Lens Formula and Magnification

Lenses form images by bending light at their curved surfaces. A full ray diagram shows where the image forms, but drawing one every time can be slow. The lens formula gives a quick way to calculate the image distance and figure out whether the image is real, virtual, magnified or diminished.

This formula works for both convex and concave lenses if I use the correct sign convention.

To avoid confusion, the same convention used for mirrors is applied to lenses:

• Distances measured opposite to incoming light are negative.
• Distances measured in the direction of incoming light are positive.
• Convex lens has a positive focal length.
• Concave lens has a negative focal length.
• Heights above the principal axis are positive; below are negative.

Once I set the signs correctly, the formula automatically tells me the nature of the image.

The mathematical relation connecting object distance \(u\), image distance \(v\), and focal length \(f\) of a lens is:

This is the most convenient form, but another commonly used version is:

Both forms mean the same thing as long as I stick to the sign convention.

Magnification shows how big or small the image is compared to the object. It also tells the orientation — upright or inverted.

The basic magnification formula is:

where \(h_i\) is image height and \(h_o\) is object height.

Magnification can also be calculated from object and image distances:

Unlike mirrors, there is no minus sign here. The sign of \(m\) itself tells me the orientation.

To analyse any lens situation, I usually follow these steps:

1. Use object distance \(u\) and focal length \(f\) in the lens formula to find image distance \(v\).
2. Use the magnification formula to find image height and orientation.
3. Check the sign of \(v\) and \(m\) to know whether the image is real/virtual and upright/inverted.

This method works for both convex and concave lenses.

Suppose an object is placed at \(u = -20\,\text{cm}\) in front of a convex lens of focal length \(f = +10\,\text{cm}\):

Step 1: Lens formula

The image is formed on the opposite side → real image.

Step 2: Magnification

The image is the same size but negative sign → inverted.

The lens and magnification formulas are helpful in understanding and designing:

• Magnifying lenses
• Eyeglasses (convex or concave)
• Camera lenses
• Microscopes
• Projectors and telescopes

Once I know \(u\) and \(f\), I can predict exactly how the image will look.