Lentile subțiri

Definiție

O asociere de 2 dioptrii și mediul transparent dintre dioptrii.

Relația punctelor conjugate

$$ \frac{1}{x_2} - \frac{1}{x_1} = (\frac{n_l}{n_m} - 1)(\frac{1}{R_1} - \frac{1}{R_2}) $$

Demonstrație

$$ \displaylines{ \frac{n_2}{x'_1} - \frac{n_1}{x_1} = \frac{n_2 - n_1}{R_1} \\ \frac{n_3}{x_2} - \frac{n_2}{-(d - x'_1)} = \frac{n_3 - n_2}{R_2} \\ \text{Dacă} \ n_3 = n_1 \implies \frac{n_1}{x_2} - \frac{n_2}{-(d - x'_1)} = \frac{n_1 - n_2}{R_2} \\ \\ \text{Pentru lentile subțiri:} \ d \ll -x_1, \ R, \ x'_1, \ etc \\ \\ \left. \begin{aligned} \frac{n_2}{x'_1} - \frac{n_1}{x_1} = \frac{n_2 - n_1}{R_1} \\ \frac{n_1}{x_2} - \frac{n_2}{-(d - x'_1)} = \frac{n_1 - n_2}{R_2} \\ \end{aligned} \right) (+) \\ \left. \frac{n_1}{x_2} - \frac{n_1}{x_1} = (n_2 - n_1)(\frac{1}{R_1} - \frac{1}{R_2}) \right\vert :n_1 \implies \\ \\ \implies \bf{\frac{1}{x_2} - \frac{1}{x_1} = (\frac{n_l}{n_m} - 1)(\frac{1}{R_1} - \frac{1}{R_2})} } $$

Distanțele focale

$$ \displaylines{ f_1=x_1, \ x_2 \rightarrow \infty \implies \frac{1}{f_1} = -(\frac{n_l}{n_m} - 1)(\frac{1}{R_1} - \frac{1}{R_2}) \\ f_2=x_2, \ x_1 \rightarrow \infty \implies \frac{1}{f_2} = (\frac{n_l}{n_m} - 1)(\frac{1}{R_1} - \frac{1}{R_2}) \\ } $$
  • OBS:\( \ \frac{1}{x_2} - \frac{1}{x_1} = \frac{1}{f} \)
  • Lentilă în aer liber:\( \ \frac{1}{f} = (n_l - 1)(\frac{1}{R_1} - \frac{1}{R_2}) \)
  • Convergența:\( \ C = \frac{1}{f} \quad [C]_{SI} = 1 \rm{d} = \text{dioptrie} = 1 \rm{m^{-1} } \)

    • Lentile convergente \( \scriptstyle \ \ f \gt 0 \)

    Lentile divergente \( \scriptstyle \ \ f \lt 0 \)