1. Why we need a mirror formula
When a mirror forms an image, I usually want to know three things — where the image is, how big it is, and whether it is upright or inverted. Instead of drawing full ray diagrams every time, there is a simple mathematical relation that connects the object distance, image distance and focal length. This relation is the mirror formula.
The formula works for both concave and convex mirrors, as long as I use the proper sign convention.
2. Sign convention recap
To apply the mirror formula correctly, I always follow these rules:
- Distances measured against the direction of incoming light are negative.
- Distances measured in the direction of incoming light are positive.
- Heights measured upward from the principal axis are positive.
- Heights measured downward are negative.
- A concave mirror has negative focal length.
- A convex mirror has positive focal length.
3. The mirror formula
The relationship among object distance \(u\), image distance \(v\), and focal length \(f\) is:
\( \dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} \)
This single formula helps me find the position of the image without drawing any ray diagram. The signs of \(u\), \(v\) and \(f\) automatically tell me whether the image is real or virtual.
3.1. Quick meaning of the signs
- If v is positive, the image is formed on the same side as the reflected light (convex mirror virtual image).
- If v is negative, the image forms on the object side (concave mirror real image).
- Negative f → concave mirror; Positive f → convex mirror.
This sign logic helps me instantly understand the nature of the image.
4. Magnification and what it tells me
Magnification tells me how large or small the image is compared to the object. It also tells whether the image is upright or inverted.
The magnification \(m\) of a mirror is defined as:
\( m = \dfrac{h_i}{h_o} \)
where \(h_i\) is the height of the image and \(h_o\) is the height of the object.
4.1. Magnification formula using distances
The height ratio can also be written using distances:
\( m = -\dfrac{v}{u} \)
The minus sign is important — it tells me the orientation of the image.
4.2. How to interpret magnification
- m > 1: Image is enlarged.
- m < 1: Image is diminished.
- m positive: Image is upright (virtual for mirrors).
- m negative: Image is inverted (real for mirrors).
5. Using both formulas together
For most mirror questions, I first use the mirror formula to find the image distance \(v\), then substitute that value into the magnification formula to get the size and orientation of the image.
This combination removes guesswork and makes it easier to analyse different cases like objects placed beyond C, at C, between C and F, or closer than F.
6. A small example to make it clearer
If an object is placed at \(u = -30\,\text{cm}\) in front of a concave mirror with focal length \(f = -15\,\text{cm}\):
Using the mirror formula:
\( \dfrac{1}{v} + \dfrac{1}{-30} = \dfrac{1}{-15} \)
\( \dfrac{1}{v} = -\dfrac{1}{15} + \dfrac{1}{30} = -\dfrac{1}{30} \)
\( v = -30\,\text{cm} \)
So the image forms at the same distance on the object's side, meaning it is real and inverted.
Magnification:
\( m = -\dfrac{v}{u} = -\dfrac{-30}{-30} = -1 \)
This says the image is the same size as the object but inverted.
7. Where this formula helps me in real setups
I find the mirror and magnification formulas very helpful when working with:
- Concave shaving/makeup mirrors (for magnified images)
- Headlights and reflectors (concave mirrors focusing light)
- Security and vehicle mirrors (convex mirrors giving diminished images)
The formulas make it easy to predict how big the image will be and where it will appear.