Stepwise Derivation of Mirror Formula with Sign Conventions and Diagram
The mirror formula and magnification are core tools in ray optics, connecting object distance, image distance, and focal length to predict how mirrors form images in the real world. Whether designing car mirrors or solving JEE Main questions, understanding these concepts allows physicists and engineers to control how images are produced, their size, and orientation. In this guide, you’ll master the mirror formula and magnification—definitions, derivations, sign conventions, worked examples, and key contrasts for perfect exam application.
The mirror formula and magnification not only help explain everyday phenomena like side-view mirrors or makeup mirrors but are tested heavily in competitive exams like JEE Main, making them essential for aspirants. Let’s break down every aspect of these formulas and see how they interlink with other physics concepts.
Mirror Formula and Magnification: Definition and Need
A spherical mirror uses a part of a sphere’s surface to reflect light. The mirror formula establishes a precise mathematical relationship: object distance (u), image distance (v), and focal length (f) for both concave and convex mirrors. It allows rapid calculation of where and how big an image forms.
The magnification formula for mirrors quantifies how much larger or smaller the image is compared to the object. This is crucial for tasks like identifying car number plates from a convex mirror or focusing light with a concave mirror.
Derivation of Mirror Formula: Stepwise for JEE Main
The mirror formula is given by:
- 1/v + 1/u = 1/f
- Where: v = image distance, u = object distance, f = focal length (all in SI units, metres).
To derive this, we use the sign convention in optics (known as the New Cartesian sign convention):
- Distances measured in the direction of incident light are positive
- Distances against incident light (towards the reflecting surface) are negative
- All distances are measured from the pole (vertex) of the mirror
With a ray diagram for a concave mirror: let an object be at distance u from the pole, image forms at v, focal length is f. By applying laws of reflection and geometry to triangles formed by incident and reflected rays, and substituting lengths as per sign, we get the final result: 1/v + 1/u = 1/f.
Magnification Formula for Mirrors: Meaning and Calculation
The magnification (m) produced by a spherical mirror is the ratio:
- m = (height of image) / (height of object) = -v/u
- The negative sign accounts for image inversion by concave mirrors.
- If |m| > 1, image is larger; if < 1, image is smaller. Negative m: inverted image. Positive m: erect image.
For convex mirrors, m is always positive and less than one, corresponding to a diminished, erect image. For concave mirrors, m can be positive or negative depending on the object's position relative to focus and center of curvature.
Mirror Formula and Magnification: Quick Reference Table
| Mirror Type | Focal Length (f) | Image Nature | Typical m | Sign |
|---|---|---|---|---|
| Concave | Negative | Real or virtual | >1 or <1 | + or – |
| Convex | Positive | Virtual | <1 | + |
Use this table to instantly identify the focal length sign, image characteristics, and likely magnification when tackling a JEE Main optics problem.
Applying the Mirror Formula and Magnification: Solved Example
Suppose an object is placed 30 cm in front of a concave mirror with focal length 15 cm. Find the image position and magnification.
- u = –30 cm (object on left); f = –15 cm (concave, by sign convention)
- Mirror formula: 1/v + 1/u = 1/f → 1/v – 1/30 = –1/15 ⇒ 1/v = –1/15 + 1/30 = –1/30
- v = –30 cm (image forms at same distance as object, but real)
- Magnification: m = –v/u = –(–30)/(–30) = –1
- Result: Magnification is –1; image is real, inverted, same size.
Watch out for the sign convention: most errors in JEE Main come from using the wrong u, v, or f sign! For more solved optics questions, check out the optics mock test series as you practice.
Concave vs Convex Mirrors: Mirror Formula and Magnification
Concave and convex mirrors use the same mirror formula, but the value and sign of focal length and image formation differ. Here’s how:
- Concave mirrors: f negative, can form real/inverted or virtual/erect images, magnification can be positive or negative.
- Convex mirrors: f positive, always form virtual, erect, diminished images, magnification always positive and less than 1.
To visualise these differences, compare result tables, use ray diagrams, or try examples with varying positions. The difference between lens and mirror article can clarify when to use each formula.
Tips, Pitfalls, and Quick Revision Points
- Always apply the correct sign convention for u, v, f.
- Use SI units throughout—convert cm to m as needed.
- For convex mirrors, m is always positive; for concave, sign depends on image.
- When magnification is negative, the image is inverted.
- Mirror formula does not apply to plane mirrors (use simpler rules).
Memorise the mirror formula and magnification equation, but focus on understanding sign logic and patterns in ray diagrams. Be careful with virtual vs real image placement and direction of measurement.
