Content

• To step
• Method 1 of 2: Determining the size of a single lens
• Method 2 of 2: Determining the magnification of several lenses in a row
• Method for two lenses
• Detailed method
• Tips

In optics, the magnification of an object such as a lens, the ratio between the height of the image of an object that you can see and its actual size. For example, a lens that makes a small object look large has a strong magnification, while a lens that makes an object look smaller is a weak magnification. The magnification of an object is generally given by the formula M = (h_i/ h_O) = - (d_i/ d_O), where M = magnification, h_i = image height, h_O = object height, and d_i and d_O = image distance and object distance.

To step

Method 1 of 2: Determining the size of a single lens

Note: A. converging lens is wider in the center than at the edge (like a magnifying glass). A diverging lens is wider at the rim and thinner in the center (like a bowl). The same rules apply to both when it comes to determining magnification, with one important exception, as you'll see below.

1. Take the equation / formula as a starting point and determine what data you have. As with other physics problems, it is a good approximation to first write down the equation you need. Then you can start looking for the missing pieces from the equation.
   • For example, suppose an action doll measuring 6 inches by two feet from one converging lens with a focal length of 20 centimeters. If we use the magnification, image size and image spacing To determine, we start by writing down the equation:

     M = (h_i/ h_O) = - (d_i/ d_O)

   • At this point we know h_O (the height of the action doll) and d_O (the distance from the action doll to the lens.) We also know the focal length of the lens, which is not included in the equation. We will now h_i, d_i and M must find.
2. Use the lens equation to d_i to decide. If you know the distance from the object you are magnifying to the lens and the focal length of the lens, determining the distance of the image is easy using the lens equation. The lens comparison is 1 / f = 1 / d_O + 1 / d_i, where f = the focal length of the lens.
   • In our example problem, we can use the lens equation to calculate d_i to decide. Enter the values ​​of f and d_O and solve:

     1 / f = 1 / d_O + 1 / d_i

     1/20 = 1/50 + 1 / d_i

     5/100 - 2/100 = 1 / d_i

     3/100 = 1 / d_i

     100/3 = d_i = 33.3 centimeters

   • The focal length of a lens is the distance from the center of the lens to the point where the light rays converge in a focal point. If you've ever tried to burn a hole in a piece of paper with a magnifying glass, you know what it means. This value is often given for physics exercises. In real life, you can sometimes find this information marked on the lens itself.
3. Solve for h_i. You know d_O and d_i, then you can find the height of the magnified image and the magnification of the lens. Notice the two equal signs in the equation (M = (h_i/ h_O) = - (d_i/ d_O)) - this means that all terms are equal, so we now have M and h_i can determine, in any order.
   • In our example problem, we determine h_i as follows:

     (h_i/ h_O) = - (d_i/ d_O)

     (h_i/6) = -(33.3/50)

     h_i = -(33.3/50) × 6

     h_i = -3,996 cm

   • Note that a negative height indicates that the image we are seeing has been flipped.
4. Solve for M. You can now solve for the last variable with - (d_i/ d_O) or with (h_i/ h_O).
   • In our example, we determine M as follows:

     M = (h_i/ h_O)

     M = (-3,996 / 6) = -0.666

   • We also get the same answer if we use the d values:

     M = - (d_i/ d_O)

     M = - (33.3 / 50) = -0.666

   • Note that magnification has no unit.
5. Interpret the value of M. Once you have found the magnification, you can predict several things about the image you will see through the lens. These are:
   • The size. The bigger the absolute value of M, the more the object will be magnified through the lens. The values ​​of M between 1 and 0 indicate that the object will look smaller.
   • The orientation. Negative values ​​indicate the image is upside down.
   • In our example, the value of M is -0.666, which means that, under the given conditions, the action doll image upside down and two thirds of its normal size.
6. For diverging lenses, use a negative focal length. Even though diverging lenses look very different from converging lenses, you can determine their magnification using the same formulas mentioned above. The only significant exception is that diverging lenses have a negative focal length to have. In a similar problem as indicated above, this will affect the value of d_i, so make sure you pay close attention to that.
   • Let's take another look at the above problem, only this time for a diverging lens with a focal length of -20 centimeters. All other initial conditions are the same.
   • First we determine d_i with the lens equation:

     1 / f = 1 / d_O + 1 / d_i

     1 / -20 = 1/50 + 1 / d_i

     -5/100 - 2/100 = 1 / d_i

     -7/100 = 1 / d_i

     -100/7 = d_i = -14.29 centimeters

   • Now we determine h_i and M with our new value for d_i.

