{"id":748,"date":"2026-08-10T02:20:40","date_gmt":"2026-08-10T02:20:40","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=748"},"modified":"2026-08-29T21:49:19","modified_gmt":"2026-08-29T21:49:19","slug":"lens-equation","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/waves\/lens-equation\/","title":{"rendered":"Lens Equation (1\/f = 1\/do + 1\/di)"},"content":{"rendered":"\n
Definition<\/div>

\n\nThe lens equation, written 1\/f = 1\/do<\/sub> + 1\/di<\/sub>, relates the focal length of a thin lens or spherical mirror to the object distance and the image distance. Rearranged, it locates an image from any two known quantities: a positive image distance means a real image, a negative one means a virtual image.\n\n<\/p><\/div>\n\n

Hold a magnifying glass over this page and the letters swell. Lift it away slowly and something odd happens \u2014 the text blurs, then flips upside down. Nothing about the glass changed. Only the distance did.<\/p>\n\n

That flip is not a quirk. It is one equation crossing zero, and the same equation tells a phone camera where to put your face, a projector where to throw a film, and an optician which lens will fix your vision.<\/p>\n\n

What Is the Lens Equation?<\/h2>\n\n

The lens equation is the formula that links an optical element’s focal length to where an object sits and where its image forms. Give it any two of those three distances and it hands you the third.<\/p>\n\n

Its real power is that it does not care what the element is. A glass lens bends light by refraction; a curved mirror bends it by reflection. Two completely different physical mechanisms \u2014 and one identical piece of algebra.<\/p>\n\n

That is why physicists usually call it the lens and mirror equation<\/strong>, or the thin-lens equation<\/strong> when a lens is specifically meant. The name changes; the maths does not.<\/p>\n\n

\"Lens<\/figure>\n\n

The three quantities in the lens equation, drawn to scale for a converging lens with the object beyond twice the focal length.<\/p>\n\n

Real images and virtual images<\/h3>\n\n

A real image<\/strong> is one that light actually reaches. Put a screen there and a picture appears \u2014 that is a cinema, a camera sensor, the retina at the back of your eye.<\/p>\n\n

A virtual image<\/strong> is where the light only appears<\/em> to come from. Your brain traces the outgoing rays backwards and places an image behind the glass. Put a screen there and you get nothing at all.<\/p>\n\n

The lens equation tells you which one you have, and it tells you with a single minus sign.<\/p>\n\n

The Lens Equation Formula<\/h2>\n\n

The equation in its standard form is short enough to memorise in one reading.<\/p>\n\n

1\/f = 1\/d_o + 1\/d_i<\/div>\n\n

Two companion formulas travel with it and you will need both. The first gives image size, the second converts a mirror’s curvature into a focal length.<\/p>\n\n

m = h_i \/ h_o = \u2212 d_i \/ d_o<\/div>\n\n
f = R \/ 2 (spherical mirror)<\/div>\n\n
\n\n\n\n\n\n\n\n\n\n\n\n
Symbol<\/th>\nQuantity<\/th>\nSI unit<\/th>\nSign rule<\/th>\n<\/tr>\n<\/thead>\n
f<\/strong><\/td>\nFocal length<\/td>\nmetre (m)<\/td>\nPositive if converging, negative if diverging<\/td>\n<\/tr>\n
do<\/sub><\/strong><\/td>\nObject distance<\/td>\nmetre (m)<\/td>\nPositive for any real object<\/td>\n<\/tr>\n
di<\/sub><\/strong><\/td>\nImage distance<\/td>\nmetre (m)<\/td>\nPositive if real, negative if virtual<\/td>\n<\/tr>\n
m<\/strong><\/td>\nMagnification<\/td>\ndimensionless<\/td>\nPositive if upright, negative if inverted<\/td>\n<\/tr>\n
ho<\/sub>, hi<\/sub><\/strong><\/td>\nObject and image height<\/td>\nmetre (m)<\/td>\nPositive above the optical axis<\/td>\n<\/tr>\n
R<\/strong><\/td>\nRadius of curvature (mirrors)<\/td>\nmetre (m)<\/td>\nPositive for a concave mirror<\/td>\n<\/tr>\n
P<\/strong><\/td>\nOptical power<\/td>\ndioptre (D) = m\u22121<\/sup><\/td>\nP = 1\/f, with f in metres<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

Note the units carefully. The equation itself is happy with centimetres, millimetres or metres \u2014 as long as all three<\/em> distances use the same one. Optical power is the exception: dioptres demand metres.<\/p>\n\n

The Three Sign Rules That Decide Every Answer<\/h2>\n\n

Nearly every wrong answer in geometric optics is a sign error, not an arithmetic one. Three rules prevent almost all of them.<\/p>\n\n

