The Mathematics of Lenses

Ray diagrams can be used to determine the image location, size, orientation and type of image formed of objects when placed at a given location in front of a lens. The use of these diagrams was demonstrated earlier in Lesson 5 for both converging and diverging lenses. Ray diagrams provide useful information about object-image relationships, yet fail to provide the information in a quantitative form. While a ray diagram may help one determine the approximate location and size of the image, it will not provide numerical information about image distance and image size. To obtain this type of numerical information, it is necessary to use the Lens Equation and the Magnification Equation. The lens equation expresses the quantitative relationship between the object distance (d_o), the image distance (d_i), and the focal length (f). The equation is stated as follows:

The magnification equation relates the ratio of the image distance and object distance to the ratio of the image height (h_i) and object height (h_o). The magnification equation is stated as follows:

These two equations can be combined to yield information about the image distance and image height if the object distance, object height, and focal length are known.

Practice Problems

As a demonstration of the effectiveness of the lens equation and magnification equation, consider the following sample problem and its solution.

Sample Problem #1

A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a double convex lens having a focal length of 15.2 cm. Determine the image distance and the image size.

Like all problems in physics, begin by the identification of the known information.

ho = 4.00 cm

do = 45.7 cm

f = 15.2 cm

Next identify the unknown quantities that you wish to solve for.

di = ???

hi = ???

To determine the image distance, the lens equation must be used. The following lines represent the solution to the image distance; substitutions and algebraic steps are shown.

1/f = 1/do + 1/d_i

1/(15.2 cm) = 1/(45.7 cm) + 1/d_i

0.0658 cm^-1 = 0.0219 cm^-1 + 1/d_i

0.0439 cm^-1 = 1/d_i

d_i = 22.8 cm

The numerical values in the solution above were rounded when written down, yet unrounded numbers were used in all calculations. The final answer is rounded to the third significant digit.

To determine the image height, the magnification equation is needed. Since three of the four quantities in the equation (disregarding the M) are known, the fourth quantity can be calculated. The solution is shown below.

h_i/h_o = - d_i/d_o

h_i /(4.00 cm) = - (22.8 cm)/(45.7 cm)

h_i = - (4.00 cm) • (22.8 cm)/(45.7 cm)

h_i = -1.99 cm

The negative values for image height indicate that the image is an inverted image. As is often the case in physics, a negative or positive sign in front of the numerical value for a physical quantity represents information about direction. In the case of the image height, a negative value always indicates an inverted image.

From the calculations in this problem it can be concluded that if a 4.00-cm tall object is placed 45.7 cm from a double convex lens having a focal length of 15.2 cm, then the image will be inverted, 1.99-cm tall and located 22.8 cm from the lens. The results of this calculation agree with the principles discussed earlier in this lesson. In this case, the object is located beyond the 2F point (which would be two focal lengths from the lens) and the image is located between the 2F point and the focal point. This falls into the category of Case 1: The object is located beyond 2F for a converging lens.

Now lets try a second sample problem:

Sample Problem #2

A 4.00-cm tall light bulb is placed a distance of 8.30 cm from a double convex lens having a focal length of 15.2 cm. (NOTE: this is the same object and the same lens, only this time the object is placed closer to the lens.) Determine the image distance and the image size.

Again, begin by the identification of the known information.

ho = 4.00 cm

do = 8.3 cm

f = 15.2 cm

Next identify the unknown quantities that you wish to solve for.

di = ???

hi = ???

To determine the image distance, the lens equation will have to be used. The following lines represent the solution to the image distance; substitutions and algebraic steps are shown.

1/f = 1/do + 1/d_i

1/(15.2 cm) = 1/(8.30 cm) + 1/d_i

0.0658 cm^-1 = 0.120 cm^-1 + 1/d_i

-0.0547 cm^-1 = 1/d_i

d_i = -18.3 cm

The numerical values in the solution above were rounded when written down, yet unrounded numbers were used in all calculations. The final answer is rounded to the third significant digit.

