Every camera, telescope, microscope and pair of glasses depends on one short equation. Move the object and the image moves with it — sometimes flipping upside down, sometimes vanishing entirely.
Convex Lens Imaging
Move the object and trace the rays to find the imageThe lens equation
1/f = 1/u + 1/v
where f is the focal length, u is the distance from object to lens, and v is the distance from lens to image. Magnification follows from it:
m = −v / u
That minus sign is doing real work. It is the reason a real image is inverted: when v is positive, m is negative, and a negative height means the image hangs below the axis.
The three regimes
- u > 2f — image is real, inverted and diminished. This is how a camera lens works: a distant scene projected small onto a sensor.
- u = 2f — real, inverted, exactly the same size. A neat symmetry worth seeing.
- f < u < 2f — real, inverted and magnified. This is a projector.
- u = f — the rays emerge parallel and no image forms at all.
- u < f — the rays never meet. Traced backwards they appear to come from a point behind the object: an upright, magnified virtual image. This is a magnifying glass.
Try this
- Press u = 2f and confirm the magnification reads exactly −1.00.
- Push the object closer than the focal length. The image flips to the same side as the object and turns upright — that is the magnifying glass.
- Set u exactly equal to f. The readout changes to “at infinity” and the rays leave the lens parallel.
- Double the focal length and watch how much less strongly the lens bends light.
Why it matters
The same equation governs the lens in your eye. Your eye has a fixed image distance — the retina — and focuses by changing the focal length of its lens instead. When that stops working well enough, the focal point lands in front of or behind the retina, and a second lens (glasses or contacts) is added to bring it back.