A converging lens of focal length 7.740 cm is 19.7 cm to the left of a diverging lens of focal length -5.51 cm. A coin is placed 18.2 cm to the right of the diverging lens.
a. Find the location of the coin's final image.
b. Find the magnification of the coin's final image.
Any help is appreciated.
To solve this problem, we can use the lens formula and magnification formula. The lens formula is given by:
1/f = 1/v - 1/u
Where:
f = focal length of the lens
v = image distance
u = object distance
We can start by finding the image distance and the object distance for each lens.
For the converging lens:
Focal length (f1) = 7.740 cm
Object distance (u1) = distance from lens to coin = 18.2 cm (to the right)
Using the lens formula, we can find the image distance (v1) for the converging lens:
1/7.740 = 1/v1 - 1/18.2
Simplifying the equation, we get:
1/v1 = 1/7.740 + 1/18.2
Now, let's solve for v1 by finding the reciprocal of both sides:
v1 = 1 / (1/7.740 + 1/18.2)
v1 ≈ 5.178 cm
So, the image distance for the converging lens is approximately 5.178 cm.
Now let's move on to the diverging lens:
Focal length (f2) = -5.51 cm (notice the negative sign indicates a diverging lens)
Lens position (pu2) = distance from the converging lens to the diverging lens = 19.7 cm (to the left)
To find the object distance (u2) for the diverging lens, we need to calculate it from the image distance (v1) of the converging lens:
u2 = pu2 - v1
u2 = 19.7 - 5.178
u2 ≈ 14.522 cm
Now that we have the object distance for the diverging lens, we can calculate the image distance (v2) using the lens formula:
1/-5.51 = 1/v2 - 1/14.522
Rearranging the equation, we get:
1/v2 = -1/5.51 + 1/14.522
Taking the reciprocal of both sides, we have:
v2 = 1 / (-1/5.51 + 1/14.522)
v2 ≈ -6.678 cm
Since the image distance (v2) of the diverging lens is negative, it indicates a virtual image formed on the same side as the object.
Now, we can find the overall image distance by adding the two image distances together:
V = v2 + v1
V ≈ -6.678 + 5.178
V ≈ -1.5 cm
The image distance (V) is approximately -1.5 cm from the diverging lens.
a. The location of the coin's final image is approximately 1.5 cm to the left of the diverging lens.
To find the magnification (m) of the final image, we can use the magnification formula:
m = -(v2/v1)
Plugging in the calculated values:
m = -(-6.678/5.178)
m ≈ -1.29
b. The magnification of the coin's final image is approximately -1.29. Since the value is negative, it indicates an inverted image.
Note: In optical calculations, it's important to pay attention to the sign conventions regarding the directions of object and image distances. The negative sign indicates that the image is formed on the same side as the object, and the positive sign indicates that the image is formed on the opposite side.