Problem 49 - University Physics with Modern Physics

Question

A camera lens has a focal length of 180.0 mm and an aperture diameter of 16.36 mm. (a) What is the \(f\)-number of the lens? (b) If the correct exposure of a certain scene is \(1\over 30\)s at \(f/\)11, what is the correct exposure at \(f/\)2.8?

Step-by-Step Solution

Answer

(a) The f-number is 11. (b) The correct exposure at f/2.8 is approximately 0.5 seconds.

Step 1: Understand the f-Number Formula

The f-number, often denoted as \( f/\) or \( N \), is calculated using the formula \( N = \frac{f}{D} \), where \( f \) is the focal length of the lens, and \( D \) is the aperture diameter.

Step 2: Calculate the f-Number

Using the given focal length \( f = 180.0 \) mm and aperture diameter \( D = 16.36 \) mm, we apply the formula: \[ N = \frac{180.0}{16.36} \approx 11.00. \] Thus, the f-number is approximately 11.

Step 3: Understand Exposure Relation

The exposure time needed for a lens is related to the square of the f-number, given a constant amount of light, by the formula \( t_2 = t_1 \left( \frac{N_2}{N_1} \right)^2 \). This formula helps us find the new exposure time given a different f-number.

Step 4: Calculate New Exposure Time

We're given \( t_1 = \frac{1}{30} \) seconds at \( f/11 \), and we need to find \( t_2 \) at \( f/2.8 \). Using the formula: \[ t_2 = \frac{1}{30} \left( \frac{11}{2.8} \right)^2 \approx \frac{1}{30} \times 15.49 \approx \frac{1}{2}. \] Thus, the correct exposure at \( f/2.8 \) is approximately \( 0.5 \) seconds.

Key Concepts

Focal Length

In the world of camera optics, understanding focal length is essential. Focal length is the distance between the lens and the image sensor when the subject is in focus. It is usually measured in millimeters (mm). This length determines the field of view and magnification of the image.

A longer focal length results in a narrower field of view, making it easier to zoom in on distant subjects. Conversely, a shorter focal length offers a wider view, suitable for capturing vast landscapes. Using a lens with a focal length of 180.0 mm, for instance, means that it's likely geared toward zoomed-in shots or portraits where the background is less emphasized.

Aperture

Aperture refers to the size of the opening in the lens through which light enters. This opening can be adjusted, much like the pupil of an eye. A larger aperture allows more light to reach the camera sensor.

Aperture size is typically expressed in terms of the diameter of the aperture opening, measured in millimeters. In the given example, the lens has an aperture diameter of 16.36 mm. This indicates how wide the lens opening is, impacting the brightness of the captured image. A larger aperture not only increases light intake but also affects the depth of field, enabling beautifully blurred backgrounds, which is especially popular in portrait photography.

Exposure Time

Exposure time, sometimes known as shutter speed, is the duration for which the camera's sensor is exposed to light. It is crucial in determining the overall brightness of the photo. Shorter exposure times freeze motion, while longer times can capture motion blur.

In our exercise, the initial exposure time is \( \frac{1}{30} \) seconds. Adjustment of exposure time is necessary when changing the f-number of the lens.Using a mathematical relationship involving the square of the f-number, we can compute the new exposure time required when the aperture size is altered to maintain consistent illumination.

f-number

The f-number, also known as the f-stop, is a critical concept in optics that influences both exposure and depth of field. It is calculated by dividing the focal length by the aperture diameter, represented as \(N = \frac{f}{D}\).

A smaller f-number (e.g., f/2.8) means a larger aperture and more light entering the camera. This is ideal for shooting in low-light conditions. On the other hand, a larger f-number (e.g., f/11) denotes a smaller aperture, suitable for brightly lit scenes.

In practical terms, understanding the f-number helps photographers adjust exposure settings correctly, as demonstrated in our exercise's calculations. Switching from an f/11 to an f/2.8 aperture requires recalculating the exposure time to ensure the photographed scene remains properly exposed.