To determine the image height, the magnification equation is needed. Since three of the four quantities in the equation (disregarding the M) are known, the fourth quantity can be calculated. The solution is shown below.

h_i/h_o = - d_i/d_o

h_i /(4.00 cm) = - (-18.3 cm)/(8.30 cm)

h_i = - (4.00 cm) • (-18.3 cm)/(8.30 cm)

h_i = 8.81 cm

The negative value for image distance indicates that the image is a virtual image located on the object's side of the lens. Again, a negative or positive sign in front of the numerical value for a physical quantity represents information about direction. In the case of the image distance, a negative value always means the image is located on the object's side of the lens. Note also that the image height is a positive value, meaning an upright image. Any image that is upright and located on the object's side of the lens is considered to be a virtual image.

From the calculations in the second sample problem it can be concluded that if a 4.00-cm tall object is placed 8.30 cm from a double convex lens having a focal length of 15.2 cm, then the image will be enlarged, upright, 8.81-cm tall and located 18.3 cm from the lens on the object's side. The results of this calculation agree with the principles discussed earlier in this lesson. In this case, the object is located in front of the focal point (i.e., the object distance is less than the focal length) and the image is located behind the lens. This falls into the category of Case 5: The object is located in front of F (for a converging lens).

The third sample problem will pertain to a diverging lens.

Sample Problem #3

A 4.00-cm tall light bulb is placed a distance of 35.5 cm from a diverging lens having a focal length of -12.2 cm. Determine the image distance and the image size.

Like all problems in physics, begin by the identification of the known information.

ho = 4.00 cm

do = 35.5 cm

f = -12.2 cm

Next identify the unknown quantities that you wish to solve for.

di = ???

hi = ???

To determine the image distance, the lens equation will have to be used. The following lines represent the solution to the image distance; substitutions and algebraic steps are shown.

1/f = 1/do + 1/d_i

1/(-12.2 cm) = 1/(35.5 cm) + 1/d_i

-0.0820 cm^-1 = 0.0282 cm^-1 + 1/d_i

-0.110 cm^-1 = 1/d_i

d_i = -9.08 cm

The numerical values in the solution above were rounded when written down, yet unrounded numbers were used in all calculations. The final answer is rounded to the third significant digit.

To determine the image height, the magnification equation is needed. Since three of the four quantities in the equation (disregarding the M) are known, the fourth quantity can be calculated. The solution is shown below.

h_i/h_o = - d_i/d_o

h_i /(4.00 cm) = - (-9.08 cm)/(35.5 cm)

h_i = - (4.00 cm) * (-9.08 cm)/(35.5 cm)

h_i = 1.02 cm

The negative values for image distance indicate that the image is located on the object's side of the lens. As mentioned, a negative or positive sign in front of the numerical value for a physical quantity represents information about direction. In the case of the image distance, a negative value always indicates the existence of a virtual image located on the object's side of the lens. In the case of the image height, a positive value indicates an upright image.

From the calculations in this problem it can be concluded that if a 4.00-cm tall object is placed 35.5 cm from a diverging lens having a focal length of 12.2 cm, then the image will be upright, 1.02-cm tall and located 9.08 cm from the lens on the object's side. The results of this calculation agree with the principles discussed earlier in this lesson. Diverging lenses always produce images that are upright, virtual, reduced in size, and located on the object's side of the lens.

Sign Conventions

The sign conventions for the given quantities in the lens equation and magnification equations are as follows:

· f is + if the lens is a double convex lens (converging lens)

· f is - if the lens is a double concave lens (diverging lens)

· d_i is + if the image is a real image and located on the opposite side of the lens.

· d_i is - if the image is a virtual image and located on the object's side of the lens.

· h_i is + if the image is an upright image (and therefore, also virtual)

· h_i is - if the image an inverted image (and therefore, also real)

Like many mathematical problems in physics, the skill is only acquired through much personal practice. Perhaps you would like to take some time to try the following problems.

Check Your Understanding

1. Determine the image distance and image height for a 5-cm tall object placed 45.0 cm from a double convex lens having a focal length of 15.0 cm.

