Question

What kind of image, virtual or real, is formed by a converging mirror when the object is placed a distance away from the mirror that is a) beyond the center of curvature of the mirror, b) between the center of curvature and half of the distance to the center of curvature, and c) closer than half of the distance to the center of curvature?

Short answer

Question: For each of the following cases, determine the nature of the image formed (real or virtual) when an object is placed in front of a converging mirror: a) Beyond the center of curvature of the mirror. b) Between the center of curvature and half of the distance to the center of curvature. c) Closer than half of the distance to the center of curvature. Answer: a) Real image b) Virtual image c) Virtual image

Step-by-step solution

Identify the object distance

The object is placed beyond the center of curvature (C) of the mirror. This is a real object, so the distance (u) is negative.

Use the mirror equation

Use the mirror equation to solve for image distance (v): 1/f = 1/u + 1/v In this case, 1/f and 1/u are both negative, so when you add them, you get a negative value. Thus, 1/v is negative, and the image distance (v) is also negative.

Conclusion for case (a)

Since the image distance (v) is negative, the image formed is a real image. b) Between the center of curvature and half of the distance to the center of curvature:

Identify the object distance

The object is placed between the center of curvature (C) and half of the distance to the center of curvature (F). This is a real object, so the distance (u) is negative.

Use the mirror equation

Use the mirror equation to solve for image distance (v): 1/f = 1/u + 1/v In this case, 1/f is negative while 1/u is more negative. The sum 1/u + 1/f yields a positive 1/v, and hence the image distance (v) is positive.

Conclusion for case (b)

Since the image distance (v) is positive, the image formed is a virtual image. c) Closer than half of the distance to the center of curvature:

Identify the object distance

The object is placed closer than half of the distance to the center of curvature (F). This is a real object, so the distance (u) is negative.

Use the mirror equation

Use the mirror equation to solve for image distance (v): 1/f = 1/u + 1/v In this case, 1/f is negative while 1/u is more negative. The sum 1/u + 1/f yields a positive 1/v, and hence the image distance (v) is positive.

Conclusion for case (c)

Since the image distance (v) is positive, the image formed is a virtual image.

Key concepts

Mirror Equation

Understanding the mirror equation is crucial for comprehending how images are formed by mirrors. In essence, the mirror equation is a mathematical relationship which connects the object distance (\( u \)), the image distance (\( v \)), and the focal length (\( f \)) of a curved mirror. It is expressed as \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
For a converging mirror, also known as a concave mirror, the focal length is considered negative as per the sign convention used in physics. If the object is real—which it typically is in optical systems—the object distance (\( u \)) is also negative.

The mirror equation allows us to classify the resulting image as real or virtual. A real image occurs when the image distance (\( v \)) is found to be negative, indicating the image forms on the same side as the reflective surface. Conversely, a positive image distance signifies a virtual image, which means the image appears to be behind the mirror. This straightforward equation is a powerful tool in determining the behavior and nature of images in curved mirror systems.

Real and Virtual Images

In the context of curved mirrors, it's important to distinguish between two types of images: real and virtual. A real image is produced when light rays converge at a point after being reflected off the mirror. This type of image can be projected onto a screen, as it exists on the same side of the mirror as the reflected light.

In contrast, a virtual image is formed when the reflected rays appear to diverge from a point behind the mirror. Despite being an optical illusion of sorts, virtual images are just as visible to an observer as real images. They simply cannot be captured on a screen because the light rays don't actually come from the location of the image.

Differentiating Based on Object Distance

• Placing the object beyond the center of curvature results in a real image, as the reflected rays actually converge in front of the mirror.
• If the object is positioned between the center of curvature and its halfway point to the focal length, a virtual image is formed because the reflected rays appear to come from a point behind the mirror.
• Lastly, placing the object closer than half the distance to the center of curvature likewise results in a virtual image for the same reason.

Understanding these principles is key for interpreting optical systems and predicting the behavior of light in various scenarios involving curved mirrors.

Center of Curvature

The center of curvature (\( C \)) is an essential concept to grasp when studying the image formation by a converging mirror. It is the point on the optical axis that lies at a distance equal to the radius of curvature from the mirror's vertex. The center of curvature is part of the mirror's sphere and is vitally important as a reference point in image formation.

The position of the object relative to the center of curvature significantly affects the nature of the image formed. For instance, objects placed beyond the center of curvature (\( u < -2f \) where \( f \) is the focal length) produce real, inverted, and diminished images. As the object moves closer to the mirror, passing through the center of curvature, the image grows in size and remains real and inverted until the object reaches the focal point, where no image is formed.

When Object is at the Center of Curvature

When the object is placed at the center of curvature itself, the image formed is real, inverted, and of the same size as the object. It's a unique point where the object distance (\( u \)) equals the image distance (\( v \)), both being equal to the radius of curvature. By understanding the relationship between the object position and the center of curvature, students can more easily predict and describe the properties of images produced by converging mirrors.