Understanding the Mirror Formula and Magnification
The size of an object that you see through your binoculars is not the same as you will see with your naked eyes. In general, your binocular focuses on the light coming from a distant object and allows you to see the object in better detail, increasing its size. However, can you calculate the distance? Well, this can be done by using the magnification and mirrors formula. Well, it may look a little complex to understand, but the formula is simple. By using the mirror formula, you can understand the relationship between the distance of the image from the mirror (v), the distance of the object from the mirror (u), and the focal length of a spherical mirror (f).
Mirror Formula
In Physics, there are many equations and theorems that play an important role in understanding nature. Similarly, in ray optics, mirror formula is one of the key concepts that helped us in understanding the behaviour of light in a better way. The mirror formula gives the relation between the focal length of the spherical surfaces, the object distances and the image distance.
We can say that the mirror formula provides a relation between the focal length of the lens or mirror, distance of the object and distance of the image. The mathematical formula of the mirror formula is given as:
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
Where, v and u are the distance of the image and object from a pole of a mirror, respectively. On the other hand, f is the primary focus distance.
You can quickly determine the magnification(m) of the object using u and v following the equation $m=-\frac{v}{u}$
When the curvature’s radius or R is twice the focal length, R=2f, or \[f = \frac{R}{2}\] , the formula can be interpreted in the following manner:
$\frac{1}{u}+\frac{1}{v}=\frac{1}{f} $
$\frac{2}{R}=\frac{1}{u}+\frac{1}{v}$
The formula is valid for different types of spherical mirrors, irrespective of the object’s position. However, while calculating, you will have to be very careful about the values and place them in an appropriate place to the sign convention given above. It may be noted that a mirror focal length will be the distance to the focal point.
What Is Magnification?
In the field of optics, magnification means a qualification to compare the image size with the object size. This allows you to know about the image in terms of how small or large is a particular image formed. It can be defined as the ratio of the object’s height and the image’s height. Magnification is generally denoted by m. The magnification formula for mirror is:
Magnification, $m=\frac{{{h}'}}{h}$.
Here, the height of the object is h, and h’ means height of the image.
Sometimes, magnification can be related to the object distance and image distance. That’s why the formula can be written as: $m=-\frac{v}{u}$.
v means the distance of the image, and the object distance is u.
Now, the final formula for magnification becomes:
$m=\frac{{{h}'}}{h}$
$m =-\frac{v}{u} $
Measuring a Magnification Mirror
When the objects are placed between a concave mirror’s focal point and the surface of the mirror, the images are considered virtual, magnified, and upright. On the other hand, if the objects are beyond the mirror’s focal point, then the images are real; however, the images are inverted. Well, a spherical mirror’s magnification can be easily evaluated analytically if the curvature centre and focal length of the mirror are known.
Here you can use the mirror equation. The formula is $\frac{1}{{{D}_{object}}}+\frac{1}{{{D}_{image}}}=\frac{1}{F}$. The letter F stands for the mirror’s focal length, ${{D}_{image}}$ means the distance of the image, and ${{D}_{object}}$ means the distance of the object. First, you need to determine the image distance using this formula, only then you will be able to determine the image magnification value.
Understanding the Sign Conventions
The sign convention for spherical mirrors has a set of rules known as the New Cartesian Sign Convention. During the calculation of magnification of mirror, the new Cartesian Sign Convention can be utilised to eliminate the confusion of ascertaining the directions of the ray. Some of the important points to consider here are:
To measure all the distances, you will have to consider the lens’s optical centre.
When you measure the distance opposite to the incident light’s direction, it should be considered negative.
When the distances of the images are calculated in the light’s directions, the result should be considered positive.
When heights are calculated upwards as well as perpendicular to the principal axis, they will be positive.
On the other hand, if the heights are measured perpendicular to the primary axis but downwards, then they will be negative.
Conclusion
The magnification formula for mirror can help you in determining the ratio of an image distance to the object distance. As per the formula, the magnification can be negative or positive. If you can analyse the signs prior to magnification, you can understand the nature of an image generated. If the image is virtual and erect, then the magnification is positive. On the other hand, if the image is inverted and real, then the magnification will be negative. So, check what is the sign in the value to determine whether the image is real or not.
FAQs on Mirror Formula and Magnification for JEE
1. What are spherical mirrors?
The curved surface that you generally see on a shining spoon can be treated as a curved mirror. The most extensively used type of mirror is the spherical mirror. As per the experts, such mirrors have a shiny reflective surface and can form a part of any sphere’s surface. The mirrors that come with a reflecting surface, which are spherical in nature, are known as spherical mirrors. You will find two different types of such mirrors where you can use the magnification and mirrors formula. These are the concave and convex mirrors.
2. What do you mean by linear magnification?
Linear magnification, also known as lateral magnification, or you can say, transverse magnification, is a ratio of the size of an object’s image generated by a lens to the true size of the object. Well, if the object, as well as the image, are in the same physical state, like air or water, the linear magnification formula is the image size divided by the object side.
$M=-\frac{I}{O}$.
Here, M stands for the magnification and ‘I’ means the height of the image, and ‘O’ means the height of the object. The “–” sign means the objects’ images created by a convex mirror appear inverted.