Problem 16 - Applied Physics

Question

Find the focal length of a mirror that forms an image \(5.66 \mathrm{~cm}\) behind a mirror of an object \(34.4 \mathrm{~cm}\) in front of the mirror.

Step-by-Step Solution

Answer

The focal length of the mirror is \(-6.77 \text{ cm}\).

Step 1: Identify Known Quantities

We are given that the image distance \( v \) is \(-5.66 \) cm (negative because the image is behind the mirror), and the object distance \( u \) is \( 34.4 \) cm (positive because the object is in front of the mirror).

Step 2: Mirror Formula Setup

To find the focal length \( f \), we use the mirror formula:\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

Step 3: Substitute Known Values

Substitute the known values into the mirror formula:\[ \frac{1}{f} = \frac{1}{34.4} + \frac{1}{-5.66} \]

Step 4: Simplify the Equation

Calculate the individual terms:\[ \frac{1}{34.4} = 0.02907 \]\[ \frac{1}{-5.66} = -0.17668 \]

Step 5: Calculate the Combined Term

Add the fractions:\[ \frac{1}{f} = 0.02907 - 0.17668 = -0.14761 \]

Step 6: Find the Focal Length

Take the reciprocal of the combined term to find the focal length:\[ f = \frac{1}{-0.14761} = -6.77 \text{ cm} \]

Key Concepts

Focal Length Calculation

To understand how we can find the focal length of a mirror, we need to start by identifying what focal length is.

Focal length, typically denoted by \( f \), is the distance between the mirror and the point where the parallel rays converge (or seem to diverge). This point is known as the focal point.

For mirrors, the focal length helps us predict how an image will be formed, depending on the position of the object. To calculate it, we need to know the image distance \( v \) (distance from the image to the mirror) and the object distance \( u \) (distance from the object to the mirror).

In the context of our problem, we have:

• Image distance \( v = -5.66 \) cm
• Object distance \( u = 34.4 \) cm

The negative sign of the image distance indicates the image is formed behind the mirror, which is typical for mirrors like concave ones, where virtual images form.

With these distances, we can proceed to the next step which involves using the mirror formula.

Mirror Formula

The mirror formula is a fundamental equation in optics that relates focal length, image distance, and object distance. This equation is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
This formula applies to spherical mirrors and is essential for solving problems involving image formation. The given form of the mirror formula assumes that all distances are measured from the mirror's central axis.

For our problem, by substituting the known values of \( u = 34.4 \) cm and \( v = -5.66 \) cm:

• Calculate \( \frac{1}{34.4} \), which results in \( 0.02907 \).
• Calculate \( \frac{1}{-5.66} \), yielding \( -0.17668 \).

By summing these fractions, we find the left-hand side of the mirror formula to be:
\[ \frac{1}{f} = 0.02907 - 0.17668 = -0.14761 \]
This step simplifies the equation so we can solve for the focal length accurately.
_next section of understanding is how the image forms and is affected by this focal length._

Image Formation in Mirrors

Understanding how images form in mirrors is crucial to grasping mirror optics. Mirrors can create virtual or real images, depending on their type and the object's position.

Real images are formed when light rays converge, such as in concave mirrors when an object is placed between focal length and infinity. These images can be projected onto screens. Conversely, virtual images appear where light rays only seem to diverge from. This occurs, for instance, when using a convex mirror or when a concave mirror has the object closer than its focal point.

In this exercise, we determined the image to be formed behind the mirror. Considering the given data, it suggests a virtual image structure, typical for an object located between a concave mirror and its focal point.

Furthermore, the negative focal length value indicates we're dealing with a concave mirror. In practical applications:

• Virtual images are often used in makeup mirrors, providing a magnified view.
• Real images, possible at other setups, are common in projectors.

Thus, image formation principles combined with focal length and mirror formula, allow us to predict where and how an image forms, enhancing our ability to design and use optical systems effectively.