflects through the focal point $f$ (Parallel ray).
    ($iii$)The ray passing through the focal point is reflected parallel to the optical axis of the mirror (Focal ray)
    ($iv$)The ray strikes the vertex of the mirror with particular angle relative to the optical axis is reflected with the same angle as the incident ray. 
In the following figures, the different situations are shown for concave mirror.

Now use these special rays and find the location of images in different places of object.
($i$) If an extended object, say arrow, is outside the center of curvature, its corresponding image is on the same side of the mirror (real image) as the incident rays and is larger and inverted as illustrated in the figure below.
($ii$) When the object is placed inside the focal length, a virtual, erect and larger image can be formed. As we can see, this image is located by the radial ray, which is reflected back on itself, and the focal ray, which is reflected parallel to the optical axis. These reflected rays intersect each other at a point in the opposite side (behind) of the mirror. 
($iii$) If the object is between the center of curvature and focal point, its image is real (in the same side as the incident rays), inverted and larger.

Using the geometrical arguments, we can find an equation that relates the object $s_o$ and image $s_i$ distances to the focal length $f$ of the spherical mirror. 
\[\frac{1}{s_o} +\frac{1}{s_i}=\frac{1}{f}\ \   ,mirror\ equation\]

Keep in mind that this equation is derived for paraxial rays and consequently is approximately correct. The other important relation for the spherical mirrors is the lateral magnification which is defined as the ratio of the image height $y_i$ to the object height $y_o$ or the ratio of the negative of image distance $s_i$ to the object distance $s_o$.
\[m=\frac{y_i}{y_o} =-\frac{s_i}{s_o}\ \    ,lateral\ magnification\ ,spherical\ mirror\]

The negative sign takes into account, as we can describe later, since in the spherical mirrors the image and object can be on opposite sides of the mirror.

For plane mirrors, the radius of curvature is infinite. Thus the focal length is $f=\frac{R}{2}=\infty$. Using the mirror equation, we find $s_i=-s_o$, indicating that the image is behind the mirror at a distance equal to the object distance. Now by applying the lateral magnification equation, we can see that $m=+1$, indicating that the image is upright and the same size as the object, as we expected.

To apply the equation above, we must adopt a sign convention that is the same as the conventions for plane mirrors but for clarity we have stated those for spherical mirror as follows
    ($i$) $s_o$ is positive if the object is on the incident-light side of the mirror. In this case we say the object is real.
    ($ii$) $s_i$ is positive if the image is on the reflected-light side of the mirror. That is, we have a real image. 
    ($iii$) The radius of curvature $R$ (and thus the focal length $f$) is positive if the mirror is concave. 
If one of the above conventions does not satisfy that parameter must be negative.

Example: the focal point of a concave mirror is $12\,{\rm cm}$. Find location and type of the image of an object that is placed on the optical axis in two following cases:
($a$)    At $20\,{\rm cm}$ in front of the mirror.
($b$)    At $4\,{\rm cm}$ in front of the mirror. 

($a$) In this case, the object is placed between the focal point and center of curvature. We expect its corresponding image must be real, inverted, larger and formed outside the center of curvature. Using the mirror equation and applying the sign conventions, we have
\[\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f} \to \frac{1}{s_i}=\frac{1}{f}-\frac{1}{s_o}\]
\[\Rightarrow \frac{1}{s_i}=\frac{1}{12}-\frac{1}{20}=\frac{5-3}{60}=\frac{1}{30}\]
\[\Rightarrow s_i=+30\ {\rm cm}\]
As we can see the positive value of $s_i$ indicates that the image is real. Now by calculating the value of lateral magnification, we can observe the inverted image $m<0$.
($b$) The object is placed inside the focal length, so use the mirror equation and lateral magnification to find the properties of the corresponding image.
\[\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f} \to \frac{1}{s_i}=\frac{1}{12}-\frac{1}{4}=\frac{1-3}{12}\]
\[\Rightarrow s_i=-6\ {\rm cm}\]
Therefore, the image is formed behind the mirror (virtual since $s_i<0$) and is upright ($m>0$) and larger, as we expected.

Convex mirrors

These mirrors are formed always a virtual, upright and smaller image. A convex mirror never forms a real image of a real object. The image is formed between the focal length and the vertex of the mirror as illustrated in the figure below.
note: the focal length of this mirror is negative i.e. $f<0$


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