An Object Is Placed 40cm In Front Of A Mirror, And An Image Is Formed 8cm Behind It. What Type Of Mirror Is It, And What Is Its Focal Length?

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In the fascinating world of physics, mirrors stand as intriguing tools that manipulate light to create images. From the simple reflection we see in a bathroom mirror to the complex optical systems in telescopes, mirrors play a crucial role in our understanding and utilization of light. This article delves into a specific scenario involving a mirror and an object, aiming to determine the type of mirror used and calculate its focal length. Understanding these concepts is fundamental to grasping the principles of optics and image formation.

Decoding the Mirror Equation and Magnification

To effectively tackle the problem at hand, let's first revisit the foundational concepts of mirror optics. The behavior of light as it interacts with mirrors is governed by the mirror equation and the magnification equation. These equations serve as powerful tools for analyzing image formation in various scenarios.

The mirror equation establishes a relationship between the object distance (u), the image distance (v), and the focal length (f) of the mirror. Mathematically, it is expressed as:

1/f = 1/u + 1/v

Where:

  • f represents the focal length of the mirror, which is the distance between the mirror's surface and its focal point.
  • u denotes the object distance, measured as the distance between the object and the mirror's surface.
  • v signifies the image distance, indicating the distance between the image and the mirror's surface.

The magnification equation, on the other hand, provides information about the size and orientation of the image relative to the object. It is defined as:

m = -v/u

Where:

  • m represents the magnification, which is the ratio of the image height to the object height.
  • A positive magnification indicates an upright image, while a negative magnification signifies an inverted image.
  • The absolute value of magnification reveals the relative size of the image compared to the object. If |m| > 1, the image is magnified; if |m| < 1, the image is diminished; and if |m| = 1, the image is the same size as the object.

By applying these equations and carefully considering the sign conventions, we can unravel the mysteries of image formation and determine the characteristics of mirrors.

Sign Conventions: Navigating the World of Optics

In the realm of optics, sign conventions play a crucial role in ensuring accurate calculations and interpretations. Adhering to these conventions allows us to distinguish between real and virtual images, concave and convex mirrors, and object and image distances. Let's outline the key sign conventions commonly used:

  • Object Distance (u): The object distance is always considered positive as the object is placed in front of the mirror.
  • Image Distance (v): The image distance is positive for real images, which are formed by the actual intersection of light rays. Real images can be projected onto a screen. Conversely, the image distance is negative for virtual images, which are formed by the apparent intersection of light rays. Virtual images cannot be projected onto a screen.
  • Focal Length (f): The focal length is positive for concave mirrors, which converge light rays. Conversely, the focal length is negative for convex mirrors, which diverge light rays.
  • Magnification (m): A positive magnification indicates an upright image, while a negative magnification signifies an inverted image.

By consistently applying these sign conventions, we can avoid confusion and obtain accurate results when analyzing mirror systems.

Problem Scenario: Unveiling the Mirror's Nature and Focal Length

Now, let's turn our attention to the specific problem scenario presented: An object is placed 40 cm in front of a mirror, and an image is formed 8 cm behind the mirror. Our task is to determine the type of mirror used and calculate its focal length.

Step 1: Identifying the Given Information

Before we dive into calculations, let's extract the key information provided in the problem statement:

  • Object distance (u) = 40 cm
  • Image distance (v) = -8 cm (The image is formed behind the mirror, indicating a virtual image, hence the negative sign)

Step 2: Determining the Type of Mirror

To determine the type of mirror, we need to analyze the nature of the image formed. The problem states that the image is formed 8 cm behind the mirror. This crucial piece of information tells us that the image is a virtual image. Virtual images are formed by the apparent intersection of light rays and cannot be projected onto a screen.

Virtual images are characteristic of convex mirrors and concave mirrors when the object is placed closer to the mirror than the focal point. Convex mirrors always produce virtual, upright, and diminished images. Concave mirrors, on the other hand, can form both real and virtual images depending on the object's position. When the object is closer to the concave mirror than its focal point, a virtual, upright, and magnified image is formed. However, if the object is placed beyond the focal point, the image formed is real and inverted.

Since the problem doesn't mention whether the image is magnified or diminished, we cannot definitively rule out a concave mirror based on this information alone. However, the fact that the image is formed behind the mirror is a strong indicator of a convex mirror.

To confirm our suspicion, we can calculate the magnification using the magnification equation:

m = -v/u = -(-8 cm) / 40 cm = 0.2

The magnification is positive, which confirms that the image is upright. Furthermore, the magnification is less than 1, indicating that the image is diminished. This aligns perfectly with the characteristics of images formed by convex mirrors. Therefore, we can confidently conclude that the mirror used in this scenario is a convex mirror.

Step 3: Calculating the Focal Length

Now that we have identified the type of mirror, let's calculate its focal length. We can use the mirror equation to achieve this:

1/f = 1/u + 1/v

Substituting the given values:

1/f = 1/40 cm + 1/(-8 cm)

To solve for f, we need to find a common denominator for the fractions:

1/f = (1 - 5) / 40 cm

1/f = -4 / 40 cm

1/f = -1 / 10 cm

Taking the reciprocal of both sides:

f = -10 cm

The focal length is -10 cm. The negative sign confirms that the mirror is a convex mirror, as convex mirrors have negative focal lengths.

Conclusion: Unraveling the Mysteries of Mirrors

In this article, we embarked on a journey to understand the principles of image formation in mirrors. We explored the mirror equation and magnification equation, which serve as fundamental tools for analyzing mirror systems. By applying these equations and adhering to sign conventions, we successfully determined the type of mirror used in a specific scenario and calculated its focal length.

We learned that the formation of an image 8 cm behind a mirror when an object is placed 40 cm in front of it indicates the use of a convex mirror. The negative focal length (-10 cm) further confirmed this conclusion. This exercise highlights the power of physics in unraveling the mysteries of the world around us and provides a solid foundation for further exploration of optics and image formation.

By understanding the behavior of mirrors and the principles that govern their operation, we gain valuable insights into the nature of light and its interactions with matter. This knowledge has far-reaching implications in various fields, including optical instruments, photography, and even medical imaging.

Keep exploring, keep questioning, and keep unraveling the fascinating world of physics!