Combination of Lenses and Mirrors

It is almost impossible to imagine a life without light

From the heavenly bodies above us to almost everything around us, light is either produced or reflected by, for us to see things

Concerning light, the most common phenomenon is Reflection or Refraction

We use Reflection and Refraction of light for a lot of our daily needs accordingly

The spectacles we use to correct defected vision uses Refraction of Light to form images

The rear-view mirrors we use in our vehicles uses Reflection of Light to form images

Let us see the image-formation process in a Mirror

The rays of light parallel to the principal axis of the curved (concave) mirror are converged to it's focus

The ray of light passing through the centre of curvature of the mirror retraces it's path

The image of an object placed at a distance $u$ from a mirror having a focal length $f$ will be formed at a distance $v$

The image distance, object distance and the focal length of the mirror are related by the mirror equation, $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

Thus, to get the image distance we need the focal length and the object distance of the mirror

While in the case of a lens the image distance and object distance are related by the lens formula, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

Let's see some of the characteristics of the image formed by the lenses and mirrors

In the case of a concave mirror, the image formed are inverted and real in nature

In the case of a convex mirror, the image formed will be erect and virtual in nature

In the case of a convex lens, the image formed is real and inverted

While in then case of a concave lens, the image formed will be virtual and erect

A real image can be obtained on a screen while a virtual image cannot be obtained on a screen

The focal length of the concave mirror and convex lens can be obtained experimentally by mirror formula and lens formula respectively

Let's determine the focal length of a concave lens using a concave lens and a concave mirror

We know that the image formed by a concave lens is virtual and hence cannot be obtained on a screen

Let us start by taking a concave mirror such that an object is placed at the centre of curvature of the mirror

The image formed by the mirror will be at the centre of curvature itself and real, hence can be obtained on a screen

Now let us place a concave lens between the mirror and the object

The image will be formed now farther than the previous case

For this combination, $\text{LI}$ = Image Distance = v and, $\text{LO}$ = Object Distance = u

Using the Lens Formula we obtain the focal length $f$ as, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

Now it is important to note that u,v will be considered positive according to the cartesian sign convention

Thus, we get the value of the focal length for the concave lens

Let's find the focal length of a concave lens using the combination of concave lens and plane mirror

Let us take a concave lens and place an object at $\text{O}$, such that a virtual image is obtained at $\text{I}$

As the image at $\text{I}$ is virtual, it cannot be obtained on a screen

So we take a plane mirror and place it such that the polished part faces the concave lens

Now, we place an object in front of the mirror such that the virtual image due to the plane mirror is obtained at $\text{I}$

Hence, Object Distance = $\text{LO}$=$\text{u}$ Image distance= $\text{LI}$ = $\text{MI}-\text{ML}$ = $\text{MQ}-\text{ML}$ = $\text{v}$

Using the Lens Formula we obtain the focal length $f$ as, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

It is important to note that $u,v$ are considered negative according to sign convention

Let us find the focal length of a convex mirror using the combination of convex mirror and convex lens

Let us place a convex mirror and a convex lens such that their principle axis coincides

An object is placed at O such that the light rays after converging through the lens are incident normally on the convex mirror

The light rays are reflected back from the mirror along the same path such that the image coincides with the object at O

Now, we remove the Convex Mirror such that an image is obtained at I

The focal length can be obtained by the formula $\text{f}=\frac{R}{2}$

$R=MI$ is the radius of curvature of the Convex Mirror

The End