    \n
  1. Object distance is always positive.<\/strong> For any real object sitting in front of the lens or mirror, do<\/sub> is greater than zero. You will rarely meet the exception (a “virtual object” inside a multi-lens system).<\/li>\n
  2. Focal length carries the element’s character.<\/strong> Converging elements \u2014 a convex lens, a concave mirror \u2014 take a positive f. Diverging elements \u2014 a concave lens, a convex mirror \u2014 take a negative f.<\/li>\n
  3. Image distance is an output, never an input.<\/strong> You do not decide its sign. The equation returns it, and its sign is the answer: positive means real, negative means virtual.<\/li>\n<\/ol>\n\n

    Rule three is the one students fight. A negative di<\/sub> feels like a mistake, so they quietly drop the minus sign and lose the whole result. It is not a mistake \u2014 it is the equation telling you the image is virtual.<\/p>\n\n

    \"Sign<\/figure>\n\n

    Where each quantity is positive and negative, for a thin lens and for a spherical mirror.<\/p>\n\n

    Why your textbook may print a different formula<\/h3>\n\n

    Search the lens equation and you will meet a rival version: 1\/v \u2212 1\/u = 1\/f<\/strong>. Many UK and Indian syllabuses use it, and so do our companion guides to the concave and convex lens<\/a> and the concave and convex mirror<\/a>.<\/p>\n\n

    Neither version is wrong. They are the same physics with different bookkeeping \u2014 the Cartesian convention measures every distance as a signed coordinate along the direction light travels, so a real object to the left of the lens gets a negative<\/em> distance. Georgia State University’s HyperPhysics reference on the thin lens equation<\/a> sets both forms side by side and notes the Cartesian one is favoured in advanced texts because it scales better to multi-lens instruments.<\/p>\n\n

    Here is the exact translation.<\/p>\n\n

    \n\n\n\n\n\n\n\n\n\n\n
    Quantity<\/th>\nThis article (real-is-positive)<\/th>\nCartesian convention<\/th>\n<\/tr>\n<\/thead>\n
    Lens equation<\/td>\n1\/f = 1\/do<\/sub> + 1\/di<\/sub><\/td>\n1\/v \u2212 1\/u = 1\/f<\/td>\n<\/tr>\n
    Mirror equation<\/td>\n1\/f = 1\/do<\/sub> + 1\/di<\/sub> (identical)<\/td>\n1\/v + 1\/u = 1\/f<\/td>\n<\/tr>\n
    Real object<\/td>\ndo<\/sub> positive<\/td>\nu = \u2212do<\/sub> (negative)<\/td>\n<\/tr>\n
    Real image, lens<\/td>\ndi<\/sub> positive<\/td>\nv = +di<\/sub> (positive)<\/td>\n<\/tr>\n
    Real image, mirror<\/td>\ndi<\/sub> positive<\/td>\nv = \u2212di<\/sub> (negative)<\/td>\n<\/tr>\n
    Magnification<\/td>\nm = \u2212di<\/sub>\/do<\/sub><\/td>\nlens: m = v\/u    mirror: m = \u2212v\/u<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

    Substitute the middle column into the right column and the two collapse into one another exactly. In practice, pick whichever convention your course uses and never mix them inside a single problem \u2014 that is where the marks go missing.<\/p>\n\n

    How to Use the Lens Equation in Four Steps<\/h2>\n\n

    Start by writing down what you know with its sign attached, before touching any algebra. The method below works for lenses and mirrors alike.<\/p>\n\n

      \n
    1. Sign your knowns.<\/strong> Object distance positive. Focal length positive for converging, negative for diverging. For a mirror, get f from f = R\/2 first.<\/li>\n
    2. Rearrange for the unknown.<\/strong> Solving for image distance gives di<\/sub> = 1 \/ (1\/f \u2212 1\/do<\/sub>).<\/li>\n
    3. Compute the reciprocals separately.<\/strong> Work out 1\/f and 1\/do<\/sub> as decimals, subtract, then invert at the very end.<\/li>\n
    4. Read the signs back.<\/strong> Positive di<\/sub> means real; negative means virtual. Then find m = \u2212di<\/sub>\/do<\/sub>: negative m means inverted, and a magnitude above 1 means enlarged.<\/li>\n<\/ol>\n\n

      Step three is where most arithmetic slips happen, because it is tempting to invert too early. If you would rather have the algebra checked for you while you learn the pattern, our Lens & Mirror Calculator<\/a> solves for whichever quantity you leave blank and shows the working line by line.<\/p>\n\n

      The simulator below is the fastest way to build intuition. Drag the object closer and watch the image distance swing to infinity and come back on the other side as a negative number.<\/p>\n\n

      <\/span>Lens Equation Lab<\/span><\/div>