Lens Formula and Magnification

This topic is a procedure topic. There is one equation — the lens formula — and one ratio — magnification. The real skill is not memorising them. It is plugging numbers into them with the signs done right. So we will state where the formula comes from, then drill the method: identify, choose, check signs, execute, sanity-check.

1. State the Goal

Here is the goal. You are given a lens and an object. You want to predict three things without drawing a single ray:

• Where the image forms (the image distance, $v$).
• How big it is (the magnification, $m$).
• What kind it is — real or virtual, upright or inverted, bigger or smaller.

Two relations do all of this:

• The lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$
• The magnification: $m = \dfrac{h'}{h} = \dfrac{v}{u}$

Let me name every symbol right now, because a weak-numeracy student gets blocked by an unglossed letter even when the words are clear:

• $u$ — the object distance. How far the object sits from the lens.
• $v$ — the image distance. How far the image forms from the lens.
• $f$ — the focal length. A fixed property of the lens itself.
• $h$ — the object height (how tall the object is).
• $h'$ — the image height (how tall the image is). The little dash is read "h-dash".
• $m$ — the magnification. Just a number — how many times bigger or smaller the image is.

You can now say, in one breath, what these two relations are for: one finds where, the other finds how big and which way up.