Question

Точечный источник света лежит на главной оптической оси тонкой собирающей линзы с фокусным расстоянием F = 85.00 см. Расстояние от источника до центра линзы равно 2F. На какое расстояние x сместится изображение источника, если линзу повернуть на угол α = 5∘ так, чтобы центр линзы остался неподвижным?

Answer 1

Ответ 13 мм
смотри решение во вложении

Answer 2

The Problem

We have a thin converging lens with a focal length of F = 85.00 cm. The light source is located on the main optical axis of the lens, and the distance from the source to the center of the lens is 2F. We want to find out how far the image of the source will be displaced (in cm) if the lens is rotated by an angle α = 5°, while keeping the center of the lens fixed.

Solution

To solve this problem, we can use the lens formula and the lens-maker's formula. The lens formula relates the object distance (u), the image distance (v), and the focal length (f) of a lens:

1/u + 1/v = 1/f

The lens-maker's formula relates the focal length (f) of a lens to the refractive index (n) of the lens material, the radius of curvature of the lens surfaces (R1 and R2), and the thickness of the lens (t):

1/f = (n - 1) * (1/R1 - 1/R2 + (n - 1) * t / (n * R1 * R2))

In this problem, we are given the focal length (F) of the lens, which is 85.00 cm. We are also given that the distance from the source to the center of the lens is 2F.

Let's calculate the object distance (u) and the image distance (v) using the lens formula. Then we can find the displacement of the image by considering the rotation of the lens.

Calculations

Given: - Focal length (f) = 85.00 cm - Distance from source to center of lens (u) = 2F = 2 * 85.00 cm = 170.00 cm - Angle of rotation (α) = 5°

First, let's calculate the object distance (u) using the given distance from the source to the center of the lens:

u = 170.00 cm

Next, let's calculate the image distance (v) using the lens formula:

1/u + 1/v = 1/f

Substituting the values:

1/170.00 + 1/v = 1/85.00

Simplifying the equation:

1/v = 1/85.00 - 1/170.00

1/v = (2 - 1) / (2 * 85.00)

1/v = 1 / (2 * 85.00)

v = 2 * 85.00 cm = 170.00 cm

Now, let's consider the rotation of the lens by an angle α = 5°. The displacement of the image can be calculated using the formula:

Displacement (x) = v * tan(α)

Substituting the values:

x = 170.00 cm * tan(5°)

Using a calculator, we find:

x ≈ 14.88 cm

Therefore, the image of the source will be displaced by approximately 14.88 cm if the lens is rotated by an angle of 5° while keeping the center of the lens fixed.

Conclusion

The image of the light source will be displaced by approximately 14.88 cm if the lens is rotated by an angle of 5° while keeping the center of the lens fixed.