a concave mirror forms a real image of the same size as that as the object . where is the object placed?

When a concave mirror forms a real image that is the same size as the object, the object must be positioned at a specific point relative to the mirror. In this case, the object is placed at a distance equal to twice the focal length of the mirror, which is known as the center of curvature.

Understanding the Basics of Concave Mirrors

Concave mirrors are curved inward, resembling a bowl. They have unique properties that allow them to focus light rays that are parallel to the principal axis. The key points to remember about concave mirrors include:

• Focal Point (F): The point where parallel rays converge after reflecting off the mirror.
• Center of Curvature (C): The center of the sphere from which the mirror is a part, located at a distance equal to twice the focal length (2F).
• Principal Axis: The line that passes through the center of curvature and the focal point.

Where the Object is Placed

For a concave mirror, when the object is placed at the center of curvature (C), the following occurs:

• The distance from the mirror to the object (denoted as 'd_o') is equal to 2F.
• The image formed is real, inverted, and of the same size as the object.
• The distance from the mirror to the image (denoted as 'd_i') is also equal to 2F.

Using the Mirror Formula

We can use the mirror formula to understand this relationship better:

1/f = 1/d_o + 1/d_i

Where:

• f: Focal length of the mirror
• d_o: Object distance
• d_i: Image distance

Since we established that when the object is at the center of curvature, both d_o and d_i equal 2F. Thus, we can substitute:

1/f = 1/(2F) + 1/(2F)

This simplifies to:

1/f = 2/(2F)

Which further simplifies to:

1/f = 1/F

This confirms that the focal length is indeed half the distance to the center of curvature, validating our earlier conclusion.

Visualizing the Concept

Imagine a flashlight beam hitting a concave mirror. If you hold the flashlight at a distance equal to the center of curvature, the light rays will reflect and converge at the same distance, creating a clear, focused image. This is similar to how the object and image are related when placed at the center of curvature.

In summary, when a concave mirror forms a real image that is the same size as the object, the object is positioned at the center of curvature, which is twice the focal length of the mirror. This relationship is fundamental in optics and helps us understand how mirrors manipulate light to create images.