Question

An object is 25.0 cm from the center of a spherical Part A silvered-glass Christmas tree ornament 6.40 cm in diameter. What is the position of its image (counting from the omament surface)? Follow the sign rules. Express your answer with the appropriate units. Part B What is the magnification of its image?

Answer 1

The magnification of the image is approximately
`\(0.068\)`.

To solve these problems, we'll consider the spherical Christmas tree ornament as a spherical mirror. Since it's silvered, we'll assume it's a convex mirror (as it's silvered on the inside), which always forms virtual, upright, and reduced images.

Part A: Image Position

For a spherical mirror, the mirror equation is given by:

`\[ (1)/(f) = (1)/(d_o) + (1)/(d_i) \]`

where:

-
`\( f \)` is the focal length,

-
`\( d_o \)` is the object distance (from the mirror's surface),

-
`\( d_i \)` is the image distance (from the mirror's surface, with virtual images having a negative value).

For a convex mirror, the focal length is positive and equal to half the radius of curvature, and the radius of curvature
`\( R \)` is half the diameter.

Given:

- Diameter of the ornament
`\( D = 6.40 \)` cm,

- Object distance from the center of the ornament
`\( d_o' = 25.0 \)` cm.

To find the object distance from the surface
`(\( d_o \)),`we subtract the radius from the distance from the center:

`\[ d_o = d_o' - (D)/(2) \]`

The focal length
`\( f \` is negative for a convex mirror and is half the radius of curvature
`\( R \):`

`\[ f = -(R)/(2) = -(D)/(4) \]`

Then, we solve for
`\( d_i \)` using the mirror equation.

Part B: Magnification

The magnification
`\( m \)` is given by:

`\[ m = -(d_i)/(d_o) \]`

The negative sign in the magnification formula for spherical mirrors indicates that the image formed by a convex mirror is virtual and upright.

Let's perform these calculations.

Part A: Position of the Image

The position of the image, counting from the ornament surface, is approximately
`\(-1.49\)` cm. According to the sign convention for mirrors, a negative image distance indicates that the image is virtual and located on the same side of the mirror as the object (i.e., inside the spherical ornament in this case).

Part B: Magnification of the Image

The magnification of the image is approximately
`\(0.068\)`. This value indicates that the image is smaller than the object and upright, as expected for a convex mirror. Since the magnification is less than 1, the image is reduced in size.

Answer 2

(a) The position of the image is 1.71 cm.

(b) The magnification of the image is 0.07.

How to calculate the position of the image?

(a) The position of the image is calculated by applying the following lens equation;

1/f = 1/v + 1/u

where;

• f is the focal length of the lens
• v is the image position
• u is the position of the object

The radius of curvature of the lens is, R = 6.4 cm / 2 = 3.2 cm

The focal length, f = 3.2 cm / 2 = 1.6 cm

The position of the image is calculated as;

1/f = 1/v + 1/u

1/1.6 = 1/v + 1/25

1/v = 1/1.6 - 1/25

1/v = 0.585

v = 1/0.585

v = 1.71 cm

(b) The magnification of the image is calculated as;

M = v/u

M = (1.71/25)

M = 0.07