Concave Mirror Image Distance And Focal Length Calculation

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Introduction

Understanding the behavior of concave mirrors is crucial in the field of optics. These mirrors, known for their ability to converge light, play a significant role in various applications, from telescopes to dental mirrors. This article delves into a specific problem involving a concave mirror, where we aim to determine the image distance and focal length given the object distance, object height, and image height. By applying the mirror equation and the magnification formula, we will systematically solve the problem, providing a clear and comprehensive explanation of the underlying principles. The principles of reflection and image formation are fundamental to understanding how concave mirrors work, and this article will serve as a practical guide to applying these concepts. We'll start by outlining the problem statement, then proceed to a step-by-step solution, ensuring clarity and accuracy in our calculations. This article is designed to be a valuable resource for students, educators, and anyone interested in the fascinating world of optics.

Problem Statement

Consider a concave mirror. An object with a height of 0.2 cm is placed 5 cm away from the mirror. A real image is formed, and its height is measured to be 1.2 cm. Our task is to determine two key parameters: the distance of the image from the mirror and the focal length of the mirror. This problem allows us to apply the fundamental principles of mirror optics, specifically the mirror equation and the magnification formula. Understanding how to solve this problem is essential for anyone studying physics or working with optical systems. The given information provides a solid foundation for our calculations. We know the object height (hâ‚€), the object distance (u), and the image height (háµ¢). From these, we can deduce the magnification and subsequently determine the image distance (v). Once we have both the object and image distances, we can use the mirror equation to find the focal length (f). This step-by-step approach will ensure a clear and accurate solution.

Solution

1. Understanding Magnification

Magnification (M) is a crucial concept in optics, describing how much larger or smaller the image is compared to the object. It's defined as the ratio of the image height (háµ¢) to the object height (hâ‚€). Mathematically, this is expressed as: M = háµ¢ / hâ‚€. In our case, the object height (hâ‚€) is 0.2 cm, and the image height (háµ¢) is 1.2 cm. Plugging these values into the formula, we get: M = 1.2 cm / 0.2 cm = 6. This tells us that the image is six times larger than the object. However, it's important to consider the sign of the magnification. Since the image is real and inverted in a concave mirror, the magnification is negative. Therefore, M = -6. The negative sign indicates the inversion, a characteristic of real images formed by concave mirrors. This understanding of magnification is a key stepping stone in determining the image distance and ultimately the focal length. The magnification not only tells us the size difference but also the orientation of the image, which is crucial for a complete understanding of image formation.

2. Determining Image Distance

The magnification (M) is also related to the object distance (u) and the image distance (v) by the formula: M = -v / u. We already know the magnification (M = -6) and the object distance (u = -5 cm). Note that we use the sign convention where the object distance is negative because the object is placed in front of the mirror. Now, we can solve for the image distance (v). Substituting the known values into the equation, we get: -6 = -v / (-5 cm). Multiplying both sides by -5 cm, we have: v = -30 cm. The negative sign for the image distance indicates that the image is real and formed on the same side of the mirror as the object. This is a characteristic of real images formed by concave mirrors when the object is placed beyond the focal point. The image distance of -30 cm means the image is formed 30 cm in front of the mirror. This value is essential for the next step, where we will determine the focal length of the mirror. Understanding the relationship between magnification, object distance, and image distance is fundamental to solving problems involving mirrors and lenses.

3. Calculating Focal Length

The focal length (f) of a mirror is a critical parameter that determines its focusing power. It is related to the object distance (u) and the image distance (v) through the mirror equation: 1/f = 1/v + 1/u. We have already determined the object distance (u = -5 cm) and the image distance (v = -30 cm). Now, we can substitute these values into the mirror equation to find the focal length (f). Plugging in the values, we get: 1/f = 1/(-30 cm) + 1/(-5 cm). To solve for f, we first find a common denominator for the fractions: 1/f = -1/30 cm - 6/30 cm. Combining the fractions, we get: 1/f = -7/30 cm. Now, we take the reciprocal of both sides to solve for f: f = -30 cm / 7. This gives us the focal length: f ≈ -4.29 cm. The negative sign indicates that the mirror is concave, as expected. The focal length of approximately 4.29 cm means that parallel rays of light incident on the mirror will converge at a point 4.29 cm in front of the mirror. This calculation completes our solution, providing us with both the image distance and the focal length of the concave mirror.

Conclusion

In this article, we successfully determined the image distance and focal length of a concave mirror given the object distance, object height, and image height. We began by understanding the concept of magnification and calculated it using the object and image heights. Then, we used the magnification formula along with the object distance to find the image distance. Finally, we applied the mirror equation to calculate the focal length. The results showed that the image distance is -30 cm and the focal length is approximately -4.29 cm. These values provide a complete picture of the image formation process for the given scenario. The negative signs for both the image distance and focal length are consistent with the properties of a concave mirror and the formation of a real, inverted image. This exercise highlights the importance of understanding and applying the mirror equation and magnification formula in solving optics problems. By following a systematic approach, we can accurately analyze and predict the behavior of light interacting with mirrors and lenses. This knowledge is crucial for various applications, including designing optical instruments and understanding the principles behind vision. Further exploration of these concepts can lead to a deeper understanding of the fascinating field of optics.