an object 5.50 cm high is placed 100 cm from a converging lens that has a focal length of 40.0cm.
a)Calculate the image distance
b)Calculate the hight of the image height
To calculate the image distance using the lens formula, we can use the following equation:
1/f = 1/v - 1/u
Where:
f = focal length of the lens
v = image distance
u = object distance
Given:
f = 40.0 cm
u = 100 cm
a) Calculating the image distance:
Plugging in the given values into the lens formula equation:
1/40 = 1/v - 1/100
To simplify the equation, let's take the common denominator:
(100 - v)/(40v) = 1/100
Cross multiplying:
100(100 - v) = 40v
Expanding:
10000 - 100v = 40v
Bringing the variables to one side:
140v = 10000
Dividing both sides by 140:
v = 10000/140
v ≈ 71.43 cm
Therefore, the image distance is approximately 71.43 cm.
b) Calculating the image height:
To find the image height, we can use the magnification formula:
m = - v/u
Where:
m = magnification
v = image distance
u = object distance
Given:
v ≈ 71.43 cm
u = 100 cm
Plugging in the values into the magnification formula:
m = - 71.43/100
Calculating the ratio:
m = -0.7143
The magnitude of the magnification is:
|m| = |-0.7143| ≈ 0.7143
Since the magnification is negative, it means the image is inverted. The magnitude of the magnification indicates that the image is smaller than the object height.
Therefore, the height of the image is approximately 0.7143 times the height of the object, or approximately 0.7143 * 5.50 cm.
Hence, the height of the image is approximately 3.93 cm.
To answer these questions, you can use the lens equation:
1/f = 1/di + 1/do
Where:
- f is the focal length of the lens
- di is the image distance (distance from the lens to the image)
- do is the object distance (distance from the lens to the object)
a) Calculate the image distance (di):
In this case, the object distance (do) is given as 100 cm, and the focal length of the lens (f) is given as 40.0 cm. Plugging these values into the lens equation, we can solve for di:
1/40 = 1/di + 1/100
Rearranging the equation:
1/di = 1/40 - 1/100
1/di = (10 - 4)/400
1/di = 6/400
di = 400/6
di = 66.67 cm
Therefore, the image distance is 66.67 cm.
b) Calculate the height of the image (hi):
The height of the image can be calculated using the magnification formula:
magnification = hi/ho = -di/do
Where:
- hi is the image height
- ho is the object height
In this case, the object height (ho) is given as 5.50 cm. The magnification can be calculated as follows:
magnification = -di/do = -66.67/100
Using the magnification formula, we can find hi:
hi/5.50 = -66.67/100
Cross-multiplying:
hi = (5.50 * -66.67) / 100
Calculating:
hi = -3.67 cm
Therefore, the height of the image is -3.67 cm. The negative sign indicates that the image is inverted.