A converging lens of focal length 4.3 m produces a magnified image 6.7 times the size of the object. What is the object distance if the image is
real?
virtual?
pleasse help!
To find the object distance, we can 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
For a converging lens, when the image is real, it forms on the opposite side of the lens from the object. In this case, the image distance (v) will be positive. Let's solve for the object distance (u) in this situation.
1/4.3 = 1/6.7 - 1/u
To simplify the equation, let's find the common denominator:
1/4.3 = (6.7 - 1/u) / 6.7
Now, cross-multiply to eliminate the fractions:
6.7/u = 6.7 - 1/4.3
Next, rearrange the equation to isolate u:
1/4.3 = 6.7 - 6.7/u
1/4.3 - 6.7 = -6.7/u
1/4.3u - 6.7u = -6.7
Combine like terms:
(1 - 6.7 * 4.3) / 4.3u = -6.7
Evaluate the left side:
(-27.41) / 4.3u = -6.7
Now, solve for u:
4.3u = (-27.41) / (-6.7)
u = (-27.41) / (-6.7 * 4.3)
u ≈ 0.573 m
So, when the image formed by the converging lens is real, the object distance is approximately 0.573 m.
Now let's consider the case when the image is virtual.
For a converging lens, when the image is virtual, it forms on the same side of the lens as the object. In this case, the image distance (v) will be negative.
Let's calculate the object distance (u) when the image is virtual.
1/4.3 = 1/(-6.7) - 1/u
Again, let's find the common denominator:
1/4.3 = (-6.7 - 1/u) / (-6.7)
Cross-multiply to eliminate the fractions:
-6.7/u = -6.7 - 1/4.3
Rearrange the equation to isolate u:
-1/4.3 = -6.7 - 6.7/u
6.7 - 6.7/u = 1/4.3
Combine like terms:
6.7u - 6.7 = 1/4.3
Multiply both sides by 4.3:
(6.7 * 4.3)u - 6.7 * 4.3 = 1
Evaluate the left side:
28.81u - 28.81 = 1
Add 28.81 to both sides:
28.81u = 1 + 28.81
28.81u = 29.81
Finally, divide both sides by 28.81 to solve for u:
u = 29.81 / 28.81
u ≈ 1.032 m
So, when the image formed by the converging lens is virtual, the object distance is approximately 1.032 m.