Spherical Refracting Surfaces — Question 2

Question 2

A curved refracting surface can leave the outgoing rays divergent

Use $n_1/p+n_2/q=(n_2-n_1)/R$ and $m=-n_1q/(n_2p)$ for paraxial refraction. Here p is positive on the incident side, q on the outgoing side, and R when the center lies on the outgoing side. The indices are n1=1.00 and n2=1.50, the radius is R=+10 cm, and a real object is only p=10 cm from the surface.

1. Find q and determine whether the image is directly projectable onto a screen in the glass.

2. Find transverse magnification and the physical separation of object and image.

Question 2 - Solution

A curved refracting surface can leave the outgoing rays divergent

(a) Method and calculation

$$1.50/q=0.50/10-1/10=-0.050\,\mathrm{cm^{-1}}\Rightarrow\boxed{q=-30\,\mathrm{cm}}.$$$$q<0\Rightarrow\boxed{\text{virtual image; no direct outgoing screen focus}}.$$

(b) Interpretation and further calculation

$$\boxed{m=-1(-30)/[1.50(10)]=+2}.$$$$x_O=-10,\ x_I=-30\Rightarrow\boxed{|x_I-x_O|=20\,\mathrm{cm}}.$$

Physical check

Both the real object and the virtual image lie on the incident side, so their physical separation is a difference of magnitudes.