An object is 18.0 cm from the center of a spherical silvered-glass Christmas tree ornament 6.00 cm in diameter. What are the position and magnification of its ?

Verified step by step guidance

1. Step 1: Identify the type of mirror formed by the spherical silvered-glass ornament. Since the ornament is silvered, it acts as a spherical mirror. The diameter of the ornament is given as 6.00 cm, so the radius of curvature (R) is half of the diameter: R = 6.00 cm / 2 = 3.00 cm.
2. Step 2: Determine the focal length (f) of the spherical mirror using the relationship between the focal length and the radius of curvature: f = R / 2. Substitute the value of R to find f.
3. Step 3: Use the mirror equation to find the position of the image (q). The mirror equation is: $\frac{1}{f}=\frac{1}{p}+\frac{1}{q}$, where p is the object distance (18.0 cm), f is the focal length, and q is the image distance. Rearrange the equation to solve for q.
4. Step 4: Calculate the magnification (M) of the image using the magnification formula: $M=\; -\frac{q}{p}$. Substitute the values of q and p to find M.
5. Step 5: Interpret the results. The sign of q will indicate whether the image is real or virtual, and the sign of M will indicate whether the image is upright or inverted. Additionally, the magnitude of M will indicate the size of the image relative to the object.

Key Concepts

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a spherical mirror. It is expressed as 1/f = 1/v + 1/u. Understanding this formula is essential for determining the position of the image formed by the ornament, which acts as a concave mirror due to its silvered surface.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed as M = -v/u. This concept is crucial for understanding how the size of the image compares to the size of the object, which is important in the context of the ornament's reflective properties.

Focal Length of a Spherical Mirror

The focal length (f) of a spherical mirror is half the radius of curvature (R). For a spherical mirror, the focal length is positive for concave mirrors and negative for convex mirrors. In this case, knowing the diameter of the ornament allows us to calculate the focal length, which is necessary for applying the mirror formula.