Mirror Formula and Magnification: Practical Uses for JEE and Everyday Life
Mastering the mirror formula and magnification prepares you for diverse optics topics in JEE Main and real situations. Applications include:
- Vehicle rearview mirrors (convex) always show erect, diminished images for traffic awareness.
- Concave mirrors in torches and reflectors focus light into a beam.
- Optical instruments (microscope mirrors, telescope reflectors) rely on predicting magnification.
- Physics experiments where precise control of image location and size is vital.
For deeper comparisons, read about lens vs mirror or explore ray diagrams for spherical mirrors and actual JEE-level problems to see the mirror formula in action. Vedantu’s curated physics resources make exam revision structured and effective for every JEE Main aspirant.
FAQs on Mirror Formula and Magnification: Concepts, Formula, and Applications
1. What is the mirror formula and why is it important?
The mirror formula relates object distance (u), image distance (v), and focal length (f) for spherical mirrors, making it essential for calculating image position and nature in optics problems.
Key points about its importance:
- It allows students to solve questions on image formation by mirrors in exams like JEE, NEET, and Board tests.
- The formula is: 1/v + 1/u = 1/f
- It works for both concave and convex mirrors when the correct sign convention is applied.
- This formula simplifies the process of predicting whether images are real or virtual, inverted or erect.
2. What is the formula for a mirror?
The standard formula for spherical mirrors is:
1/v + 1/u = 1/f
Where:
- v = Image distance from mirror
- u = Object distance from mirror
- f = Focal length of the mirror
3. How do you calculate magnification in a mirror?
Magnification for a mirror quantifies how much larger or smaller the image is compared to the object. It is calculated using:
Magnification (m) = -v/u
Key points:
- v = Image distance from the mirror
- u = Object distance from the mirror
- The negative sign indicates inversion of image for some cases (e.g., real images are inverted)
4. What is the sign convention used in mirror formula numericals?
The sign convention in mirror formula follows the New Cartesian Sign Convention, crucial for correct calculations in optics:
- All distances are measured from the pole of the mirror (reference point).
- Distances measured in the direction of incident light are positive.
- Distances measured against the direction of incident light are negative.
- Focal length (f) is negative for concave mirrors and positive for convex mirrors.
- Image distance (v) is negative if image forms on the same side as the object, positive for the opposite side.
5. What are the differences between concave and convex mirror magnification?
Concave and convex mirrors differ in the type of images and magnification they produce:
- Concave Mirror: Can produce both real, inverted (magnification negative) or virtual, erect (magnification positive) images depending on object position.
- Convex Mirror: Always produces virtual, erect, and diminished images (magnification always positive but less than 1).
6. Can a mirror have a magnification greater than 1?
Yes, a mirror can show magnification greater than 1 especially with concave mirrors when the object is placed between the pole and focus.
- Magnification > 1 means the image is larger than the object.
- This occurs in concave mirrors for specific object placements (e.g., makeup mirrors).
- Convex mirrors never give magnification above 1 – they always make images smaller.
7. Why is the minus sign used in the magnification formula for mirrors?
The minus sign in the magnification formula (m = -v/u) indicates the orientation of the image:
- If m is negative, the image is inverted relative to the object
- If m is positive, the image is erect
- This helps distinguish between real and virtual images in exams and numericals
8. Are u and v related in magnification for a mirror?
Yes, in mirrors, magnification (m) is directly related to the object distance (u) and image distance (v). It is given by:
m = -v/u
- If you know two of the distances (u or v) and m, you can calculate the third.
- This relationship is vital for solving numerical optics questions in school exams.
9. What happens if I forget the sign convention while solving a problem?
If you ignore the sign convention in mirror formula or magnification problems, it's likely you'll get an incorrect answer (wrong image position, size, or nature).
- Always use negative for distances to the left of the mirror (incident light), positive to the right.
- Be careful with focal length: negative for concave mirrors, positive for convex.
- Consistent application ensures full marks in Board and JEE/NEET numericals.
10. Is the mirror formula valid for all types of mirrors and images?
The mirror formula (1/v + 1/u = 1/f) is valid for both concave and convex spherical mirrors and works for real and virtual images when the proper sign convention is followed.
- It applies only to spherical mirrors (not plane mirrors or other shapes).
- Always check if you're using the correct sign for focal length and distances as per the mirror type and image nature.
11. Why does the image sometimes appear inverted or erect depending on the mirror?
Whether an image is inverted or erect depends on the mirror type and the object's placement relative to the mirror's focal point:
- Concave mirrors: can form both inverted (real) and erect (virtual) images depending on where the object is placed.
- Convex mirrors: always form erect (virtual) images.
- The sign and value of magnification help identify if the image is inverted (negative m) or erect (positive m).
