If the thimble is 2.50 cm tall and has a focal length of 11.0 cm, how tall is the image of the thimble? What is its orientation? Draw a diagram
To determine the height and orientation of the image of the thimble, we can use the thin lens equation and the magnification formula. Let's go through the steps:
Step 1: Understand the given information.
- Thimble height (object height) = 2.50 cm
- Focal length of the lens = 11.0 cm
Step 2: Determine the image height using the thin lens equation.
The thin lens equation is given by: (1/f) = (1/d_o) + (1/d_i)
where:
- f is the focal length of the lens,
- d_o is the object distance from the lens (positive for objects on the same side as the incident light),
- d_i is the image distance from the lens (positive for images formed on the opposite side of the lens as the incident light).
In this case, since the object is positioned on the same side as the incident light, the object distance (d_o) is negative.
Step 2a: Calculate the object distance (d_o).
Given that the object height (h_o) is 2.50 cm, and the object is the same distance from the lens as its height, the object distance (d_o) is -2.50 cm.
Step 2b: Calculate the image distance (d_i).
Using the thin lens equation, we can rearrange it to solve for d_i:
(1/f) = (1/d_o) + (1/d_i)
(1/11.0 cm) = (1/-2.50 cm) + (1/d_i)
(1/11.0 cm) = (-2/5) + (1/d_i)
(1/11.0 cm) = (-2 + 5/5) + (1/d_i)
(1/11.0 cm) = (3/5) + (1/d_i)
To solve for d_i, we can equate the numerators:
1 = (3/5)(11.0 cm) + (5/5)(d_i)
1 = (33/5) cm + (5/5)(d_i)
1 = (33/5) cm + d_i
d_i = 1 cm - (33/5) cm
d_i ≈ -4.4 cm
Note: The negative sign indicates that the image is formed on the opposite side of the lens as the incident light.
Step 3: Calculate the magnification (m).
The magnification (m) is given by the formula: m = -d_i/d_o
m = (-4.4 cm) / (-2.50 cm)
m ≈ 1.76
Step 4: Determine the height of the image (h_i).
The height of the image (h_i) can be calculated using the magnification formula:
m = -h_i/h_o
1.76 = -h_i / 2.50 cm
Solving for h_i:
h_i = (1.76)(2.50 cm)
h_i ≈ 4.40 cm
Therefore, the height of the image of the thimble is approximately 4.40 cm. The orientation of the image is inverted, as indicated by the negative values of the image distance (d_i) and magnification (m).
Apologies, but I'm unable to draw a diagram as a text-based AI. However, you can easily visualize it by considering the thimble (object) on one side of the lens and the image formed on the opposite side, with the image being upside down.