Magnification Formula: A Closer Look

magnification formulaMagnification is the method of enlarging the appearance of an object, not its actual size. A calculated number called ‘magnification’ quantifies this enlargement. When the number is not more than one, it refers to size reduction, sometimes called de-magnification or minification. Usually, magnification involves scaling up images and visuals to see more details with greater resolution such as in digital processing, printing techniques and when you use a microscope. For more math formulas, here is an article about Newton’s laws.

There are 2 basic formulas for magnification, the magnification equation and the lens equation. To compute for an object’s magnification by a convex lens, both formulas are required. The magnification equation relates to the distance and height of an image or object and defines what M is, which is magnification. The lens equation has to do with the projected image, the distances between objects, the shape of the lens and focal length, which you can learn more about by taking this Physics course.

The Magnification Equation for Photography

For photography, the magnification equation is:

M= Mi/Ho= -Di/Do where Do is the distance from the lens to the object, Di is the distance from the image to the lens, Ho is the object’s height and Hi is the image height. M stands for magnification, as we mentioned earlier. When you see a ‘minus’ sign such as in an equation like this one, this tells you that there will be an inverted image. The 2 equal signs mean that there are 3 immediate forms namely:

Hi/Ho = -Di/Do

M = -Di/Do

M = Hi/Ho

At times, the distance between the image detector and the specimen tested is increased in order to obtain image magnification. When the details are very small, including the parts being inspected, magnification becomes particularly useful, as you shall see when you take this Photography course. The farther the specimen is from the detector of the image, the greater the magnification done. You can also calculate magnification amounts with this formula:

M- (a + b)/a

Where :

a= distance from object to source

b= distance from detector to object

and

M= magnification

You can use this particular equation when you need to calculate the geometric magnification when the object to detector distance is thirty centimeters and the source to object distance is eighty centimeters.

M= (a+b) /a

M= (80 cm + 30 cm)/80 cm

M= 1.375

The Magnification Equation for a Microscope

The equation used for calculating a microscope’s magnification is:

MA= Mo * Me

Where the magnification of the objective is Mo and the eyepiece magnification is Me.

The magnification of the objective depends on the distance (d) between the eyepiece and the focal plane and on its focal length (fo):

Mo= d/fo

The eyepiece magnification is calculated with the same equation used for a magnifying glass (see below_ and depends upon its focal length. Remember that both simple and astronomical telescopes produce inverted images. Thus, the equation for magnification with a microscope or a telescope is frequently accompanied by a ‘minus’ sign.

The Magnification Equation for a Magnifying Glass

Compared to the naked eye, a magnifying glass’ maximum angular magnification depends on how the object and the glass are held in relation to your eye. When the lens is held far from the object so that its frontal focused point is on the viewed object, the eye can see the image with angular magnification, which you can read about in this course about vision enhancement. The formula for this is:

MA= 25 cm/f

The constant 25cm is an eye distance ‘near point’ estimate, which is the nearest distance that healthy eyes are able to focus. F here is the lens’ focal length in centimeters. In cases like this the angular magnification is independent from the distances kept between the magnifying glass and the eye. When holding the lens very close to the eye, the object is positioned nearer the lens, larger angular magnification can be obtained with this formula:

MA= (25cm/f) +1

In this case, the magnifying glass alters the eye’s diopter causing it to be myopic so that there is larger angular magnification when objects are placed nearer the eyes.

The Magnification Equation for a Single Lens

The equation to compute for lens tells you not just what kind of lens to use when you already know the distance, but also tells you how far the image will be from the lens and the object.

The lens equation is:

1/Do + 1/Di = 1/f

Where Do is the distance from the length to the object, Di is the distance from the in-focus projected image to the lens and f is the lens’ focal length. This lens equation form results in 3 more useful computational forms by computing for three variables using algebraically straightforward solutions.

These forms are:

Di = (Do*f)/(Do-f).

Do=(Di*f)/(D i-f)

F= (Do * Di)/(Do + Di)

When you need to compute for the third variable and already have two of the variables, these three equations become simple to use.

Using These Equations:

These equations can tell you what type of lens to utilize if you know how far the distance happens to be. For instance, if a camera shoots from ten feet and projects onto a film half a foot away, the lens’ focal length should be:

Using one of the magnification equations, you can calculate the image size of an object on the camera film:

Hi = – (di* Ho)/ Do

= -(0.5 * Ho)/10

= -(1/20) * Ho

Thus the film image will be one-twentieth of the photographed image size. The sign for ‘minus’ tells you that the image is going to be inverted.

Using the lens equation:

F= (10 * 0.5) / (10 + 0.5)

= 5/ 10.5

=0.476

Magnification Examples

Other examples of magnification include using a slide projector, a microscope, a telescope and a magnifying glass. Slide projectors project larger images of smaller slides on a projector screen. Microscopes make smaller objects appear to be larger at a comfortable viewing distance. Telescopes use a big objective lens in order to create images of distant objects, allowing users to closely examine images with a small eyepiece lens to make objects seem larger. A magnifying glass uses a convex positive lens to make objects appear larger by letting users hold them nearer to the eye, which comes in handy when you are learning this Photoshop Master course.