See Answer

Answer: d_i = 22.5 cm and h_i = -2.5 cm

Given: f =15 cm and d_o = 45 cm and h_o= 5 cm

Use 1 / f = 1 / d_o + 1 / d_i to solve for d_i

1 / (15 cm) = 1 / (45 cm) + 1 / d_i

1 / (15 cm) - 1 / (45 cm) = 1 / d_i

0.0444 cm^-1 = 1 / d_i

d_i = 1 / (0.0444 cm^-1)

d_i = 22.5 cm

Then use h_i / h_o = - d_i / d_o to solve for h_i

h_i / (5 cm) = - (22.5 cm) / (45 cm)

h_i = - (5 cm) • (22.5 cm) / (45 cm)

h_i = -2.5 cm

NOTE: this is "Case 1"

2. Determine the image distance and image height for a 5-cm tall object placed 30.0 cm from a double convex lens having a focal length of 15.0 cm.

See Answer

Answer: d_i = 30.0 cm and h_i = -5.0 cm

Given: f =15 cm and d_o = 30 cm and h_o = 5 cm

Use 1 / f = 1 / d_o + 1 / d_i

1 / (15 cm) = 1 / (30 cm) + 1 / d_i

1 / (15 cm) - 1 / (30 cm) = 1 / d_i

0.0.333 cm^-1 = 1 / d_i

d_i = 1 / (0.0333 cm^-1)

d_i = 30 cm

Then use h_i / h_o = -d_i / d_o

h_i / (5 cm) = - (30 cm) / (30 cm)

h_i = - (5 cm) • (30 cm) / (30 cm)

h_i = -5.0 cm

NOTE: this is "Case 2"

3. Determine the image distance and image height for a 5-cm tall object placed 20.0 cm from a double convex lens having a focal length of 15.0 cm.

See Answer

Answer: d_i = 60.0 cm and h_i = -15.0 cm

Given: f = 15 cm and d_o = 20 cm and h_o = 5 cm

Use 1 / f = 1 / d_o + 1 / d_i

 1 / (15 cm) = 1 / (20 cm) + 1 / d_i

1 / (15 cm) - 1 / (20 cm) = 1 / d_i

0.0167 cm^-1 = 1 / d_i

d_i = 1 / (0.0167 cm^-1)

d_i = 60.0 cm

Then use h_i / h_o = -d_i / d_o

h_i / (5 cm) = - (60 cm) / (20 cm)

h_i = - (5 cm) • (60 cm) / (20 cm)

h_i = -15.0 cm

NOTE: this is "Case 3"

4. Determine the image distance and image height for a 5-cm tall object placed 10.0 cm from a double convex lens having a focal length of 15.0 cm.

See Answer

Answer: d_i = - 30.0 cm and h_i = +15.0 cm

Given: f = 15 cm and d_o = 10.0 cm and h_o = 5 cm

Use 1 / f = 1 / d_o + 1 / d_i

 1 / (15 cm) = 1 / (10 cm) + 1 / d_i

1 / (15 cm) - 1 / (10 cm) = 1 / d_i

-0.0333 cm^-1 = 1 / d_i

d_i = 1 / (-0.0333 cm^-1)

d_i = - 30.0 cm

Then use h_i / h_o = - d_i / d_o

h_i / (5 cm) = - (-30 cm) / (10 cm)

h_i = - (5 cm) • (-30 cm) / (10 cm)

h_i = 15.0 cm

NOTE: this is "Case 5"

5. A magnified, inverted image is located a distance of 32.0 cm from a double convex lens with a focal length of 12.0 cm. Determine the object distance and tell whether the image is real or virtual.

See Answer

Answer: d_i = 19.2 cm and Real

Given: f = 12 cm and d_i = + 32 cm (inverted images are real and have + image distances)

Use 1 / f = 1 / d_o + 1 / d_i

 1 / (12 cm) = 1 / d_o + 1 / (32 cm)

1 / (12 cm) - 1 / (32 cm) = 1 / d_o

0.052083 cm^-1 = 1 / d_o

d_o = 1 / (0.052083 cm^-1)

d_o = 19.2 cm

All inverted images produced by lenses are real images.

