An object 0.600 cm tall is placed 16.5 cm to the left of the vertex of a concave spherical mirror having a radius of curvature of 22.0 cm. Determine the position, size, orientation, and nature (real or virtual) of the image.

Verified step by step guidance

1. Step 1: Understand the problem and identify the given values. The object height is 0.600 cm, the object distance (distance from the mirror) is 16.5 cm, and the radius of curvature of the concave mirror is 22.0 cm. Recall that the focal length (f) of a spherical mirror is half the radius of curvature, so f = R/2.
2. Step 2: Use the mirror equation to find the image distance (di). The mirror equation is: $\frac{1}{f}=\frac{1}{\mathrm{do}}+\frac{1}{\mathrm{di}}$, where f is the focal length, do is the object distance, and di is the image distance. Rearrange the equation to solve for di.
3. Step 3: Determine the magnification (M) using the magnification formula: $M=-\frac{\mathrm{di}}{\mathrm{do}}$. The magnification will help you find the size of the image by multiplying the magnification by the object height.
4. Step 4: Analyze the sign of di and M to determine the orientation and nature of the image. If di is positive, the image is real and located on the same side as the reflected rays. If di is negative, the image is virtual and located on the opposite side of the mirror. Similarly, if M is positive, the image is upright; if M is negative, the image is inverted.
5. Step 5: Combine all the results to describe the position, size, orientation, and nature of the image. Use the calculated di to find the position, the magnification to find the size, and the signs of di and M to determine the orientation and nature.

Key Concepts

Concave Mirror

A concave mirror is a spherical mirror that curves inward, resembling a portion of a sphere. It can converge light rays that are parallel to its principal axis, allowing for the formation of real or virtual images depending on the object's position relative to the focal point. The focal length of a concave mirror is half its radius of curvature, which is crucial for image formation calculations.

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror, expressed as 1/f = 1/v + 1/u. This equation is essential for determining the position of the image formed by the mirror. The sign conventions for distances must be carefully applied, where distances measured in the direction of the incoming light are negative.

Magnification

Magnification is the ratio of the height of the image (h') to the height of the object (h), given by the formula magnification (m) = h'/h = -v/u. It indicates how much larger or smaller the image is compared to the object and also provides information about the orientation of the image. A positive magnification indicates an upright image, while a negative value indicates an inverted image.