Thin Lenses — Question 10

Question 10

Material dispersion separates the image planes of two colors

Use the thin-lens relations $1/p+1/q=1/f$ and $m=-q/p$ in one surrounding medium. Distances p and q are positive for real objects and real images; f is positive for a converging lens. A thin lens in air has refractive index 1.50 and focal length 20 cm for red light. Its blue-light index is 1.52, with unchanged geometry. A real object is 60 cm away. Use the lensmaker dependence $1/f\propto n-1$.

1. Find the blue focal length and both image distances.

2. Find the longitudinal separation of the color images and state which lies nearer the lens.

Question 10 - Solution

Material dispersion separates the image planes of two colors

(a) Method and calculation

$$f_B=f_R\frac{n_R-1}{n_B-1}=20\frac{0.50}{0.52}=\boxed{250/13\,\mathrm{cm}}.$$$$\boxed{q_R=20(60)/(60-20)=30\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$\boxed{q_B=\frac{(250/13)60}{60-250/13}=1500/53=28.3019\,\mathrm{cm}}.$$$$\boxed{q_R-q_B=90/53=1.69811\,\mathrm{cm}}\quad\text{blue focuses nearer}.$$

Physical check

The larger blue index increases positive power and shortens both the focal length and the real image distance for this fixed object.