Question

Источник света находится на расстоянии L = 90 см от экрана. Тонкая собирающая линза, помещённая между источником света и экраном, даёт чёткое изображение источника при двух положениях.Найдите фокусное расстояние, если расстояние между положениями линзы 30 см. Ответ- 0.2!!! Нужно решение, поподробнее.

Answer 1

 

F расстояние фокусное

x расстояние свет/линза

y расстояние   линза/экран

Источник света находится на расстоянии L = 90 см от экрана.

x+y = 90  ;  y = 90 -x

формула тонкой линзы    

для первого случая

1/F = 1/x + 1/y  ;  1/F = 1/x + 1/(90 -x) ;     (1)

для второго случая

если расстояние между положениями линзы 30 см.

x2=x+30   ;   y2=y-30

1/F = 1/x2 + 1/y2  ; 1/F = 1/(x+30) + 1/(y-30) ; 1/F = 1/(x+30) + 1/(90 -x-30)

1/F = 1/(x+30) + 1/(60 -x)      (2)

 

приравняем  (1) (2)

1/x + 1/(90 -x) =  1/(x+30) + 1/(60 -x)  <-----это  арифметика

x = 30 см

у = 90 - 30 =60 см

1/F = 1/30+ 1/60  <-----это  арифметика

F = 20 см   =  0.2  м

ОТВЕТ 0.2 м

   

Answer 2

Problem Statement

We are given that a light source is located at a distance of L = 90 cm from a screen. A thin converging lens is placed between the light source and the screen, and it produces a clear image of the source at two different positions. The distance between the lens positions is 30 cm. We need to find the focal length of the lens.

Solution

To find the focal length of the lens, 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 from the lens, and - u is the object distance from the lens.

Let's assume the first position of the lens is at a distance x from the light source. The object distance u1 is the distance between the light source and the first lens position, which is L - x. The image distance v1 is the distance between the lens and the screen, which is x.

Similarly, for the second position of the lens, the object distance u2 is the distance between the light source and the second lens position, which is L + (30 - x). The image distance v2 is still the distance between the lens and the screen, which is 30 - x.

We are given that the lens produces a clear image at both positions, which means the image is formed on the screen. In this case, the image distance v is positive.

Using the lens formula for the first position: 1/f = 1/v1 - 1/u1

Substituting the values: 1/f = 1/x - 1/(L - x) ----(1)

Using the lens formula for the second position: 1/f = 1/v2 - 1/u2

Substituting the values: 1/f = 1/(30 - x) - 1/(L + (30 - x)) ----(2)

Since the lens produces a clear image at both positions, the focal length f should be the same for both equations (1) and (2).

Now, we can solve these two equations simultaneously to find the value of x, and then substitute it back into either equation to find the focal length f.

Let's solve the equations:

From equation (1): 1/f = 1/x - 1/(L - x)

Multiplying through by x(L - x): x(L - x)/f = (L - x) - x

Simplifying: x(L - x)/f = L - 2x

Expanding: Lx - x^2 = L - 2x

Rearranging: x^2 - Lx + 2x - L = 0

Combining like terms: x^2 - (L - 2)x - L = 0 ----(3)

From equation (2): 1/f = 1/(30 - x) - 1/(L + (30 - x))

Multiplying through by (30 - x)(L + (30 - x)): f = (L + (30 - x)) - (30 - x)

Simplifying: f = L + 30 - x - 30 + x

Simplifying further: f = L

So, we have found that the focal length f is equal to L.

Now, let's solve equation (3) to find the value of x:

x^2 - (L - 2)x - L = 0

Using the quadratic formula: x = (-(-L + 2) ± sqrt((-L + 2)^2 - 4(-L)(-L))) / 2

Simplifying: x = (L - 2 ± sqrt(L^2 - 4L + 4 + 4L^2)) / 2

Simplifying further: x = (L - 2 ± sqrt(5L^2 - 4L + 4)) / 2

Since the distance x cannot be negative, we take the positive value: x = (L - 2 + sqrt(5L^2 - 4L + 4)) / 2

Substituting the value of L = 90 cm: x = (90 - 2 + sqrt(5(90)^2 - 4(90) + 4)) / 2

Simplifying: x = (88 + sqrt(40500 - 360 + 4)) / 2

Simplifying further: x = (88 + sqrt(40544)) / 2

Calculating the square root: x = (88 + 201.35) / 2

Simplifying: x = 289.35 / 2

Calculating: x = 144.675

Now, we can substitute the value of x back into equation (1) to find the focal length f:

1/f = 1/x - 1/(L - x)

Substituting the values: 1/f = 1/144.675 - 1/(90 - 144.675)

Calculating: 1/f = 0.0069 - (-0.0069)

Simplifying: 1/f = 0.0069 + 0.0069

Calculating: 1/f = 0.0138

Taking the reciprocal: f = 1/0.0138

Calculating: f ≈ 72.46 cm

Therefore, the focal length of the lens is approximately 72.46 cm.

Please note that the answer may vary slightly due to rounding errors during calculations.

Conclusion

The focal length of the thin converging lens placed between the light source and the screen is approximately 72.46 cm. This focal length allows the lens to produce a clear image of the source at two different positions, with the lens placed 30 cm apart.