6. ZINGER: An inverted image is magnified by 2 when the object is placed 22 cm in front of a double convex lens. Determine the image distance and the focal length of the lens.

See Answer

=Answer: d_i = 44 cm and f =14.7 cm and Real

Given: d_o = 22 cm and M = -2 (inverted images have negative image heights and therefore negative magnification values)

The magnification of an image is both the h_i / h_o ratio and the -d_i / d_o ratio. Setting the -d_i / d_o ratio equal to -2 allows one to determine the image distance:

M = -d_i / d_o = -2

-d_i / (22 cm) = -2

d_i = 2 • (22 cm)

d_i = 44 cm

Now substitute the di and do values into the lens equation 1 / f = 1 / d_o + 1 / d_i to solve for the focal length.

 1 / f = 1 / (22 cm) + 1 / (44 cm)

 1 / f = 0.06818 cm^-1

f = 1 / (0.06818 cm^-1)

f = 14.7 cm

7. A double concave lens has a focal length of -10.8 cm. An object is placed 32.7 cm from the lens's surface. Determine the image distance.

See Answer

Answer: d_i = -8.12 cm

Given: f = -10.8 cm and d_o = +32.7 cm

Use the equation 1 / f = 1 / d_o + 1 / d_i

Substitute and solve for d_i.

 1 / (-10.8 cm) = 1 / (+32.7 cm) + 1 / d_i

1 / (-10.8 cm) - 1 / (+32.7 cm) = 1 / d_i

-0.1232 cm^-1 = 1 / d_i

d_i = 1 / (-0.1232 cm^-1)

d_i = -8.12 cm

8. Determine the focal length of a double concave lens that produces an image that is 16.0 cm behind the lens when the object is 28.5 cm from the lens.

See Answer

Answer: f = -36.5 cm

Given: d_o = 28.5 cm and d_i = -16.0 cm

(Careful: image distances for double concave lens are always negative)

Use the equation 1 / f = 1 / d_o + 1 / d_i where

Substitute and solve for f.

 1 / f = 1 / (28.5 cm) + 1 / (-16.0 cm)

 1 / f = -0.02741 cm^-1

f = 1 / (-0.02741 cm^-1)

f = -36.5 cm

9. A 2.8-cm diameter coin is placed a distance of 25.0 cm from a double concave lens that has a focal length of -12.0 cm. Determine the image distance and the diameter of the image.

See Answer

=Answer: d_i = -8.11 cm and h_i = 0.908 cm

Given: f = -12.0 cm and d_o = +25.0 cm and h_o = 2.8 cm

Use the equation 1 / f = 1 / d_o + 1 / d_i

Substitute and solve for d_i.

 1 / (-12.0) = 1 / (25.0) + 1 / d_i

1 / (-12.0) - 1 / (25.0) = 1 / d_i

-0.1233 cm^-1 = 1 / d_i

d_i = 1 / (-0.1233 cm^-1)

d_i = -8.11 cm

Then use h_i / h_o = - d_i / d_o to solve for h_i.

h_i / (2.8 cm) = - (-8.11 cm) / (25 cm)

h_i = - (2.8 cm) • (-8.11 cm) / (25 cm)

h_i = 0.908 cm

10. The focal point is located 20.0 cm from a double concave lens. An object is placed 12 cm from the lens. Determine the image distance.

See Answer

Answer: d_i = -7.5 cm

Given: f = -20.0 cm and d_o = +12.0 cm

(Careful: double concave lens have focal lengths which are negative.)

Use the equation 1 / f = 1 / d_o + 1 / d_i

Substitute and solve for d_i.

 1 / (-20.0) = 1 / (12.0) + 1 / d_i

1 / (-20.0) - 1 / (12.0) = 1 / d_i

-0.1333 cm^-1 = 1 / d_i

d_i = 1 / (-0.1233 cm^-1)

d_i = -7.5 cm