Key Equations

Key Equations

Image distance in a plane mirror	$d_{\text{o}}=\text{−}{d}_{\text{i}}$
Focal length for a spherical mirror	$f=\frac{R}{2}$
Mirror equation	$\frac{1}{d_{\text{o}}}+\frac{1}{d_{\text{i}}}=\frac{1}{f}$
Magnification of a spherical mirror	$m=\frac{h_{\text{i}}}{h_{\text{o}}}=\text{−}\frac{d_{\text{i}}}{d_{\text{o}}}$
Sign convention for mirrors
Focal length f	$\begin{array}{c}+\text{for concave mirror}\hfill \\ -\text{for convex mirror}\hfill \end{array}$
Object distance do	$\begin{array}{c}+\text{for real object}\hfill \\ -\text{for virtual object}\hfill \end{array}$
Image distance di	$\begin{array}{c}+\text{for real image}\hfill \\ -\text{for virtual image}\hfill \end{array}$
Magnification m	$\begin{array}{c}+\text{for upright image}\hfill \\ -\text{for inverted image}\hfill \end{array}$
Apparent depth equation	$h_{\text{i}}=\left(\frac{n_2}{n_1}\right)h_{\text{o}}$
Spherical interface equation	$\frac{n_1}{d_{\text{o}}}+\frac{n_2}{d_{\text{i}}}=\frac{n_2-n_1}{R}$
The thin-lens equation	$\frac{1}{d_{\text{o}}}+\frac{1}{d_{\text{i}}}=\frac{1}{f}$
The lens maker’s equation	$\frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
The magnification m of an object	$m\equiv \frac{h_{\text{i}}}{h_{\text{o}}}=\text{−}\frac{d_{\text{i}}}{d_{\text{o}}}$
Optical power	$P=\frac{1}{f}$
Optical power of thin, closely spaced lenses	$P_{\mathrm{total}}=P_{\text{lens}1}+P_{\text{lens}2}+P_{\text{lens}3}+\text{⋯}$
Angular magnification M of a simple magnifier	$M=\frac{{\theta }_{\text{image}}}{{\theta }_{\text{object}}}$
Angular magnification of an object a distance L from the eye for a convex lens of focal length f held a distance ℓ from the eye	$M=\left(\frac{25\text{cm}}{L}\right)\left(1+\frac{L-\mathcal{\ell }}{f}\right)$
Range of angular magnification for a given lens for a person with a near point of 25 cm	$\frac{25\text{cm}}{f}\le M\le 1+\frac{25\text{cm}}{f}$
Net magnification of compound microscope	$M_{\text{net}}=m^{\text{obj}}{M}^{\text{eye}}=\text{−}\frac{d_{\text{i}}^{\text{obj}}\left(f^{\text{eye}}+25\text{cm}\right)}{f^{\text{obj}}{f}^{\text{eye}}}$