Thin Lens Equation Calculator

The fundamental formula governing thin lenses relates the focal length ($f$), the object distance ($d_o$), and the image distance ($d_i$) through the standard reciprocal relationship: 1/f = 1/d_o + 1/d_i. Additionally, optical magnification ($M$) measures the relative size of the formed image compared to the original object, calculated using the absolute ratio of image distance to object distance: M = |d_i| / d_o. Sign conventions are crucial here: convex converging lenses feature positive focal lengths, while concave diverging lenses use negative focal lengths.

Consider a converging convex lens with a known focal length of $f = 10\text{ cm}$. Suppose we place an object at a distance of $d_o = 30\text{ cm}$ in front of the lens. To find the image distance ($d_i$), we rearrange the thin lens formula to 1/d_i = 1/f - 1/d_o. Substituting our values gives 1/d_i = 1/10 - 1/30. Finding a common denominator yields 1/d_i = 3/30 - 1/30 = 2/30, which simplifies to 1/15. Inverting this gives an image distance of $d_i = 15\text{ cm}$. Next, to calculate the magnification, we apply $M = d_i / d_o = 15 / 30 = 0.5$. This indicates that the resulting real image is inverted and half the size of the original object.

Always maintain consistent units throughout your calculations; mixing centimeters and meters is a frequent source of error. Pay close attention to sign conventions, particularly for virtual images where the image distance will yield a negative value. When checking your work manually, remember that a real image formed by a single lens is always inverted, whereas virtual images are always upright.