To solve this problem, we will use the mirror formula and follow the steps required to determine the final position of the image formed by the system of mirrors.
Step 1: Calculate the image position for the concave mirror.
For any spherical mirror (concave or convex), the mirror equation is given by:
[tex]\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \][/tex]
where:
- [tex]\( f \)[/tex] is the focal length of the mirror.
- [tex]\( d_o \)[/tex] is the distance of the object from the mirror.
- [tex]\( d_i \)[/tex] is the distance of the image from the mirror.
Given:
- The focal length of the concave mirror [tex]\( f = 20 \)[/tex] cm (since it's a concave mirror, we take it as negative according to the sign convention: [tex]\( f = -20 \)[/tex] cm).
- The object distance [tex]\( d_o = 40 \)[/tex] cm (since the object is in front of the mirror, we take it as negative for the same reason: [tex]\( d_o = -40 \)[/tex] cm).
Using the formula, we calculate the image distance [tex]\( d_i \)[/tex]:
[tex]\[ \frac{1}{-20} = \frac{1}{-40} + \frac{1}{d_i} \][/tex]
[tex]\[ \frac{1}{d_i} = \frac{1}{-20} - \frac{1}{-40} \][/tex]
[tex]\[ \frac{1}{d_i} = \frac{-2 + 1}{40} \][/tex]
[tex]\[ \frac{1}{d_i} = \frac{-1}{40} \][/tex]
[tex]\[ d_i = -40 \][/tex] cm
The negative sign indicates that the image is formed on the same side of the mirror as the object, which is characteristic of a real image formed by a concave mirror.
Step 2: Consider the effect of the plane mirror.
Now, we have an image at a distance of 40 cm behind the concave mirror, which will act as a "virtual object" for the plane mirror. Note that plane mirrors always form virtual images that are the same distance behind the mirror as the object is in front of it.
Since the plane mirror is placed 20 cm in front of the concave mirror, the virtual object is effectively 40 cm + 20 cm = 60 cm in front of the plane mirror. Hence, the plane mirror will form a virtual image 60 cm behind itself.
Step 3: Locate the final image relative to the concave mirror.
To find the final image position relative to the concave mirror, we must account for the distance between the two mirrors. The image is 60 cm behind the plane mirror, so from the concave mirror's point of view, this would be:
Final image position [tex]\(= \)[/tex] Plane mirror distance from concave mirror [tex]\( + \)[/tex] Distance of plane mirror's image behind itself
Final image position [tex]\( = 20 \)[/tex] cm [tex]\( + 60 \)[/tex] cm
Final image position [tex]\( = 80 \)[/tex] cm
This final image is virtual and is located 80 cm behind the concave mirror. Thus, the final position of the image is 80 cm from the concave mirror.
