Question

A camera uses a 200.0 -mm focal length telephoto lens to take pictures from a distance of infinity to as close as 2.0 \(\mathrm{m}\). What are the minimum and maximum distances from the lens to the film?

Short answer

Answer: The minimum distance from the lens to the film is 222.22 mm and the maximum distance from the lens to the film is 200 mm.

Step-by-step solution

Identify the Thin Lens Equation

For this problem, we will use the Thin Lens Equation: 1/f = 1/u + 1/v Where f is the focal length of the lens, u is the object distance, and v is the image distance.

Plug in the given values of the focal length and object distance for minimum distance

We are given the focal length as f = 200 mm. For the minimum distance from the lens to the film, the object is at the closest possible distance, which is u = 2.0 m (or, 2,000 mm). We can now plug in these values into the Thin Lens Equation: 1/200 = 1/2000 + 1/v

Solve for the minimum distance from the lens to the film (v_min)

Rearrange the equation to get the value of v_min: 1/v_min = 1/200 - 1/2000 Now, to calculate v_min, you can take the inverse of the result: v_min = 1 / (1/200 - 1/2000) = 1 / (9/2000) = 222.22 mm So, the minimum distance from the lens to the film is 222.22 mm.

Plug in the given values of the focal length and object distance for maximum distance

For the maximum distance from the lens to the film, the object is at a distance of infinity so we can assume u is approaching infinity. Plug in the given values: 1/200 = 1/infinity + 1/v

Solve for the maximum distance from the lens to the film (v_max)

As the object distance (u) is approaching infinity, 1/u will approach 0. So the equation becomes: 1/200 = 0 + 1/v_max v_max = 200 mm So, the maximum distance from the lens to the film is 200 mm.

Conclude with the results

The minimum distance from the lens to the film is 222.22 mm and the maximum distance from the lens to the film is 200 mm.