1a. Two converging lenses, each of focal length 14.4 cm, are placed 60.6 cm apart, and an object is placed 26.9 cm in front of the first. Where is the final image relative to the second lens formed?
b. What is the magnification of the system?
To solve this problem, we will use the lens formula and the lens makers formula. The lens formula is given by:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens,
- v is the image distance from the lens, and
- u is the object distance from the lens.
The lens makers formula relates the focal length of the lens to the refractive index and the radii of curvature of the lens surfaces. However, for your question, we don't need to use the lens makers formula.
Let's solve part (a) first:
1. Identify the given values:
- Focal length of each lens (f): 14.4 cm
- Distance between the two lenses (d): 60.6 cm
- Object distance (u) from the first lens: 26.9 cm
2. Determine the image distance (v) from the first lens:
Using the lens formula for the first lens:
1/f1 = 1/v1 - 1/u1
Substituting the values we have:
1/14.4 = 1/v1 - 1/26.9
Rearranging the equation, we get:
1/v1 = 1/14.4 + 1/26.9
Calculating the sum:
1/v1 = 0.0694 + 0.0372
1/v1 = 0.1066
Taking the reciprocal:
v1 = 9.38 cm
This means that the image formed by the first lens is located 9.38 cm away from it.
3. Determine the object distance (u2) from the second lens:
Since the first image becomes the object for the second lens, the distance between the lenses (d) is equal to the object distance (u2) for the second lens:
u2 = d = 60.6 cm
4. Determine the final image distance relative to the second lens (v2):
Using the lens formula for the second lens:
1/f2 = 1/v2 - 1/u2
Substituting the values we have:
1/14.4 = 1/v2 - 1/60.6
Rearranging the equation, we get:
1/v2 = 1/14.4 + 1/60.6
Calculating the sum:
1/v2 = 0.0694 + 0.0165
1/v2 = 0.0859
Taking the reciprocal:
v2 = 11.62 cm
Therefore, the final image formed by the second lens is located 11.62 cm away from it.
Now let's move on to part (b) to determine the magnification:
The magnification (m) can be calculated using the formula:
m = -v1/u1 * v2/u2
Substituting the values we have:
m = - (9.38 cm) / (26.9 cm) * (11.62 cm) / (60.6 cm)
Calculating the magnification:
m = -0.347
The negative sign indicates that the image formed is inverted.
So, the final image formed by the second lens is located 11.62 cm away from it, and the magnification of the system is -0.347.