Two converging lenses are placed 38.5 cm apart. The focal length of the lens on the right is 15.5 cm , and the focal length of the lens on the left is 11.0 cm . An object is placed to the left of the 11.0 cm focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses.
How far to the left of the 11.0 cm focal-length lens is the original object?
To find the distance to the left of the 11.0 cm focal-length lens where the original object is placed, we can use the lens formula and the concept of the thin lens equation.
1. Start by labeling the given values:
- Distance between the lenses (d) = 38.5 cm
- Focal length of the right lens (fR) = 15.5 cm
- Focal length of the left lens (fL) = 11.0 cm
2. Let's assume the distance to the left of the 11.0 cm focal-length lens where the original object is placed is x cm.
3. Now, we can use the thin lens equation for the right lens to find the position of the intermediate image formed by the right lens. The thin lens equation is:
1/fR = 1/vR - 1/uR,
where fR is the focal length of the right lens, vR is the image distance formed by the right lens, and uR is the object distance from the right lens.
Since the final image is formed halfway between the two lenses, the distance between the intermediate image formed by the right lens and the left lens is equal to (d/2) = 19.25 cm.
4. Now, we can calculate the image distance formed by the right lens (vR):
vR = (d/2) - uR
5. Next, we can substitute the values in the thin lens equation for the right lens:
1/15.5 = 1/((d/2) - x) - 1/x
6. Simplify the equation:
1/15.5 = (2x - d)/(x((d/2) - x))
7. Cross-multiply and simplify further:
15.5 * (d/2 - x) = x * (2x - d)
8. Expand and rearrange the equation:
(15.5 * d)/2 - 15.5 * x = 2x^2 - dx
9. Rearrange the equation to the quadratic form:
2x^2 - dx + 15.5 * x - (15.5 * d)/2 = 0
10. Solve for x using the quadratic formula:
x = (-(-d) ± √((-d)^2 - 4 * 2 * (15.5 * d)/2))/(2 * 2)
11. Simplify the equation and calculate x:
x = (d + √(d^2 + 31 * d))/4
12. Substitute the given values:
x = (38.5 + √(38.5^2 + 31 * 38.5))/4
Using a calculator, you can find the value of x to determine the distance to the left of the 11.0 cm focal-length lens where the original object is placed.