Concave & Convex Mirrors – Interactive Calculator

Uses Cartesian sign convention. Numbers can be entered as magnitudes; the script assigns signs correctly.

Calculator A

Given \(f\) and \(u\) → find \(v\), \(m\), \(h_i\)

Mirror Type

Focal length \(f\) (magnitude, cm)

Object distance \(u\) (magnitude, cm)

Object height \(h_o\) (optional, cm)

Output appears here.

Formulae used: \( \displaystyle \frac{1}{f}=\frac{1}{v}+\frac{1}{u}\), \( m=\frac{v}{u}=\frac{h_i}{h_o}\). Sign assignment: concave \(f<0\), convex \(f>0\), real object \(u<0\).

Calculator B

Given \(f\) and magnification \(m\) → find \(u\), \(v\)

Mirror Type

Focal length \(f\) (magnitude, cm)

Target magnification \(m\) (use sign: − for inverted, + for erect)

Output appears here.

Relations: \( m=\dfrac{v}{u}\) and \( \dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}\Rightarrow u=\dfrac{(1+m)}{m}\,f,\quad v=m\,u\). Signs are applied per mirror type.

Notes

Cartesian Sign Convention & Interpretation

Take the pole \(P\) as origin, principal axis along +x to the right (into the mirror).
Real object in front: \(u<0\). Real image in front of mirror: \(v<0\). Virtual image behind mirror: \(v>0\).
Concave mirror: \(f<0\); Convex mirror: \(f>0\).
Magnification \(m=\dfrac{v}{u}\):
- \(m<0\) → inverted (real) image.
- \(0<m<1\) → erect, diminished (typical convex or concave–virtual).

Output panel states sign, nature (real/virtual), and location relative to \(P\).