A concave spherical mirror is used to create an image of an object 5.00 cm tall that is located

step 1 Geometric Optics Chapter and in which we have a concave spherical mirror having radius of curvat

A concave spherical mirror is used to create an image of an...

A concave spherical mirror is used to create an image of an object $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm},$ which is $20.0 \mathrm{~cm}$ from point $C$, the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature of the mirror is $10.0 \mathrm{~cm} .$ Calculate the position $x_{\mathrm{i}}$ where the image is formed. Use the coordinate system given in the drawing. What is the height $h_{\mathrm{i}}$ of the image? Is the image upright (pointing up) or inverted (pointing down)? Is it real or virtual?

A concave spherical mirror is used to create an image of an object $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm},$ which is $20.0 \mathrm{~cm}$ from point $C$, the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature of the mirror is $10.0 \mathrm{~cm} .$ Calculate the position $x_{\mathrm{i}}$ where the image is formed. Use the
coordinate system given in the drawing. What is the height $h_{\mathrm{i}}$ of the image? Is the image upright (pointing up) or inverted (pointing down)? Is it real or virtual?

Key Concepts

Sign Conventions

Understanding sign conventions is essential in optics. For spherical mirrors, distances are measured from the mirror's vertex, and choosing the proper sign for the object distance, image distance, and focal length determines whether the formed image is real or virtual, and whether it is upright or inverted.

Magnification

Magnification in mirror systems is given by m = -d_i/d_o, which is also equal to the ratio of the image height to the object height. The negative sign indicates image inversion. Magnification helps determine not just the size of the image relative to the object, but also whether it is upright or inverted.

Real and Virtual Images

Real images occur when light rays converge after reflection, making them projectable onto a screen; these images are typically inverted. In contrast, virtual images form when light rays appear to diverge from a point behind the mirror, and they are upright. Differentiating between these two types of images is crucial when analyzing mirror systems.

Mirror Equation

The mirror equation, expressed as 1/f = 1/d_o + 1/d_i, connects the object distance (d_o), the image distance (d_i), and the focal length (f) of a mirror. It is a fundamental relation used in geometrical optics to determine the image location when two of these variables are known.

Focal Length

The focal length of a spherical mirror is defined as half the radius of curvature (f = R/2). This value is central in determining how the mirror converges or diverges light, and it plays a key role in the mirror equation as well as in establishing the nature of the formed image.

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