How to Calculate Focal Length of a Lens

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Before the 1590s, simple lenses dating back as far as the Romans and Vikings allowed limited magnification and simple eyeglasses. Zacharias Jansen and his father combined lenses from simple magnifying glasses to build microscopes and, from there, microscopes and telescopes changed the world. Understanding the focal length of lenses was crucial to combining their powers.

Types of Lenses

There are two basic types of lenses: convex and concave. Convex lenses are thicker in the middle than on the edges and cause light rays to converge to a point. Concave lenses are thicker on the edges than in the middle and cause light rays to diverge.

Convex and concave lenses come in different configurations. Plano-convex lenses are flat on one side and convex on the other while bi-convex (also called double-convex) lenses are convex on both sides. Plano-concave lenses are flat on one side and concave on the other side while bi-concave (or double-concave) lenses are concave on both sides.

A combined concave and convex lens called concavo-convex lenses is more commonly called the positive (converging) meniscus lens. This lens is convex on one side with a concave surface on the other side, and the radius on the concave side is greater than the radius of the convex side.

A combined convex and concave lens called a convexo-concave lens is more commonly called a negative (divergent) meniscus lens. This lens, like the concavo-convex lens, has a concave side and a convex side, but the radius on the concave surface is less than the radius on the convex side.

Focal Length Physics

The focal length of a lens ​f​ is the distance from a lens to the focal point ​F​. Light rays (of a single frequency) traveling parallel to the optical axis of a convex or a concavo-convex lens will meet at the focal point.

A convex lens converges parallel rays to a focal point with a positive focal length. Because the light goes through the lens, positive image distances (and real images) are on the opposite side of the lens from the object. The image will be inverted (up-side down) relative to the actual image.

A concave lens diverges parallel rays away from a focal point, has a negative focal length and forms only virtual, smaller images. Negative image distances form virtual images on the same side of the lens as the object. The image will be oriented the same direction (right-side up) as the original image, just smaller.

Focal Length Formula

Finding focal length uses the focal length formula and requires knowing the distance from the original object to the lens ​u​ and the distance from the lens to the image ​v​. The lens formula says that the inverse of the distance from the object plus the distance to the image equals the inverse of the focal distance ​f​. The equation, mathematically, is written:

\frac{1}{u}+\frac{1}{v}=\frac{1}{f}

Sometimes the focal length equation is written as:

\frac{1}{o}+\frac{1}{i}=\frac{1}{f}

where ​o​ refers to the distance from the object to the lens, ​i​ refers to the distance from the lens to the image and ​f​ is the focal length.

The distances are measured from the object or the image to the pole of the lens.

Focal Length Examples

To find the focal length of a lens, measure the distances and plug the numbers into the focal length formula. Be sure all measurements use the same measurement system.

Example 1​: The measured distance from a lens to the object is 20 centimeters and from the lens to the image is 5 centimeters. Completing the focal length formula yields:

\frac{1}{20}+\frac{1}{5}=\frac{1}{f} \\ \text{or}\; \frac{1}{20}+\frac{4}{20}=\frac{5}{20} \\ \text{Reducing the sum gives }\frac{5}{20}=\frac{1}{4}

The focal length is therefore 4 centimeters.

Example 2​: The measured distance from a lens to the object is 10 centimeters and the distance from the lens to the image is 5 centimeters. The focal length equation shows:

\frac{1}{10}+\frac{1}{5}=\frac{1}{f} \\ \text{Then}\; \frac{1}{10}+\frac{2}{10}=\frac{3}{10}

Reducing this gives:

\frac{3}{10}=\frac{1}{3.33}

The focal length of the lens is therefore 3.33 centimeters.

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