A converging lens (f1 = 24.0 cm) is located 56.0 cm to the left of a diverging lens (f2 = -28.0 cm). An object is placed to the left of the converging lens, and the final image produced by the two-lens combination lies 20.1 cm to the left of the diverging lens. How far is the object from the converging lens?
To determine the distance of the object from the converging lens, we can use the lens formula and the lensmaker's formula. Here's how to do it:
Step 1: Identify the given values:
f1 = 24.0 cm (focal length of the converging lens)
f2 = -28.0 cm (focal length of the diverging lens)
d1 = 56.0 cm (distance of the converging lens from the object)
d2 = -20.1 cm (distance of the final image from the diverging lens)
Step 2: Apply the lens formula for both lenses:
For the converging lens:
1/f1 = 1/d1 - 1/i1
For the diverging lens:
1/f2 = 1/i1 - 1/d2
Step 3: Solve the first equation for i1 (the distance of the image from the converging lens):
1/f1 = 1/d1 - 1/i1
Rearrange the equation:
1/i1 = 1/f1 - 1/d1
Substitute the given values:
1/i1 = 1/24 - 1/56
Step 4: Calculate the value of i1:
Substitute the equation into a common denominator:
1/i1 = (56 - 24) / (24 * 56)
1/i1 = 32 / (24 * 56)
i1 = (24 * 56) / 32
i1 = 42 cm
Step 5: Calculate the distance of the object from the converging lens (d1):
d1 = d2 + i1 (as the image produced by the converging lens becomes the object for the diverging lens)
Substitute the given values:
d1 = -20.1 + 42
d1 = 21.9 cm
Therefore, the distance of the object from the converging lens is 21.9 cm.