Understanding Sign Conventions for Lenses
Introduction
In optics, sign conventions are essential for analyzing and solving problems involving lenses. These conventions are rules that dictate how to assign positive or negative signs to the various parameters, such as focal length, object distance, and image distance. Sign conventions make calculations systematic, allowing us to determine the nature, position, and size of images formed by lenses.
The Cartesian Sign Convention
The Cartesian sign convention is commonly used for lens problems in optics. According to this convention:
- The principal axis (horizontal axis passing through the center of the lens) is the reference line for measurements.
- Distances are measured from the optical center of the lens.
- Positive direction: to the right of the lens (along the direction of the incident light).
- Negative direction: to the left of the lens (opposite to the direction of the incident light).
Rules for Cartesian Sign Convention
- Object Distance (u): Always taken as negative because the object is usually placed to the left of the lens.
- Image Distance (v): Positive if the image is formed on the opposite side of the object (real image) and negative if the image is formed on the same side as the object (virtual image).
- Focal Length (f): Positive for convex (converging) lenses and negative for concave (diverging) lenses.
- Height of Image/Object (h): Positive if measured above the principal axis and negative if measured below.
Lens Formula and Magnification
To calculate image distances and magnification in lens problems, we use the lens formula and the magnification formula. These formulas incorporate sign conventions directly, so it’s essential to apply them accurately.
Lens Formula
The lens formula is:
1/f = 1/v – 1/u
where:
- f = focal length of the lens
- v = image distance from the lens
- u = object distance from the lens
Magnification Formula
Magnification (m) is given by:
m = hi / ho = v / u
where:
- hi = height of the image
- ho = height of the object
- v and u = image and object distances, respectively
A positive magnification indicates an upright image, while a negative magnification indicates an inverted image.
Examples of Sign Conventions for Lenses
Example 1: A convex lens has a focal length of 10 cm. An object is placed 15 cm to the left of the lens. Determine the image distance and nature of the image.
Solution:
Given:
- f = +10 cm (convex lens)
- u = -15 cm (object to the left of the lens)
Using the lens formula:
1/f = 1/v – 1/u
1/10 = 1/v + 1/15
Solving, v ≈ +30 cm
The positive sign of v indicates a real image formed 30 cm to the right of the lens.
Example 2: A concave lens with a focal length of 20 cm forms a virtual image. If the object is placed 25 cm from the lens, find the image distance and the magnification.
Solution:
Given:
- f = -20 cm (concave lens)
- u = -25 cm (object to the left of the lens)
Using the lens formula:
1/f = 1/v – 1/u
1/(-20) = 1/v + 1/25
Solving, v ≈ -11.11 cm
The negative sign of v indicates a virtual image on the same side as the object.
Using the magnification formula:
m = v / u = -11.11 / -25 ≈ 0.444
The positive magnification (less than 1) shows the image is upright and smaller than the object.