Two plus lenses have focal lengths in the ratio 3:1. When placed together in close contact the focal length of the combination is +8.33cm. What are the focal lengths of the two lenses?
To find the focal lengths of the two lenses, we'll use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the image distance
- u is the object distance
Let's assign variables to the focal lengths of the lenses. Let's say the focal lengths of the two lenses are 3x and x, respectively.
Since the lenses are placed in close contact, the combination acts as a single lens. We can calculate the focal length of the combination using the formula for lenses in contact:
1/f_combination = 1/f1 + 1/f2
Given that the focal length of the combination is +8.33 cm, we can substitute the values:
1/f_combination = 1/(3x) + 1/x
Now, we'll solve this equation to find the value of x:
1/8.33 cm = 1/(3x) + 1/x
Taking the reciprocal of both sides:
8.33 cm = (3x + x) / x(3x)
Multiplying both sides by x(3x):
8.33 cm * x(3x) = 4x + 3x
24.99 cm^2 = 7x
Dividing both sides by 7:
x = 3.57 cm
Now that we have the value of x, we can substitute it back into the equations to find the focal lengths of the two lenses:
Focal length of the first lens = 3x = 3 * 3.57 cm = 10.71 cm
Focal length of the second lens = x = 3.57 cm
Therefore, the focal lengths of the two lenses are 10.71 cm and 3.57 cm, respectively.