A plano convex lens of focal length 30cm has its plane surface silvered. An object is placed 40cm from the lens on the convex side. The distance of the image from the lens is: v1−u1=f1 or, v1=301−401 or, v1=1204−3 or, v=+120cm Now, this image acts as the object for the lens. So, u=+120cm,f=30cm v1−u1=f1 or, v1=301+1201 or, v=+24cm
Image formation from compound lenses
A converging lens of focal length 30cm and diverging lens of focal length 20cm are kept 15cm apart with their principal axes coinciding. Where shall an object be placed to form an image at infinity?
Two cases are possible. Case 1: Final image is formed by the concave lens. For concave lens v=∞; f=−20cm ⇒u=+20cm Now, v=20+15=35cm serves as image distance for the convex lens. So, v=+35cm; f=+30cm v1−u1=f1 ⇒351−u1=301 ⇒u=−210cm=210cm from converging lens.
Case 2: Final image is formed by convex lens. So,v=∞; f=+30cm ⇒u=−30cm for convex lens to form image at infinity So for the concave lens, the image distance is v=−30−(−15)=−15cm; f=−20cm v1−u1=f1 ⇒−151−u1=−201 or, u=−60cm, i.e, 60cm from diverging lens.
Image formation by broken lenses
A point object O is placed at a distance 30cm from a convex lens (focal length 20cm) cut into two halves each of which is displaced by 0.05cm perpendicular to the principal axis. What is the distance between the two images formed?
v1−u1=f1 or, v1−−301=20−1 or, v=60cm So, d=3×h=0.1×3=0.3cm m=uv=hohi=3060=2=0.05hi or, hi=0.1cm
Formation of image from a combination of lenses, mirrors and glass slabs
Example: A convex lens of focal length 40cm is held at a distance 12cm co-axially above a concave mirror of focal length 18cm. If the convex lens is replaced by a glass plate of thickness 6cm, refractive index μ=23 and gives rise to an image coincident with itself, then what will be the value of d?
Solution: When the ray from O passes through the slab of refractive index (μ), then there will be shift of point O to I1 and then this point I1 will act as source for the concave mirror. Shift = (1−μ1)t=(1−32)6 ⟹ shift =2cm, i.e., the object will appear to look closer by 2cm. Now as the final image is formed at a point O itself, so the ray from point I1 will retrace its own path (i-e, I1 should be at R of concave mirror). So, d−2+12=2×f1=2×18=36 ⟹d=26cm