Two identical diverging lenses are separated by 15 cm. The focal length of each lens is -8.0 cm. An object is located 4.0 cm to the left of the lens that is on the left. Determine the final image distance relative to the lens on the right.

For the first divergent lens

F1= - 8 cm,
object distance u = 4 cm.
Apply
1/F1 =1/v+1/u,
plug the value of F1 and u, and
solve for v
-1/8 = 1/4 + 1/v,
the image distance for first lens (to the left ofthe first lens)
v = -2.67cm.
Distance between two lences
d= 15 cm.
Image formed by the first lens is object to 2nd lens,
then object distance for 2nd lens is
u1 =15+v =17.67 cm (here take only magnitude,
because we have taken sign conventions)
Focal length of 2nd lens
F2 = - 8cm
apply
1/F2 =1/v1 +1/u1
plug values of F1 and u1 we get v1(that is image distance to the left of 2nd lens)
v1 =- 5.5 cm
-represants towards left of second lens.

Oh boy, we've got some lenses and objects playing hide-and-seek! Let me crunch some numbers and bring out the answers!

To solve this, we can use the lens formula:

1/f = 1/v - 1/u

where f is the focal length, v is the image distance, and u is the object distance. Since we have two identical lenses, they have the same focal length of -8.0 cm.

The object distance (u) is given as 4.0 cm. Let's calculate the image distance for the lens on the left (v1) using the formula:

1/(-8.0) = 1/v1 - 1/4.0

Calculating this, we find v1 = 7.2 cm.

Now, to find the final image distance relative to the lens on the right (v2), we need to consider the distance between the two lenses, which is 15 cm.

Think of it this way, the image formed by the lens on the left becomes the object for the lens on the right. So, the object distance (u2) for the lens on the right is (15 cm - 7.2 cm) = 7.8 cm.

Using the lens formula once more:

1/(-8.0) = 1/v2 - 1/7.8

Calculating this, we find v2 ≈ 21.3 cm.

So, the final image distance relative to the lens on the right is approximately 21.3 cm.

Remember, in the world of lenses, things can get a bit distorted. But hey, let's put on our clown noses and have some fun with it!

To solve this problem, we can follow these steps:

Step 1: Determine the location of the image formed by the lens on the left.
Using the lens formula:
1/f = 1/v - 1/u
where f is the focal length of the lens and u is the object distance.
Given:
f = -8.0 cm (negative sign indicates a diverging lens)
u = -4.0 cm (negative sign indicates the object is on the left side of the lens)
Solving for v:
1/v = 1/f - 1/u
1/v = 1/(-8.0) - 1/(-4.0)
1/v = -1/8 + 1/4
1/v = (-1+2)/8
1/v = 1/8
v = 8.0 cm (positive sign indicates the image is formed on the same side as the object)

Step 2: Determine the location of the virtual object formed by the lens on the right.
Since the image formed by the lens on the left is located at 8.0 cm to the right of it, the virtual object formed by the lens on the right will be located at the same distance to the right of it. Therefore, the virtual object distance is -8.0 cm.

Step 3: Determine the final image distance relative to the lens on the right.
Using the lens formula again, but this time with the focal length of the lens on the right:
1/f = 1/v - 1/u
where v is the final image distance and u is the object distance.
Given:
f = -8.0 cm (negative sign indicates a diverging lens)
u = -8.0 cm (distance to the virtual object formed by the lens on the right)
Solving for v:
1/v = 1/f - 1/u
1/v = 1/(-8.0) - 1/(-8.0)
1/v = (-1+1)/(-8)
1/v = 0
v = infinity (The final image is formed at infinity relative to the lens on the right.)

Therefore, the final image distance relative to the lens on the right is infinity.

To determine the final image distance relative to the lens on the right, we can use the lens formula:

1/f = 1/v - 1/u

Where:
- f is the focal length of the lens
- v is the image distance relative to the lens
- u is the object distance relative to the lens

Given that the focal length of each lens is -8.0 cm and the object distance relative to the lens on the left is -4.0 cm, we can calculate the image distance relative to the lens on the left using the lens formula.

1/f = 1/v1 - 1/u1

Substituting the values:

1/-8.0 = 1/v1 - 1/-4.0

Simplifying the equation:

-1/8.0 = 1/v1 + 1/4.0

To find v1, we need to solve for it. Cross-multiply and simplify the equation:

4.0v1 - 8.0 = -8.0v1

12.0v1 = 8.0

v1 = 8.0 / 12.0

v1 = 0.67 cm

Now, to determine the final image distance relative to the lens on the right, we need to consider the separation between the two lenses. In this case, the lenses are separated by 15 cm.

To find the final image distance relative to the lens on the right, we can use the lens formula again:

1/f = 1/v2 - 1/u2

Where:
- f is the focal length of the lens
- v2 is the image distance relative to the lens on the right
- u2 is the object distance relative to the lens on the right

Given that the focal length of each lens is -8.0 cm and the object distance relative to the lens on the right is -15.0 cm, we can calculate the image distance relative to the lens on the right using the lens formula.

1/-8.0 = 1/v2 - 1/-15.0

Simplifying the equation:

-1/8.0 = 1/v2 + 1/15.0

To find v2, we need to solve for it. Cross-multiply and simplify the equation:

15.0v2 - 8.0 = -8.0v2

23.0v2 = 8.0

v2 = 8.0 / 23.0

v2 ≈ 0.35 cm

Therefore, the final image distance relative to the lens on the right is approximately 0.35 cm.