9.15: Use the mirror equation to deduce that:
(a) an object placed between f and 2f of a concave mirror produces a real image beyond 2f.
(b) a convex mirror always produces virtual image independent of the location of the object.
(C) the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
(d) an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.
[Note: This exercise helps you deduce algebraically properties of images that one obtains from explicit ray diagrams.]
(a)
Hint: Use mirror equation.
Step 1: Find the mirror equation for a concave mirror.
For a concave mirror, the focal length, f<0.
When the object is placed on the left side of the mirror, the object distance, u<0.
For image distance v,
1v-1u=1f1v=1f-1u ⋯(1)
Step 2: Find position of the image.
The object lies between f and 2f.
∴ 2f<u<f (∵ u and f are negative)12f>1u>1f-12f<-1u<-1f1f-12f<1f-1u<0 ⋯(2)
Using equation (1),
12f<1v<0
⇒1v=-ve⇒v=-ve
12f<1v⇒2f>v⇒-v>-2f.
Therefore, the image lies beyond 2f.
(b)
Hint: Focal length of the convex mirror is positive.
Step: Find the position of the image.
When the object is placed on the left side of the mirror, the object distance (u) is negative.
For image distance v,
1v+1u=1f
1v=1f-1u
As u is always negative and f is always positive for a convex mirror,
1v=1f+1u=+ve
v=+ve
Thus, the image is formed on the backside of the mirror.
Hence, a convex mirror always produces a virtual image, regardless of the object's distance.
(c)
Hint: m=−vu
Step 1: Find the image distance.
For image distance v,
1v+1u=1f
1v=1f-1u
As u=-ve
1v=1f+1u⇒1v>1f
⇒v<f
So the image always lies between the focus and the pole.
As 1v=1f-1u=u-fuf⇒v=ufu-f
So the image formed by a convex mirror is always diminished.
Hence, the image formed is diminished and is located between the focus (f) and the pole.
(d)
Hint: Use mirror equation.
Step 1: Find the nature of the image
When it is placed between the focus (f) and the pole:
∴ f<u (Both u and f are -ve)
1f>1u
1f-1u>0
For image distance v,
1v+1u=1f
1v=1f-1u>0⇒1v=+ve⇒v=+ve
The image is formed on the right side of the mirror. Hence, it is a virtual image.
Step 2: Find the magnification.
Magnification=-vu=-fu-f
As u and f are -ve for a concave mirror,
m=ff-u>1
Hence, the formed image is enlarged.