Spherical Lenses

To solve the problem, we need to understand the impact of the interactions and the redistribution of charges among the spheres.

1. Initially, let's assume that the charge on spheres and is each. Since they repel each other with force , we know that: , where is Coulomb's constant, and is the distance between the spheres.
2. Sphere is initially uncharged.
3. When sphere is brought into contact with sphere , the charge is shared between them. Since both spheres are identical, the charge will redistribute equally: for both and .
4. Next, sphere is brought into contact with sphere . Again, the charge is shared equally between and . So, the charge on each sphere becomes: and .
5. Now, sphere is placed at the midpoint between spheres and .
6. The net force on sphere due to spheres and must be considered. The force between sphere and is: . The force between sphere and is the same due to symmetry: .
7. Adding these forces gives: .

Thus, the force experienced by sphere is . Therefore, the correct answer is .

From 1st lens
1/v + 1/6 = 1/24
1/v = 1/24 - 1/6 = -1/8
v = -8 cm

From 2nd lens
1/v + 1/18 = 1/9
1/v = 1/9 - 1/18 = 1/18
v = 18 cm

So distance between object and its image:
6 + 10 + 18 = 34 cm

Solution:
Given:

Focal length of each lens,

Separation between the lenses,

The object is at a considerable distance (assumed to be at infinity).

Objective: Find the distance of the final image from the first lens.

Approach: Step 1: Image Formation by the First Lens For the first lens ( ), the object is at infinity. Using the lens formula: where:

is the object distance (infinity),

is the image distance,

is the focal length.

Plugging in the values: So, the first lens forms an image at 20 cm from itself.

Step 2: Object for the Second Lens The image formed by the first lens acts as the object for the second lens. The separation between the lenses is 60 cm, and the first image is 20 cm from the first lens. Therefore, the distance of this image (object for the second lens) from the second lens is: Since the image is on the same side as the incoming light for the second lens, we consider as negative in the lens formula (real image for the first lens acts as a virtual object for the second lens).

Step 3: Image Formation by the Second Lens Using the lens formula for the second lens ( ): Plugging in the values: Solving for : The positive sign indicates that the final image is formed on the opposite side of the second lens from where the light is coming.

Step 4: Distance of the Final Image from the First Lens The final image is 40 cm from the second lens. Since the lenses are 60 cm apart, the distance from the first lens is:

Conclusion: The image of the distant object formed by the combination of the two convex lenses is located at \boxed{100 \, \text{cm}} from the first lens.