How do I keep the sign conventions straight in ray optics for mirrors and lenses?

4 answers & replies

• [+][–]CPCommons Physics DeskTeam· 20d ago

  Use one convention for everything, the new Cartesian convention.

  • Measure all distances from the pole (mirror) or optical centre (lens).
  • Distances in the direction of incident light are positive; against it are negative.
  • Heights above the principal axis are positive; below are negative.

  Draw the object on the left, with light travelling to the right. Then a real object has $u < 0$ always.

  	Formula	Magnification
  Mirror	$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$	$m = -\dfrac{v}{u}$
  Lens	$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$	$m = \dfrac{v}{u}$

  Focal lengths: concave mirror $f < 0$, convex mirror $f > 0$, convex lens $f > 0$, concave lens $f < 0$.

  Example 1: concave mirror, $f = -15$ cm, $u = -30$ cm. Then $\tfrac{1}{v} = -\tfrac{1}{15} + \tfrac{1}{30} = -\tfrac{1}{30}$, so $v = -30$ cm and $m = -\dfrac{-30}{-30} = -1$. The image is real, inverted and the same size.

  Example 2: convex lens, $f = +10$ cm, $u = -5$ cm. Then $\tfrac{1}{v} = \tfrac{1}{10} + \tfrac{1}{-5} = -\tfrac{1}{10}$, so $v = -10$ cm and $m = +2$: virtual, upright, magnified (a magnifying glass).

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• [+][–]CMCommons Mock & PYQ DeskTeam· 20d ago

  In the paper, optics questions are often solved faster by a ray-diagram sketch plus a sign check than by pure algebra.

  Before substituting numbers, write "mirror" or "lens" at the top and write the formula with its signs. This single habit prevents the commonest slip, which is using the lens formula on a mirror problem.

  Then use the result's sign to check the story. A concave mirror with the object beyond $f$ must give a real image, so $v$ should be negative. If you got positive, something has flipped.

  Other items that repeat in papers:

  • Lensmaker's formula: $\dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$, where $n$ is relative to the surrounding medium.
  • Power $P = \dfrac{1}{f}$ with $f$ in metres (dioptre). Thin lenses in contact: powers add.
  • Critical angle: $\sin C = \dfrac{1}{n}$ for a medium to air.

  Mark all ray-optics misses in a mock as "sign", "formula choice", or "concept". You will probably find "sign" and "formula choice" cover most of them.

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• [+][–]CSCommons Strategy DeskTeam· 20d ago

  Make a single-page optics sheet and redraw it from memory every few days.

  Left half: the sign convention with a small sketch (object on the left, axis, positive direction arrow). Right half: the mirror and lens formulas side by side with their $m$ formulas, and the four focal-length signs.

  Then practise in a structured way:

  1. Plane mirror and refraction basics (a day or two).
  2. Spherical mirrors (ten or so questions, writing the sign of every quantity before calculating).
  3. Lenses, then lenses in combination and lens in a different medium.
  4. Prism and optical instruments (microscope, telescope), which need the earlier formulas but add their own.

  For each question write the data table first: $u = \ ?$, $v = \ ?$, $f = \ ?$. Only then substitute.

  Check the NTA syllabus to confirm which topics are currently in scope for optical instruments and wave optics.

  If a sign trick has worked well for you, share it below.

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• [+][–]CDCommons Dropper DeskTeam· 20d ago

  In a second attempt, optics is a good chapter to rebuild because it is mostly method. There is not much theory to forget, so the gain comes from a clean routine.

  Try this test: take ten old optics questions, solve them again using only the routine (write convention, write the table, substitute, check the sign story), and note every question where a sign still goes wrong. Those are the specific situations to drill, for example "virtual object" or "lens immersed in water".

  Be strict about the habit of writing signs explicitly rather than "doing it in your head". Many repeated errors survive only because the sign step was never written down.

  If you tried this and it helped, tell us what the biggest improvement was.

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