An astronomical telescope of magnifying power 7, consist of two thin lenses 40 cm apart in normal adjustment. Find the focal length of the lenses.
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To find the focal length of the lenses, we can use the formula for the magnifying power of a telescope, which is given by:
M = -f1/f2
Where M is the magnifying power, f1 is the focal length of the objective lens, and f2 is the focal length of the eyepiece.
Given that the magnifying power (M) is 7, we can rearrange the equation to solve for f2:
f2 = -f1/M
Since the two lenses are 40 cm apart in normal adjustment, the focal length of the objective (f1) plus the focal length of the eyepiece (f2) equals 40 cm:
f1 + f2 = 40
Now we can substitute the value of f2 from the first equation into the second equation:
f1 - (f1/7) = 40
Multiplying both sides of the equation by 7 to eliminate the fraction:
7f1 - f1 = 280
Combining like terms:
6f1 = 280
Dividing both sides of the equation by 6:
f1 = 46.67 cm
Now we can substitute the value of f1 back into the equation f1 + f2 = 40 to find f2:
46.67 + f2 = 40
Subtracting 46.67 from both sides of the equation:
f2 = 40 - 46.67
f2 = -6.67 cm
Therefore, the focal length of the objective lens is approximately 46.67 cm, and the focal length of the eyepiece is approximately -6.67 cm.
To find the focal length of the lenses, we can use the formula for the magnifying power of a telescope:
Magnifying power (M) = - (focal length of the objective lens / focal length of the eyepiece lens)
In this case, the magnifying power is given as 7. Therefore, we have:
7 = - (focal length of the objective lens / focal length of the eyepiece lens)
Since the two lenses are 40 cm apart in normal adjustment, the sum of their focal lengths will be equal to 40 cm:
focal length of the objective lens + focal length of the eyepiece lens = 40 cm
Let's denote the focal length of the objective lens as "f1" and the focal length of the eyepiece lens as "f2". We can rewrite the above equation as:
f1 + f2 = 40 cm
Now we have a system of two equations:
7 = - (f1 / f2)
f1 + f2 = 40 cm
To solve this system of equations, let's substitute f1 from the second equation into the first equation:
7 = - ((40 - f2) / f2)
Now we can solve for f2:
7f2 = - (40 - f2)
7f2 + f2 = -40
8f2 = -40
f2 = -40 / 8
f2 = -5 cm
We have found that the focal length of the eyepiece lens (f2) is -5 cm. Since focal lengths cannot be negative, it is likely that there was a mistake made in the question or calculations. Please verify the provided numbers and adjust if necessary.