Problem 2 - College Physics

Question

\(\cdot\) Camera \(A\) has a lens with an aperture diameter of 8.00 \(\mathrm{mm}\) . It photographs an object, using the correct exposure time of \(\frac{1}{30}\) s. What exposure time should be used with camera \(\mathrm{B}\) in photographing the same object with the same film if camera \(B\) has a lens with an aperture diameter of 23.1 \(\mathrm{mm}\) ?

Step-by-Step Solution

Answer

Camera B should use an exposure time of approximately 1/250 seconds.

Step 1: Understand the Problem

We need to determine the correct exposure time for Camera B, given the exposure time for Camera A and the apertures of both cameras. The exposure time is influenced by the amount of light entering through the aperture.

Step 2: Use the Aperture Formula

The aperture affects the exposure time according to the formula \( t_2 = t_1 \left( \frac{d_1}{d_2} \right)^2 \), where \( t_1 \) and \( t_2 \) are the exposure times for cameras A and B, respectively, and \( d_1 \) and \( d_2 \) are the aperture diameters.

Step 3: Plug in Known Values

We know \( t_1 = \frac{1}{30} \) s, \( d_1 = 8.00 \) mm, \( d_2 = 23.1 \) mm. Substituting these into the formula, we get: \[ t_2 = \frac{1}{30} \left( \frac{8.00}{23.1} \right)^2 \]

Step 4: Calculate the Ratio

First, calculate the ratio \( \frac{8.00}{23.1} \). This gives: \( 0.346 \). Now square the ratio: \( 0.346^2 = 0.1197 \).

Step 5: Calculate the Exposure Time for Camera B

Now multiply the squared ratio by the exposure time of Camera A: \( t_2 = \frac{1}{30} \times 0.1197 \approx 0.004 \) seconds. Thus, the exposure time for Camera B should be approximately \( 1/250 \) seconds.

Key Concepts

Camera Exposure Calculation

Calculating the correct exposure for a photograph is key to getting a well-exposed image. Exposure is the amount of light that reaches the camera's sensor, which is influenced by three factors: aperture, shutter speed, and ISO. In this context, we focus on the first two—aperture and shutter speed.
The formula used to calculate the required exposure time for different apertures is:

• \( t_2 = t_1 \left( \frac{d_1}{d_2} \right)^2 \)

Here, \( t_1 \) and \( t_2 \) represent the exposure times for two different scenarios, typically involving two lenses with different aperture diameters, \( d_1 \) and \( d_2 \). The exposure time is shortened or lengthened based on how much the aperture diameter changes.
Hence, understanding and applying the formula is crucial when switching lenses or cameras with different apertures. It ensures that the photographed object remains properly exposed, preventing it from being too bright or too dark.

Aperture Diameter

Aperture diameter is one of the most critical components in photography, especially concerning exposure. It affects how much light enters the lens, which directly influences the detail and brightness of a photograph.
In this problem, Camera A and Camera B have different aperture diameters, 8.00 mm and 23.1 mm, respectively. The larger the aperture diameter, the more light hits the sensor. This might be likened to opening a larger window in a dark room, allowing more sunlight to fill the room.
Since Camera B has a larger aperture, it gathers more light than Camera A for the same amount of time. Consequently, its exposure time needs to be adjusted to compensate for this increased light intake. This adjustment helps in achieving the perfect balance between light and the photograph's final look.

• Smaller apertures require longer exposure times
• Larger apertures can work with shorter exposure times

Being aware of these aspects allows photographers to make informed decisions and achieve desired photographic effects irrespective of changing lighting conditions.

Photography in Physics

Photography beautifully intersects with physics when capturing light and creating images. Understanding these principles allows photographers to manipulate elements like light and time effectively.
Physics demonstrates how the interaction between camera settings such as aperture and exposure time can affect a photograph's outcome. The aperture operates much like a pupil in the human eye, expanding and contracting to regulate light intake.
An understanding of exposure and aperture in a physical context translates into practical know-how in photography. Utilizing concepts like the inverse square law, physics explains why doubling the distance of a light source decreases its intensity to a quarter, which parallels the idea of aperture and exposure manipulation in photography.
Emphasizing these scientific principles bridges the gap between formulating the exposure calculation and its practical application, ensuring photographers of all experience levels can produce well-balanced images.