     (h_i/ h_O) = - (d_i/ d_O)

     (h_i/6) = -(-14.29/50)

     h_i = -(-14.29/50) × 6

     h_i = 1.71 centimeters

     M = (h_i/ h_O)

     M = (1.71 / 6) = 0.285

Method 2 of 2: Determining the magnification of several lenses in a row

Method for two lenses

1. Determine the focal length for both lenses. When dealing with a device that uses two lenses in a row (such as in a telescope or part of binoculars), all you need to know is the focal length of both lenses to get the final magnification of the image. determine.You do this with the simple equation M = f_O/ f_e.
   • In the equation, f_O to the focal length of the lens and f_e to the focal length of the eyepiece. The objective is the large lens at the end of the device, while the eyepiece is the part that you look through.
2. Use this data in the equation M = f_O/ f_e. Once you've found the focal length for both lenses, solving the problem becomes easy; you can find the ratio by dividing the focal length of the lens by that of the eyepiece. The answer is the magnification of the device.
   • For example: suppose we have a small telescope. If the focal length of the lens is 10 centimeters and the focal length of the eyepiece is 5 centimeters, then the 10/5 = 2.

Detailed method

1. Determine the distance between the lenses and the object. If you place two lenses in front of an object, it is possible to determine the magnification of the final image, provided you know the ratio of the distance of the lenses from the object, the size of the object, and the focal length of both lenses. You can deduce everything else.
   • For example, suppose we have the same setup as in the example of Method 1: a 6 centimeter object at a distance of 50 centimeters from a converging lens with a focal length of 20 centimeters. Now we place a second converging lens with a focal length of 5 centimeters behind the first lens (100 centimeters away from the action doll.) In the following steps, we will use this information to find the magnification of the final image.
2. Determine the image distance, height, and magnification for lens number 1. The first part of any problem that involves multiple lenses is the same as that with just one lens. Start with the lens closest to the object, and use the lens equation to find the distance of the image; now use the magnification equation to find the height and magnification of the image.
   • Through our work in Method 1, we know that the first lens produces an image of -3,996 centimeters high, 33.3 centimeters behind the lens, and with a magnification of -0.666.
3. Use the image of the first as the object for the second. Now determining the magnification, height, etc. for the second lens is easy; just use the same techniques as used for the first lens. Only this time you use the image instead of the object. Remember that the image will usually be at a different distance from the second lens compared to the distance between the object and the first lens.
   • In our example this is 50-33.3 = 16.7 centimeters for the second, because the image is 33.3 inches behind the first lens. Let's use this, along with the focal length of the new lens, to find the image from the second lens.

     1 / f = 1 / d_O + 1 / d_i

     1/5 = 1 / 16.7 + 1 / d_i

     0.2 - 0.0599 = 1 / d_i

     0.14 = 1 / d_i

     d_i = 7.14 centimeters

   • Now we can h_i and calculate M for the second lens:

     (h_i/ h_O) = - (d_i/ d_O)

     (h_i/-3.996) = -(7.14/16.7)

     h_i = -(0,427) × -3.996

     h_i = 1.71 centimeters

     M = (h_i/ h_O)

     M = (1.71 / -3.996) = -0,428

4. Continue like this with any additional lenses. The standard approach is the same whether you place 3, 4 or 100 lenses in front of an object. For each lens, consider the image from the previous lens as an object and then use the lens equation and magnification equation to calculate the answer.
   • Don't forget that the following lenses can turn your image around again. For example, the magnification we calculated above (-0.428) indicates that the image is about 4/10 the size of the image from the first lens, but upright, because the image from the first lens was reversed.

Tips

• Binoculars are usually indicated by a multiplication of two numbers. For example, binoculars can be specified as 8x25 or 8x40. The first number is the magnification of the binoculars. The second number is the sharpness of the image.
• Note that for single lens magnification, this magnification is a negative number if the distance to the object is greater than the focal length of the lens. This does not mean that the object appears smaller, but that the image is perceived in reverse.